Radicals and Rational Exponents

Elizabeth Pople

21 Radicals and Rational Exponents

Topics Covered:^[1]

In case you missed something in class, or just want to review a specific topic covered in this Module, here is a list of topics covered:

Simplify Expressions with Roots

Remember that when a real number n is multiplied by itself, we write n² and read it ‘n squared’. This number is called the square of n, and n is called the square root. For example,

13² is read “13 squared”

169 is called the square of 13, since 13² = 169

13 is a square root of 169

Square and Square Root of a number

Square

If n²= m, then m is the square of n.

Square Root

If n² = m, then n is a square root of m.

Notice (−13)² = 169 also, so −13 is also a square root of 169. Therefore, both 13 and −13 are square roots of 169.

So, every positive number has two square roots—one positive and one negative. What if we only wanted the positive square root of a positive number? We use a radical sign, and write, $\displaystyle\sqrt{m}$ , which denotes the positive square root of m. The positive square root is also called the principal square root.

We also use the radical sign for the square root of zero. Because $\displaystyle 0^2 = 0,\ \sqrt{0} = 0$ . Notice that zero has only one square root.

Square Root Notation

$\displaystyle \sqrt{m}$ is read “the square root of m”.

If n² = m, then $\displaystyle n = \sqrt{m}$ , for n ≥ 0.

$\displaystyle\textcolor{mypurple1}{\boldsymbo{\text{radical sign }}} \textcolor{mypurple1}{\boldsymbol{\longrightarrow}} \boldsymbol{\sqrt{m}}\textcolor{mypurple1}{\boldsymbol{\longleftarrow}} \textcolor{mypurple1}{\boldsymbol{\text{ radicand}}}$

We know that every positive number has two square roots and the radical sign indicates the positive one. We write $\displaystyle \sqrt{169} = 13$ . If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, $\displaystyle -\sqrt{169} = -13$ .

Try it!

Simplify: a. $\displaystyle \sqrt{144}$ b. $\displaystyle -\sqrt{289}$

Solution A (click to reveal)

a.

Steps

Algebraic

Problem

$\boldsymbol{\displaystyle \sqrt{144}}$

Since 12² = 144.

12

Solution B (click to reveal)

b.

Steps

Algebraic

Problem

$\boldsymbol{\displaystyle -\sqrt{289}}$

Since 17² = 289 and the negative is in front of the radical sign.

-17

Can we simplify $\displaystyle \sqrt{-49}$ ? Is there a number whose square is −49?

( )² = −49

Any positive number squared is positive. Any negative number squared is positive. There is no real number equal to $\displaystyle \sqrt{-49}$ . The square root of a negative number is not a real number.

Try it!

Simplify: a. $\displaystyle \sqrt{-196}$ b. $\displaystyle -\sqrt{64}$

Solution (click to reveal)

a.

Steps

Algebraic

Problem

$\boldsymbol{\displaystyle \sqrt{-196}}$

There is no real number whose square is −196.

$\boldsymbol{\displaystyle \sqrt{-196}}$ is not a real number.

b.

Steps

Algebraic

Problem

$\boldsymbol{\displaystyle -\sqrt{64}}$

The negative is in front of the radical.

-8

But what if we want to estimate $\displaystyle \sqrt{500}$ ? If we simplify the square root first, we’ll be able to estimate it easily. There are other reasons to simplify square roots as you’ll see later in this section.

Simplified Square Root

$\displaystyle \sqrt{a}$ is considered simplified if a (radicand) has no perfect square factors.

So $\displaystyle \sqrt{31}$ is simplified. But $\displaystyle \sqrt{32}$ is not simplified, because 16 is a perfect square factor of 32.

Use Product Property to Simplify Square Roots

The properties we will use to simplify expressions with square roots are similar to the properties of exponents. We know that (ab)^m= a^mb^m. The corresponding property of square roots says that $\dispaystyle\sqrt{ab} =\sqrt{a}\cdot \sqrt{b}$ .

Product Property of Square Roots

If a, b are non-negative real numbers, then

$\dispaystyle\sqrt{ab} =\sqrt{a}\cdot \sqrt{b}$ and $\dispaystyle\sqrt{a}\cdot \sqrt{b} = \sqrt{ab}$

We use the Product Property of Square Roots to remove all perfect square factors from a radical.

Try it!

Simplify: $\displaystyle \sqrt{50}$ .

Solution (click to reveal)

Steps

Algebraic

Problem:

$\boldsymbol{\displaystyle \sqrt{50}}$

Find the largest perfect square factor of the radicand by rewriting the radicand as a product using the perfect square factor.

25 is the largest perfect square factor of 50.
$\boldsymbol{\displaystyle 50=25\cdot 2}$

Always write the perfect square factor first.

$\boldsymbol{\displaystyle \sqrt{25\cdot 2}}$

Use the product rule to rewrite the radical as the product of two radicals.

$\boldsymbol{\displaystyle \sqrt{25}\cdot\sqrt{2}}$

Simplify.

$\boldsymbol{\displaystyle 5\sqrt{2}}$

Notice in the previous example that the simplified form of $\displaystyle \sqrt{50}$ is $\displaystyle 5\sqrt{2}$ , which is the product of an integer and a square root. We always write the integer in front of the square root.

Simplify a square root using the product property.

Find the largest perfect square factor of the radicand. Rewrite the radicand as a product using the perfect-square factor.
Use the product rule to rewrite the radical as the product of two radicals.
Simplify the square root of the perfect square.

Try it!

Simplify: $\displaystyle\sqrt{500}$ .

Solution (click to reveal)

Steps

Algebraic

Problem

$\boldsymbol{\displaystyle\sqrt{500}}$

Rewrite the radicand as a product using the largest perfect square factor.

$\boldsymbol{\displaystyle\sqrt{100\cdot 5}}$

Rewrite the radical as the product of two radicals.

$\boldsymbol{\displaystyle\sqrt{100}\cdot \sqrt{5}}$

Simplify.

$\boldsymbol{\displaystyle 10 \sqrt{5}}$

The next example is much like the previous examples, but with variables.

Try it!

1. Simplify: $\displaystyle \sqrt{x^3}$ .

Solution (click to reveal)

Steps

Algebraic

Problem

$\boldsymbol{\displaystyle \sqrt{x^3}}$

Rewrite the radicand as a product using the largest perfect square factor.

$\boldsymbol{\displaystyle \sqrt{x^2 \cdot x}}$

Rewrite the radical as the product of two radicals.

$\boldsymbol{\displaystyle \sqrt{x^2}\cdot \sqrt{x}}$

Simplify.

$\boldsymbol{\displaystyle x\sqrt{x}}$

fy: $\displaystyle \sqrt{25y^5}$ .

Solution (click to reveal)

Steps

Algebraic

Problem

$\boldsymbol{\displaystyle \sqrt{25y^5}}$

Rewrite the radicand as a product using the largest perfect square factor.

$\boldsymbol{\displaystyle \sqrt{25y^4\cdot y}}$

Rewrite the radical as the product of two radicals.

$\boldsymbol{\displaystyle \sqrt{25y^4}\cdot \sqrt{y}}$

Simplify.

$\boldsymbol{\displaystyle 5y^2\sqrt{y}}$

So we can multiply $\displaystyle \sqrt{3}\cdot \sqrt{5}$ in this way:

$\displaystyle \sqrt{3}\cdot \sqrt{5}$

$\displaystyle \sqrt{3\cdot 5}$

$\displaystyle \sqrt{15}$

Even when the product is not a perfect square, we must look for perfect-square factors and simplify the radical whenever possible.

Try it!

Simplify: a. $\displaystyle \sqrt{2}\cdot \sqrt{6}$ b. $\displaystyle \left(4\sqrt{3}\right)\left(2\sqrt{12}\right)$ .

Solution A (click to reveal)

a.

Steps

Algebraic

Problem

$\boldsymbol{\displaystyle \sqrt{2}\cdot \sqrt{6}}$

Multiply using the Product Property.

$\boldsymbol{\displaystyle \sqrt{12}}$

Simplify the radical.

$\boldsymbol{\displaystyle \sqrt{4}\cdot \sqrt{3}}$

Simplify.

$\boldsymbol{\displaystyle 2\sqrt{3}}$

Solution B (click to reveal)

b.

Steps

Algebraic

Problem

$\boldsymbol{\displaystyle \left(4\sqrt{3}\right)\left(2\sqrt{12}\right)}$

Multiply using the Product Property.

$\boldsymbol{\displaystyle 8\sqrt{36}\right)}$

Simplify the radical.

$\boldsymbol{\displaystyle 8 \cdot 6}$

Simplify.

$\boldsymbol{\displaystyle 48}$

Notice that we multiplied the coefficients and multiplied the radicals. Also, we did not simplify $\displaystyle \sqrt{12}$ . We waited to get the product and then simplified.

Try it!

Simplify: a. $\displaystyle \left(\sqrt{8x^3}\right)\left(\sqrt{3x}\right)$ b. $\displaystyle \left(\sqrt{20y^2}\right)\left(\sqrt{5y^3}\right)$

Solution A (click to reveal)

a.

Steps

Algebraic

Problem

$\boldsymbol{\displaystyle \left(\sqrt{8x^3}\right)\left(\sqrt{3x}\right)}$

Multiply using the Product Property.

$\boldsymbol{\displaystyle \sqrt{24x^4}}$

Simplify the radical.

$\boldsymbol{\displaystyle \sqrt{4x^4}\cdot \sqrt{6}}$

Simplify.

$\boldsymbol{\displaystyle 2x^2\sqrt{6}}$

Solution B (click to reveal)

b.

Steps

Algebraic

Problem

$\boldsymbol{\displaystyle \left(\sqrt{20y^2}\right)\left(\sqrt{5y^3}\right)}$

Multiply using the Product Property.

$\boldsymbol{\displaystyle \sqrt{100y^5}}$

Simplify the radical.

$\boldsymbol{\displaystyle 10y^2\cdot \sqrt{y}}$

Quotient Property with Roots

Whenever you have to simplify a square root, the first step you should take is to determine whether the radicand is a perfect square. A perfect square fraction is a fraction in which both the numerator and the denominator are perfect squares.

Try it!

Simplify: $\displaystyle \sqrt{\frac{9}{64}$

Solution (click to reveal)

Steps

Algebraic

Problem

$\boldsymbol{\displaystyle \frac{9}{64}}$

Since $\boldsymbol{\displaystyle \left(\frac{3}{8}\right)^2 = \frac{9}{64}}$

$\boldsymbol{\displaystyle \frac{3}{8}}$

If the numerator and denominator have any common factors, remove them. You may find a perfect square fraction!

Try it!

Simplify: $\displaystyle \sqrt{\frac{45}{80}$

Solution (click to reveal)

Steps

Algebraic

Problem

$\boldsymbol{\displaystyle \sqrt{\frac{45}{80}}}$

Simplify inside the radical first.

$\boldsymbol{\displaystyle \sqrt{\frac{5\cdot 9}{5\cdot 16}}}$

Rewrite showing the common factors of the numerator and denominator.

$\boldsymbol{\displaystyle \sqrt{\frac{9}{16}}}$

Simplify. $\boldsymbol{\displaystyle \left(\frac{3}{4}\right)^2 =\frac{9}{16}}$

$\boldsymbol{\displaystyle \frac{3}{4}}$

In the last example, our first step was to simplify the fraction under the radical by removing common factors. In the next example, we will use the Quotient Property to simplify under the radical. We divide the like bases by subtracting their exponents, $\displaystyle \frac{a^m}{a^n} = a^{m-n},\qquad a\ne 0$ .

Try it!

Simplify: $\displaystyle \left(\sqrt{\frac{m^6}{m^4}\right)$

Solution (click to reveal)

Steps

Algebraic

Problem

$\boldsymbol{\displaystyle \sqrt{\frac{m^6}{m^4}}}$

Simplify the fraction inside the radical first: Divide the like bases by subtracting the exponents.

$\boldsymbol{\displaystyle \sqrt{m^2}}$

Simplify.

$\boldsymbol{\displaystyle m}$

Remember the Quotient to a Power Property? It said we could raise a fraction to a power by raising the numerator and denominator to the power separately.

$\displaystyle \left(\frac{a}{b}\right)^m = \frac{a^m}{b^m},\qquad b \ne 0$

We can use a similar property to simplify a square root of a fraction. After removing all common factors from the numerator and denominator, if the fraction is not a perfect square, we simplify the numerator and denominator separately.

Quotient Property of Square Roots

If a, b are non-negative real numbers and b ≠ 0, then

$\displaystyle \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\quad \text{ and }\quad \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$

Try it!

Simplify: $\displaystyle \left(\sqrt{\frac{21}{64}\right)$ .

Solution (click to reveal)

Steps

Algebraic

Problem

$\boldsymbol{\displaystyle \sqrt{\frac{21}{64}}}$

We cannot simplify the fraction inside the radical. Rewrite using the quotient property.

$\boldsymbol{\displaystyle\frac{\sqrt {21}}{\sqrt {64}}}$

Simplify the square root of 64. The numerator cannot be simplified.

$\boldsymbol{\displaystyle \frac{\sqrt {21}}{8}}$

We will use the Quotient Property for Exponents, $\displaystyle \frac{a^m}{a^n}$ = $\displaystyle a^{m-n}$ , when we have variables with exponents in the radicands.

Try it!

Simplify: $\displaystyle \sqrt{\frac{27 \cdot m^3}{196}$ .

Solution (click to reveal)

Steps

Algebraic

Step 1. Simplify the fraction in the radicand, if possible. In this case, $\displaystyle \boldsymbol{\frac{27m^3}{196}}$ cannot be simplified.

$\displaystyle\boldsymbol{\sqrt{\frac{27m^3}{196}}}$

Step 2. Use the Quotient Property to rewrite the radicals as the quotient of two radicals. Meaning: We rewrite $\displaystyle\boldsymbol{ \sqrt{\frac{27m^3}{196}}}$ as the quotient of $\displaystyle \boldsymbol{\sqrt{27m^3}}$ and $\displaystyle \boldsymbol{\sqrt{196}}$ .

$\displaystyle \boldsymbol{\frac{\sqrt{27m^3}}{\sqrt{196}}}$

Step 3. Simplify the radicals in the numerator and enumerator. Note that $\displaystyle\boldsymbol{ 9m^2}$ and $\displaystyle\boldsymbol{ 196}$ are perfect squares.

$\begin{gather*} \displaystyle \boldsymbol{\frac{\sqrt{9m^2} \cdot \sqrt{3m}}{\sqrt{196}}} \\ \displaystyle \boldsymbol{\frac{3m\sqrt{3m}}{14}} \end{gather*}$

Simplify: $\displaystyle \sqrt{\frac{45x^5}{y^4}$ .

Solution (click to reveal)

Steps

Algebraic

Problem

$\boldsymbol{\displaystyle\sqrt{\frac{45x^5}{y^4}}}$

We cannot simplify the fraction in the radicand. Rewrite using the Quotient Property.

$\boldsymbol{\displaystyle \frac{\sqrt{45x^5}}{\sqrt{y^4}}}$

Simplify the radicals in the numerator and the denominator.

$\boldsymbol{\displaystyle\frac{ \sqrt {9x^4} \cdot \sqrt{5x}}{y^2}}$

Simplify.

$\boldsymbol{\displaystyle \frac{3x^2 \sqrt{5x}}{y^2}}$

Simplify: $\displaystyle \sqrt{\frac{81d^9}{25d^4}$ .

Solution (click to reveal)

Steps

Algebraic

Problem

$\boldsymbol{\displaystyle \sqrt{\frac{81d^9}{25d^4}}}$

Simplify the fraction in the radicand.

$\boldsymbol{\displaystyle \sqrt{\frac{81d^5}{25}}}$

Rewrite using the Quotient Property

$\boldsymbol{\displaystyle \frac{\sqrt{81 d^5}}{\sqrt{25}}}$

Simplify the radicals in the numerator and the denominator.

$\boldsymbol{\displaystyle\frac{ \sqrt {81d^4} \cdot \sqrt{d}}{5}}$

Simplify.

$\boldsymbol{\displaystyle \frac{9d^2\sqrt{d}}{5}}$

Simplify: $\displaystyle \frac{\sqrt{6y^5}}{\sqrt{2y}}$ .

Solution (click to reveal)

Steps

Algebraic

Problem

$\boldsymbol{\displaystyle \frac{\sqrt{6y^5}}{\sqrt{2y}}}$

Neither radicand is a perfect square, so rewrite using the quotient property of square roots.

$\boldsymbol{\displaystyle \sqrt{\frac{6y^5}{2y}}}$

Remove common factors in the numerator and denominator.

$\displaystyle\boldsymbol{\sqrt{\frac{\cancel{\color{myred1}{2}}\cdot 3\cdot y^4\cdot \cancel{\color{myblue1}{y}}}{\cancel{\color{myred1}{2}}\cdot \cancel{\color{myblue1}{y}}}}}$

Simplify.

$\boldsymbol{\displaystyle \sqrt{3y^4}}$

Simplify the radical.

$\boldsymbol{\displaystyle y^2 \cdot \sqrt{3}}$

We will use the Quotient Property of Square Roots ‘in reverse’ when the fraction we start with is the quotient of two square roots, and neither radicand is a perfect square. When we write the fraction in a single square root, we may find common factors in the numerator and denominator.

Try it!

Simplify: $\displaystyle \frac{\sqrt{27}}{\sqrt{75}}$ .

Solution (click to reveal)

Steps

Algebraic

Problem

$\boldsymbol{\displaystyle \frac{\sqrt{27}}{\sqrt{75}}}$

Neither radicand is a perfect square, so rewrite using the quotient property of square roots.

$\boldsymbol{\displaystyle \sqrt{\frac{27}{75}}}$

Remove common factors in the numerator and denominator.

$\displaystyle\boldsymbol{\sqrt{\frac{\cancel{\color{myred1}{3}}\cdot 9}{\cancel{\color{myred1}{3}}\cdot 25}}}$

Simplify.

$\boldsymbol{\displaystyle \sqrt{\frac{9}{25}}}$

Solution

$\boldsymbol{\displaystyle \frac{3}{5}}$

Add, Subtract Square Roots

We know that we must follow the order of operations to simplify expressions with square roots. The radical is a grouping symbol, so we work inside the radical first. We simplify $\displaystyle \sqrt {2 + 7}$ in the table below:

Steps

Algebraic

Expression

$\displaystyle \sqrt {2 + 7}$

Add inside the radical.

$\displaystyle \sqrt {9}$

Simplify.

$\displaystyle 3$

So if we have to add $\displaystyle \sqrt {2} + \sqrt{7}$ , we must not combine them into one radical.

$\displaystyle \sqrt {2} + \sqrt{7}$ ≠ $\displaystyle \sqrt {2 + 7}$

Trying to add square roots with different radicands is like trying to add unlike terms.

But, just like we can add x + x = 2x, we can add $\displaystyle \sqrt {3} + \sqrt{3} = 2\sqrt{3}$ .

Adding square roots with the same radicand is just like adding like terms. We call square roots with the same radicand like square roots to remind us they work the same as like terms.

Square roots with the same radicand are called like square roots.

We add and subtract like square roots in the same way we add and subtract like terms. We know that 3x + 8x is 11x.

Similarly we add $\displaystyle 3 \sqrt {x} + 8 \sqrt{x}$ and the result is $\displaystyle 11 \sqrt{x}$

Think about adding like terms with variables as you do the next few examples. When you have like radicands, you just add or subtract the coefficients. When the radicands are not like, you cannot combine the terms.

Try it!

Simplify: $\displaystyle 2 \sqrt {2} - 7 \sqrt{2}$ .

Solution (click to reveal)

Steps

Algebraic

Expression

$\boldsymbol{\displaystyle 2 \sqrt {2} - 7 \sqrt{2}}$

Since the radicals are like, we subtract the coefficients

$\boldsymbol{\displaystyle -5 \sqrt{2}}$

2. Simplify: $\displaystyle 5 \sqrt {13} + 4 \sqrt{13} + 2\sqrt{13}$ .

Solution (click to reveal)

Steps

Algebraic

Expression

$\boldsymbol{\displaystyle 5 \sqrt {13} + 4 \sqrt{13} + 2\sqrt{13}}$

Since the radicals are like, we subtract the coefficients

$\boldsymbol{\displaystyle 11 \sqrt{13}}$

Remember that we always simplify square roots by removing the largest perfect-square factor. Sometimes when we have to add or subtract square roots that do not appear to have like radicals, we find like radicals after simplifying the square roots.

Try it!

Simplify: $\displaystyle \sqrt {20} + 3 \sqrt{5}$

Solution (click to reveal)

Steps

Algebraic

Expression

$\displaystyle \boldsymbol{\sqrt {20} + 3 \sqrt{5}}$

Simplify the radicals, when possible.

$\begin{gather*} \boldsymbol{\sqrt{4} \cdot \sqrt{5} + 3\sqrt{5}} \\ \boldsymbol{2\sqrt{5} + 3\sqrt{5}} \end{gather*}$

Combine the like radicals.

$\displaystyle \boldsymbol{5\sqrt{5}}$

Tip!

Just like we use the Associative Property of Multiplication to simplify 5(3x) and get 15x, we can simplify $\displaystyle 5(3\sqrt{x})$ and get $\displaystyle 15\sqrt{x}$ .

We will use the Associative Property to do this in the next example.

Try it!

Simplify: $\displaystyle 5\sqrt {18} - 2 \sqrt{8}$

Solution (click to reveal)

Steps

Algebraic

Example

$\displaystyle \boldsymbol{5\sqrt {18} - 2 \sqrt{8}}$

Simplify the radicals.

$\begin{gather*} \boldsymbol{5\cdot\sqrt{9}\cdot\sqrt{2}-2\cdot\sqrt{4}\cdot\sqrt{2}} \\ \boldsymbol{5\cdot 3\cdot\sqrt{2}-2\cdot 2\cdot\sqrt{2}}\\ \boldsymbol{15\sqrt{2}-4\sqrt{2}} \end{gather*}$

Combine the like radicals.

$\displaystyle\boldsymbol{ 11\sqrt{2}}$

Try it!

Simplify: $\displaystyle \sqrt {18n^5} - \sqrt{32n^5}$

Solution (click to reveal)

Steps

Algebraic

ExampleExample

$\displaystyle \boldsymbol{\sqrt {18n^5} - \sqrt{32n^5}}$

Simplify the radicals.

$\begin{gather*} \boldsymbol{\sqrt{9n^4}\cdot\sqrt{2n}-\sqrt{16n^4}\cdot\sqrt{2n}} \\ \boldsymbol{3n^2\sqrt{2n}-4n^2\sqrt{2n}}\\ \end{gather*}$

Combine the like radicals.

$\displaystyle \boldsymbol{-n^2\sqrt{2n}}$

Rationalize Denominators

Before the calculator became a tool of everyday life, approximating the value of a fraction with a radical in the denominator was a very cumbersome process!

For this reason, a process called rationalizing the denominator was developed. A fraction with a radical in the denominator is converted to an equivalent fraction whose denominator is an integer. Square roots of numbers that are not perfect squares are irrational numbers. When we rationalize the denominator, we write an equivalent fraction with a rational number in the denominator.

The process of converting a fraction with a radical in the denominator to an equivalent fraction whose denominator is an integer is called rationalizing the denominator.

*Example*
Steps	Algebraic
Suppose we need an approximate value for the fraction.	$\displaystyle \frac{1}{\sqrt{2}}$
A five decimal place approximation to $\displaystyle \sqrt{2}$ is 1.41421.	$\displaystyle \frac{1}{1.41421}$
Without a calculator, would you want to do this division?	$\displaystyle 1.41421\overline{\smash{)}1.0}$

But we can find a fraction equivalent to $\displaystyle \frac {1}{\sqrt{2}}$ by multiplying the numerator and denominator by $\displaystyle \sqrt{2}$ .

$\begin{gather*} \boldsymbol{\frac{1}{\sqrt{2}}} \\ \boldsymbol{\frac{1\cdot\color{myred1}{\sqrt{2}}}{\sqrt{2}\cdot\color{myred1}{\sqrt{2}}}}\\ \boldsymbol{\frac{\sqrt{2}}{2}} \end{gather*}$

Now if we need an approximate value, we divide $\displaystyle 2\overline{\smash{)}1.41421}$ . This is much easier.

Even though we have calculators available nearly everywhere, a fraction with a radical in the denominator still must be rationalized. It is not considered simplified if the denominator contains a square root.

A square root is considered simplified if there are

no perfect-square factors in the radicand
no fractions in the radicand
no square roots in the denominator of a fraction

To rationalize a denominator with a square root, we use the property that $\displaystyle \left(\sqrt{a}\right)^2 = a$ . If we square an irrational square root, we get a rational number.

Try it!

Simplify: a. $\displaystyle \frac {4}{\sqrt{3}}$ b. $\displaystyle \sqrt{\frac {3}{20}$ c. $\displaystyle \frac {3}{\sqrt{6x}}$

Solution A (click to reveal)

To rationalize a denominator with one term, we can multiply a square root by itself. To keep the fraction equivalent, we multiply both the numerator and denominator by the same factor.

a.

Steps

Algebraic

Example

$\boldsymbol{\displaystyle \frac{4}{\sqrt{3}}}$

Multiply both the numerator and denominator by $\boldsymbol{\displaystyle \sqrt{3}}$ .

$\boldsymbol{\displaystyle \frac{4 \cdot {\color{myred1}\sqrt{3}}}{\sqrt{3} \cdot {\color{myred1}\sqrt{3}}}}$

Simplify.

$\boldsymbol{\displaystyle \frac{4\sqrt{3}}{3}}$

Solution B (click to reveal)

b. We always simplify the radical in the denominator first, before we rationalize it. This way the numbers stay smaller and easier to work with.

Steps

Algebraic

Example

$\boldsymbol{\displaystyle \sqrt{\frac{3}{20}}}$

The fraction is not a perfect square, so rewrite using the Quotient Property.

$\boldsymbol{\displaystyle \frac{\sqrt{3}}{\sqrt{20}}}$

Simplify the denominator.

$\boldsymbol{\displaystyle \frac{\sqrt{3}}{2\sqrt{5}}}$

Multiply the numerator and denominator by $\displaystyle \sqrt{5}$ .

$\boldsymbol{\displaystyle \frac{\sqrt{3} \cdot {\color{myred1}\sqrt{5}}}{2\sqrt{5} \cdot {\color{myred1}\sqrt{5}}}}$

Simplify.

$\boldsymbol{\displaystyle \frac{\sqrt{15}}{2\cdot 5}}$

Solution

$\boldsymbol{\displaystyle \frac{\sqrt{15}}{10}}$

Solution C (click to reveal)

c.

Steps

Algebraic

Example

$\boldsymbol{\displaystyle \frac{3}{\sqrt{6x}}}$

Multiply the numerator and denominator by $\displaystyle \sqrt{6x}$ .

$\boldsymbol{\displaystyle \frac{3 \cdot {\color{myred1} \sqrt{6x}}}{\sqrt{6x} \cdot {\color{myred1} \sqrt{6x}}}}$

Simplify.

$\boldsymbol{\displaystyle \frac{3\sqrt{6x}}{6x}}$

Solution

$\boldsymbol{\displaystyle \frac{\sqrt{6x}}{2x}}$

When the denominator of a fraction is a sum or difference with square roots, we use the Product of Conjugates pattern to rationalize the denominator.

**Product of Conjugates**
Algebraic Property	Example
(a − b)(a + b)	$\displaystyle (2 - \sqrt{5})(2 + \sqrt{5})$
a² − b²	$\displaystyle 2^2 -(\sqrt{5})^2$
Simplify	$\displaystyle 4 - 5$
Simplify	-1

Try it!

Simplify: $\displaystyle \frac{4}{4 + \sqrt{2}}$

Solution (click to reveal)

Steps

Algebraic

Example

$\boldsymbol{\displaystyle \frac{4}{4 + \sqrt{2}}}$

Multiply the numerator and denominator by the conjugate of the denominator.

$\boldsymbol{\displaystyle \frac{4 \cdot {\color{myred1} (4 -\sqrt{2})}}{(4 + \sqrt{2})\cdot {\color{myred1} (4 -\sqrt{2})}}}$

Multiply the conjugates in the denominator.

$\boldsymbol{\displaystyle \frac{4(4-\sqrt{2})}{4^2 - (\sqrt{2})^2}}$

Simplify the denominator.

$\boldsymbol{\displaystyle \frac{4(4-\sqrt{2})}{16 - 2}}$

Simplify the denominator.

$\boldsymbol{\displaystyle \frac{4(4-\sqrt{2})}{14}}$

Remove common factors from the numerator and denominator.

We leave the numerator in factored form to make it easier to look for common factors after we have simplified the denominator.

$\boldsymbol{\displaystyle \frac{2(4-\sqrt{2})}{7}}$

Simplify: $\displaystyle \frac{\sqrt{3}}{\sqrt{u} -\sqrt{6}}$

Solution (click to reveal)

Steps

Algebraic

Example

$\boldsymbol{\displaystyle \frac{\sqrt{3}}{\sqrt{u} -\sqrt{6}}}$

Multiply the numerator and denominator by the conjugate of the denominator.

$\boldsymbol{\displaystyle \frac{\sqrt{3} \cdot {\color{myred1} (\sqrt{u} + \sqrt{6})}}{(\sqrt{u} -\sqrt{6}) \cdot {\color{myred1} (\sqrt{u} + \sqrt{6})}}}$

Multiply the conjugates in the denominator.

$\boldsymbol{\displaystyle \frac{\sqrt{3}(\sqrt{u} + \sqrt{6})}{(\sqrt{u})^2 -(\sqrt{6})^2}}$

Simplify the denominator.

$\boldsymbol{\displaystyle \frac{\sqrt{3}(\sqrt{u} + \sqrt{6})}{u - 6}}$

Simplify Variable Expressions with Roots

So far we have only talked about squares and square roots. Let’s now extend our work to include higher powers and higher roots.

Let’s review some vocabulary in the table below.

We write:

We say:

n²

n squared

n³

n cubed

n⁴

n to the fourth power

n⁵

n to the fifth power

The terms ‘squared’ and ‘cubed’ come from the formulas for area of a square and volume of a cube.

It will be helpful to have a table of the powers of the integers from −5 to 5.

*Positive numbers*
Number (n)	Square (n²⁾	Cube (n³)	Fourth Power (n⁴)	Fifth Power (n⁵)
1	1	1	1	1
2	4	8	16	32
3	9	27	81	243
4	16	64	256	1024
5	25	125	625	3125
x	x²	x³	x⁴	x⁵
x²	x⁴	x⁶	x⁸	x¹⁰

*Negative numbers*
Number (n)	Square (n²⁾	Cube (n³)	Fourth Power (n⁴)	Fifth Power (n⁵)
-1	1	-1	1	-1
-2	4	-8	16	-32
-3	9	-27	81	-243
-4	16	-64	256	-1024
-5	25	-125	625	-3125

Notice the signs in the table. All powers of positive numbers are positive, of course. But when we have a negative number, the even powers are positive and the odd powers are negative. We’ll copy the row with the powers of −2 to help you see this in the table below.

$\displaystyle\boldsymbol{\color{myred1}{n}}$

n²

$\displaystyle\boldsymbol{\color{myred1}{n^3}}$

n⁴

$\displaystyle\boldsymbol{\color{myred1}{n^5}}$

-2

4

-8

16

32

Even power Positive result

$\displaystyle\textcolor{myred1}{\text{\boldsymbol{Odd}}}$ power $\displaystyle\textcolor{myred1}{\text{\boldsymbol{Negative}}}$ result

We will now extend the square root definition to higher roots.

n^th Root of a Number

If bⁿ= a, then b is an n^th root of a.

The principal n^th root of a is written $\displaystyle \sqrt[n]{a}$ .

n is called the index of the radical.

Just like we use the word ‘cubed’ for b³, we use the term ‘cube root’ for $\displaystyle \sqrt[3]{a}$ .

We can refer to the table below to see the powers of the integers from -5 to 5 to help find higher roots.

Example in bⁿ= a format

Nth root format

$\displaystyle 4^3 = 64$

$\displaystyle \sqrt[3]{64} = 4$

$\displaystyle 3^4 = 81$

$\displaystyle \sqrt[4]{81} = 3$

$\displaystyle (-2)^5 = -32$

$\displaystyle \sqrt[5]{-32} = -2$

Could we have an even root of a negative number? We know that the square root of a negative number is not a real number. The same is true for any even root. Even roots of negative numbers are not real numbers. Odd roots of negative numbers are real numbers.

Properties of $\displaystyle \boldsymbol{\sqrt[n]{a}}$

When n is an even number and

a ≥ 0, then $\displaystyle \sqrt[n]{a}$ is a real number.
a < 0, then $\displaystyle \sqrt[n]{a}$ is not a real number.

When n is an odd number, $\displaystyle \sqrt[n]{a}$ is a real number for all values of a.

We will apply these properties in the next two examples.

Try it!

Simplify: a. $\displaystyle \sqrt[3]{64}$ b. $\displaystyle \sqrt[4]{81}$ c. $\displaystyle \sqrt[5]{32}$ .

Solution A (click to reveal)

a.

Steps

Algebraic

Example

$\boldsymbol{\displaystyle \sqrt[3]{64}}$

Since $\boldsymbol{\displaystyle 4^3 = 64}$

$\boldsymbol{\displaystyle 4}$

Solution B (click to reveal)

b.

Steps

Algebraic

Example

$\boldsymbol{\displaystyle \sqrt[4]{81}}$

Since $\boldsymbol{\displaystyle 3^4 = 81}$

$\boldsymbol{\displaystyle 3}$

Solution C (click to reveal)

c.

Steps

Algebraic

Example

$\boldsymbol{\displaystyle \sqrt[5]{32}}$

Since $\boldsymbol{\displaystyle 2^5 = 32}$

$\boldsymbol{\displaystyle 2}$

In this example be alert for the negative signs as well as even and odd powers.

Try it!

Simplify: a. $\displaystyle \sqrt[3]{-125}$ b. $\displaystyle \sqrt[4]{-16}$ c. $\displaystyle \sqrt[5]{-243}$ .

Solution A (click to reveal)

a.

Steps

Algebraic

Example

$\boldsymbol{\displaystyle \sqrt[3]{-125}}$

Since $\boldsymbol{\displaystyle (-5)^3 = -125}$

$\boldsymbol{\displaystyle -5}$

Solution B (click to reveal)

b.

Steps

Algebraic

Example

$\boldsymbol{\displaystyle \sqrt[4]{-16}}$

Think, (?)⁴= −16. No real number raised to the fourth power is negative.

Not a real number.

Solution C (click to reveal)

c.

Steps

Algebraic

Example

$\boldsymbol{\displaystyle \sqrt[5]{-243}}$

Since $\boldsymbol{\displaystyle (-3)^5 = -243}$

$\boldsymbol{\displaystyle -3}$

The odd root of a number can be either positive or negative. In the example below we can see the starting value under the radical has the same sign (positive or negative) as the simplified result. These are demonstrated through the use of red font.

$\begin{gather*} \boldsymbol{\sqrt[3]{\color{myred1}{4}\color{black}{^3}}\quad\quad\text{and}\quad\quad\sqrt[3]{\left(\color{myred1}{-4}\color{black}{\right)^3}}} \\ \boldsymbol{\sqrt[3]{64}\quad\quad\quad\quad\quad\sqrt[3]{-64}}\\ \boldsymbol{\color{myred1}{4}\quad\quad\quad\quad\quad\color{myred1}{-4}}\\ \boldsymbol{\text{In either case, when }n\text{ is odd, }\sqrt[n]{\color{myred1}{a}\color{black}{^n}}=\color{myred1}{a}\text{.}} \end{gather*}$

But what about an even root? We want the principal root, so $\displaystyle \sqrt[4]{625} =5$ .

But notice, that the sign of the starting value will not always result in the same sign when simplified. This difference is demonstrated through the use of red font.

$\begin{gather*} \boldsymbol{\sqrt[4]{\color{myred1}{5}\color{black}{^4}}\quad\quad\text{and}\quad\quad\sqrt[4]{\left(\color{myred1}{-5}\color{black}{\right)^4}}} \\ \boldsymbol{\sqrt[4]{625}\quad\quad\quad\quad\quad\sqrt[4]{625}}\\ \boldsymbol{\color{myred1}{5}\quad\quad\quad\quad\quad\color{myred1}{5}}\\ \boldsymbol{\text{Here we see that sometimes when }n\text{ is even, }\sqrt[n]{\color{myred1}{a}\color{black}{^n}}\neq\color{myred1}{a}\text{.}} \end{gather*}$

How can we make sure the fourth root of −5 raised to the fourth power is 5? We can use the absolute value. |−5| = 5. So we say that when n is even $\displaystyle \sqrt[n]{a^n} = |a|$ . This guarantees the principal root is positive.

For any integer n ≥ 2,

**Simplifying Odd and Even Roots (n^th root of Perfect n^th power)**
Descriptive Rule	Algebraic
when the index n is odd	$\displaystyle \sqrt[n]{a^n} = a$
when the index n is even	$\displaystyle \sqrt[n]{a^n} = \|a\|$

We must use the absolute value signs when we take an even root of an expression with a variable in the radical.

Try it!

Simplify: a. $\displaystyle \sqrt{x^2}$ b. $\displaystyle \sqrt[3]{n^3}$ c. $\displaystyle \sqrt[4]{p^4}$ d. $\displaystyle \sqrt[5]{y^5}$

Solution A (click to reveal)

a. We use the absolute value to be sure to get the positive root.

Steps

Algebraic

Example

$\boldsymbol{\displaystyle \sqrt{x^2}}$

Since the index n is even, $\boldsymbol{\displaystyle\sqrt[n]{a^n}=|a|}$

$\boldsymbol{\displaystyle |x|}$

Solution B (click to reveal)

b. This is an odd indexed root so there is no need for an absolute value sign.

Steps

Algebraic

Example

$\boldsymbol{\displaystyle \sqrt[3]{m^3}}$

Since the index n is odd, $\boldsymbol{\displaystyle\sqrt[n]{a^n}=a}$

$\boldsymbol{\displaystyle m}$

Solution C (click to reveal)

c.

Steps

Algebraic

Example

$\boldsymbol{\displaystyle \sqrt[4]{p^4}}$

Since the index n is even, $\boldsymbol{\displaystyle\sqrt[n]{a^n}=|a|}$

$\boldsymbol{\displaystyle |p|}$

Solution D (click to reveal)

d.

Steps

Algebraic

Example

$\boldsymbol{\displaystyle \sqrt[5]{y^5}}$

Since the index n is odd, $\boldsymbol{\displaystyle\sqrt[n]{a^n}=a}$

$\boldsymbol{\displaystyle y}$

Simplify: a. $\displaystyle \sqrt{x^6}$ b. $\displaystyle \sqrt{y^{16}}$ .

Solution A (click to reveal)

a.

Steps

Algebraic

Example

$\boldsymbol{\displaystyle \sqrt{x^6}}$

Since $\boldsymbol{\displaystyle\left(x^3\right)^2=x^6}$

$\boldsymbol{\displaystyle \sqrt{(x^3)^2}}$

Since the index n is even, $\boldsymbol{\displaystyle\sqrt[n]{a^n}=|a|}$

$\boldsymbol{\displaystyle |x^3|}$

Solution B (click to reveal)

b.

Steps

Algebraic

Example

$\boldsymbol{\displaystyle \sqrt{y^{16}}}$

Since $\boldsymbol{\displaystyle\left(y^8\right)^2=y^{16}}$

$\boldsymbol{\displaystyle \sqrt{(x^8)^2}}$

Since the index n is even, $\boldsymbol{\displaystyle\sqrt[n]{a^n}=|a|}$

In this case the absolute value sign is not needed as $\boldsymbol{\displaystyle y^8}$ is positive.

$\boldsymbol{\displaystyle y^8}$

The next example uses the same idea for higher roots.

Try it!

Simplify: a. $\displaystyle \sqrt[3]{y^{18}}$ b. $\displaystyle \sqrt[4]{z^8}$ .

Solution A (click to reveal)

a.

Steps

Algebraic

Example

$\boldsymbol{\displaystyle \sqrt[3]{y^{18}}}$

Since $\boldsymbol{\displaystyle\left(y^6\right)^3=y^{18}}$

$\boldsymbol{\displaystyle \sqrt{(y^6)^3}}$

Since the index n is odd, $\boldsymbol{\displaystyle\sqrt[n]{a^n}=a}$

$\boldsymbol{\displaystyle y^6}$

Solution B (click to reveal)

b.

Steps

Algebraic

Example

$\boldsymbol{\displaystyle \sqrt[4]{z^8}}$

Since $\boldsymbol{\displaystyle\left(z^2\right)^4=z^8}$

$\boldsymbol{\displaystyle \sqrt[4]{(z^2)^4}}$

Since $\boldsymbol{\displaystyle z^2}$ is positive, we do not need an absolute value sign.

$\boldsymbol{\displaystyle z^2}$

Access this online resource for additional instruction and practice with simplifying expressions with roots.

Simplifying Variables Exponents with Roots using Absolute Values

Use Product and Quotient Rule to Simplify

We will simplify radical expressions in a way similar to how we simplified fractions. A fraction is simplified if there are no common factors in the numerator and denominator. To simplify a fraction, we look for any common factors in the numerator and denominator.

A radical expression, $\displaystyle \sqrt[n]{a}$ is considered simplified if it has no factors of mⁿ. So, to simplify a radical expression, we look for any factors in the radicand that are powers of the index.

Simplified Radical Expression

For real numbers a and m, and n ≥ 2,

$\displaystyle \sqrt[n]{a}$ is considered simplified if a has no factors of mⁿ

For example, $\displaystyle \sqrt{5}$ is considered simplified because there are no perfect square factors in 5. But $\displaystyle \sqrt{12}$ is not simplified because 12 has a perfect square factor of 4.

Similarly, $\displaystyle \sqrt[3]{4}$ is simplified because there are no perfect cube factors in 4. But $\displaystyle \sqrt[3]{24}$ is not simplified because 24 has a perfect cube factor of 8.

To simplify radical expressions, we will also use some properties of roots. The properties we will use to simplify radical expressions are similar to the properties of exponents. We know that (ab)ⁿ= aⁿbⁿ. The corresponding Product Property of Roots says that $\displaystyle \sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$ .

Product Property of n^th Roots

If $\displaystyle \sqrt[n]{a}$ and $\displaystyle \sqrt[n]{b}$ are real numbers, and $\displaystyle n \ge 2$ is an integer, then

$\displaystyle \sqrt[n]{ab} = \sqrt[n]{a} \cdot\sqrt[n]{b} \quad \text{ and }\quad \sqrt[n]{a}\cdot \sqrt[n]{b} = \sqrt[n]{ab}$

Try it!

Simplify: $\displaystyle \sqrt{98}$

Solution (click to reveal)

Steps

Algebraic

Step 1. Find the largest factor in the radicand that is a perfect power of the index.

Rewrite the radicand as a product of two factors, using that factor.

$\begin{gather*} \boldsymbol{98 = 49 \cdot 2} \end{gather*}$

We see that 49 is the largest factor of 98 that has a power of 2.

In other words, 49 is the largest perfect square factor of 98.

$\begin{gather*} \boldsymbol{\sqrt{49 \cdot 2}} \end{gather*}$

Step 2. Use the product rule to rewrite the radical as the product of two radicals. Always write the perfect square factor first.

$\displaystyle \boldsymbol{\sqrt{49} \cdot \sqrt{2}}$

Step 3. Simplify the root of the perfect power.

$\displaystyle\boldsymbol{7\sqrt{2}}$

Simplify a radical expression using the Product Property.

Find the largest factor in the radicand that is a perfect power of the index. Rewrite the radicand as a product of two factors, using that factor.
Use the product rule to rewrite the radical as the product of two radicals.
Simplify the root of the perfect power.

Try it!

Simplify: a. $\displaystyle \sqrt{x^3}$ b. $\displaystyle \sqrt[3]{x^4}$ c. $\displaystyle \sqrt[4]{x^7}$

Solution A (click to reveal)

a.

Steps

Algebraic

Example

$\boldsymbol{\displaystyle \sqrt{x^3}}$

Rewrite the radicand as a product using the largest perfect square factor.

$\boldsymbol{\displaystyle \sqrt{x^2 \cdot x}}$

Rewrite the radical as the product of two radicals.

$\boldsymbol{\displaystyle \sqrt{x^2}\cdot \sqrt{x}}$

Simplify.

$\boldsymbol{\displaystyle |x|\sqrt{x}}$

Solution B (click to reveal)

b.

Steps

Algebraic

Example

$\boldsymbol{\displaystyle \sqrt[3]{x^4}}$

Rewrite the radicand as a product using the largest perfect cube factor.

$\boldsymbol{\displaystyle \sqrt[3]{x^3 \cdot x}}$

Rewrite the radical as the product of two radicals.

$\boldsymbol{\displaystyle \sqrt[3]{x^3}\cdot \sqrt[3]{x}}$

Simplify.

$\boldsymbol{\displaystyle x\sqrt[3]{x}}$

Solution C (click to reveal)

c.

Steps

Algebraic

Example

$\boldsymbol{\displaystyle \sqrt[4]{x^7}}$

Rewrite the radicand as a product using the greatest perfect fourth power factor.

$\boldsymbol{\displaystyle \sqrt[4]{x^4 \cdot x^3}}$

Rewrite the radical as the product of two radicals.

$\boldsymbol{\displaystyle \sqrt[4]{x^4}\cdot \sqrt[4]{x^3}}$

Simplify.

$\boldsymbol{\displaystyle |x|\sqrt[4]{x^3}}$

Using Quotient Property to Simplify

Whenever you have to simplify a radical expression, the first step you should take is to determine whether the radicand is a perfect power of the index. If not, check the numerator and denominator for any common factors, and remove them. You may find a fraction in which both the numerator and the denominator are perfect powers of the index.

Try it!

Simplify: a. $\displaystyle \sqrt{\frac{45}{80}}$ b. $\displaystyle \sqrt[3]{\frac{16}{54}}$ c. $\displaystyle \sqrt[4]{\frac{5}{80}}$

Solution A (click to reveal)

a.

Steps

Algebraic

Example

$\displaystyle \boldsymbol{\sqrt{\frac{45}{80}}}$

Simplify inside the radical first.

Rewrite showing the common factors of the numerator and denominator.

$\displaystyle \boldsymbol{\sqrt{\frac{\cancel{\color{myred1}{5}} \cdot 9 }{\cancel{\color{myred1}{5}} \cdot 16}}}$

Simplify the fraction by removing common factors.

$\displaystyle\boldsymbol{ \sqrt{\frac{9}{16}}}$

Simplify. Note $\boldsymbol{\displaystyle\left(\frac{3}{4}\right)^2=\frac{9}{16}}$

$\displaystyle\boldsymbol{ \frac{3}{4}}$

Solution B (click to reveal)

b.

Steps

Algebraic

Example

$\displaystyle\boldsymbol{ \sqrt[3]{\frac{16}{54}}}$

Simplify inside the radical first.

Rewrite showing the common factors of the numerator and denominator.

$\displaystyle \boldsymbol{\sqrt[3]{\frac{\cancel{\color{myred1}{2}} \cdot 8 }{\cancel{\color{myred1}{2}} \cdot 27}}}$

Simplify the fraction by removing common factors.

$\displaystyle\boldsymbol{ \sqrt[3]{\frac{8}{27}}}$

Simplify. Note $\boldsymbol{\displaystyle\left(\frac{2}{3}\right)^3=\frac{8}{27}}$

$\displaystyle\boldsymbol{ \frac{2}{3}}$

Solution C (click to reveal)

c.

Steps

Algebraic

Example

$\displaystyle \boldsymbol{\sqrt[4]{\frac{5}{80}}}$

Simplify inside the radical first.

Rewrite showing the common factors of the numerator and denominator.

$\displaystyle \boldsymbol{\sqrt[4]{\frac{\cancel{\color{myred1}{5}} \cdot 1 }{\cancel{\color{myred1}{5}} \cdot 16}}}$

Simplify the fraction by removing common factors.

$\displaystyle \boldsymbol{\sqrt[4]{\frac{1}{16}}}$

Simplify. Note $\boldsymbol{\displaystyle\left(\frac{1}{2}\right)^4=\frac{1}{16}}$

$\boldsymbol{\displaystyle \frac{1}{2}}$

In the next example we will use the Quotient Property to simplify under the radical. We divide the like bases by subtracting their exponents,

$\displaystyle \frac{a^m}{a^n} = a^{m -n},\qquad a\ne 0$

Try it!

Simplify: a. $\displaystyle \sqrt{\frac{m^6}{m^4}}$ b. $\displaystyle \sqrt[3]{\frac{a^8}{a^5}}$ c. $\displaystyle \sqrt[4]{\frac{a^{10}}{a^2}}$

Solution A (click to reveal)

a.

Steps

Algebraic

Example

$\boldsymbol{\displaystyle \sqrt{\frac{m^6}{m^4}}}$

Simplify the fraction inside the radical first.

Divide the like bases by subtracting the exponents.

$\boldsymbol{\displaystyle \sqrt{m^2}}$

Simplify.

$\boldsymbol{\displaystyle |m|}$

Solution B (click to reveal)

b.

Steps

Algebraic

Example

$\boldsymbol{\displaystyle \sqrt[3]{\frac{a^8}{a^5}}}$

Use the Quotient Property of exponents to simplify the fraction under the radical first.

$\boldsymbol{\displaystyle \sqrt[3]{a^3}}$

Simplify.

$\boldsymbol{\displaystyle a}$

Solution C (click to reveal)

c.

Steps

Algebraic

Example

$\boldsymbol{\displaystyle \sqrt[4]{\frac{a^{10}}{a^2}}}$

Use the Quotient Property of exponents to simplify the fraction under the radical first.

$\boldsymbol{\displaystyle \sqrt[4]{a^8}}$

Rewrite the radicand using perfect fourth power factors

$\boldsymbol{\displaystyle \sqrt[4]{(a^2)^4}}$

Simplify.

$\boldsymbol{\displaystyle a^2}$

Remember the Quotient to a Power Property? It said we could raise a fraction to a power by raising the numerator and denominator to the power separately.

$\displaystyle \left(\frac{a}{b}\right)^m = \frac{a^m}{b^m},\qquad b\ne 0$

We can use a similar property to simplify a root of a fraction. After removing all common factors from the numerator and denominator, if the fraction is not a perfect power of the index, we simplify the numerator and denominator separately.

Quotient Property of Radical Expressions

If $\displaystyle \sqrt[n]{a}$ and $\displaystyle \sqrt[n]{b}$ are real numbers, b ≠ 0, and for any integer n ≥ 2 then,

$\displaystyle \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \quad \text{ and }\quad \frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}$

Simplify the fraction in the radicand, if possible.
Use the Quotient Property to rewrite the radical as the quotient of two radicals.
Simplify the radicals in the numerator and the denominator.

Try it! – How to Simplify the Quotient of Radical Expressions

Simplify: $\displaystyle \sqrt{\frac{27m^3}{196}}$

Solution (click to reveal)

Steps

Algebraic

Step 1. Simplify the fraction in the radicand, if possible. $\displaystyle\boldsymbol{\frac{27m^3}{196}}$ cannot be simplified.

$\displaystyle\boldsymbol{\sqrt{\frac{27m^3}{196}}}$

Step 2. Use the Quotient Property to rewrite the radicals as the quotient of two radicals. We rewrite $\displaystyle\boldsymbol{\sqrt{\frac{27m^3}{196}}}$ as the quotient of $\displaystyle\boldsymbol{\sqrt{27m^3}}$ and $\displaystyle\boldsymbol{\sqrt{196}}$ .

$\displaystyle\boldsymbol{\frac{\sqrt{27m^3}}{\sqrt{196}}}$

Step 3. Simplify the radicals in the numerator and enumerator. Note: $\displaystyle\boldsymbol{9m^2}$ and $\displaystyle\boldsymbol{196}$ are perfect squares.

$\begin{gather*} \boldsymbol{\frac{\sqrt{9m^2}\cdot\sqrt{3m}}{\sqrt{196}}} \\ \boldsymbol{\frac{3m\sqrt{3m}}{14}} \end{gather*}$

Simplify: a. $\displaystyle \frac{\sqrt[3]{-108}}{\sqrt[3]{2}}$ b. $\displaystyle \frac{\sqrt[4]{96x^7}}{\sqrt[4]{3x^2}}$

Solution A (click to reveal)

a.

Steps

Algebraic

Example

$\boldsymbol{\displaystyle \frac{\sqrt[3]{-108}}{\sqrt[3]{2}}}$

Neither radicand is a perfect cube, so use the Quotient Property to write as one radical.

$\boldsymbol{\displaystyle \sqrt[3]{\frac{-108}{2}}}$

Simplify the fraction under the radical.

$\boldsymbol{\displaystyle \sqrt[3]{-54}}$

Rewrite the radicand as a product using perfect cube factors.

$\boldsymbol{\displaystyle \sqrt[3]{(-3)^3 \cdot 2}}$

Rewrite the radical as the product of two radicals.

$\boldsymbol{\displaystyle \sqrt[3]{(-3)^3} \cdot \sqrt[3]{2}}$

Simplify.

$\boldsymbol{\displaystyle -3\sqrt[3]{2}}$

Solution B (click to reveal)

b.

Steps

Algebraic

Example

$\boldsymbol{\displaystyle \frac{\sqrt[4]{96x^7}}{\sqrt[4]{3x^2}}}$

Neither radicand is a perfect cube, so use the Quotient Property to write as one radical.

$\boldsymbol{\displaystyle \sqrt[4]{\frac{96x^7}{3x^2}}}$

Simplify the fraction under the radical.

$\boldsymbol{\displaystyle \sqrt[4]{32x^5}}$

Rewrite the radicand as a product using perfect fourth power factors.

$\boldsymbol{\displaystyle \sqrt[4]{2^4 \cdot x^4 \cdot 2x}}$

Rewrite the radical as the product of two radicals.

$\boldsymbol{\displaystyle \sqrt[4]{(2x)^4} \cdot \sqrt[4]{2x}}$

Simplify.

$\boldsymbol{\displaystyle 2|x|\sqrt[4]{2x}}$

Simplify Expressions with $\displaystyle \boldsymbol{a^{\frac{1}{n}}}$

Rational exponents are another way of writing expressions with radicals. When we use rational exponents, we can apply the properties of exponents to simplify expressions.

The Power Property for Exponents says that $\displaystyle (a^m)^n = a^{m\cdot n}$ when m and n are whole numbers. Let’s assume we are now not limited to whole numbers.

Suppose we want to find a number p such that $\displaystyle (8^p)^3 = 8$ . We will use the Power Property of Exponents to find the value of p in the table below.

Steps

Algebraic

Example

$\displaystyle (8^p)^3 = 8$

Multiply the exponents on the left.

$\displaystyle 8^{3p} = 8$

Write the exponent 1 on the right

$\displaystyle 8^{3p} = 8^1$

Since the bases are the same, the exponents must be equal.

$\displaystyle 3p = 1$

Solve for p.

$\displaystyle p = \frac{1}{3}$

So $\displaystyle \left(8 ^{\frac{1}{3}}\right)^3 =8$ . But we know also $\displaystyle \left(\sqrt[3]{8}\right)^3 = 8$ . Then it must be that $\displaystyle 8^{\frac{1}{3}} = \sqrt[3]{8}$ .

This same logic can be used for any positive integer exponent n to show that $\displaystyle a^{\frac{1}{n}} = \sqrt[n]{a}$ .

Rational Exponent $\displaystyle \boldsymbol{a^{\frac{1}{n}}}$

If $\displaystyle \sqrt[n]{a}$ is a real number and n ≥ 2, then

$\displaystyle a^{\frac{1}{n}} = \sqrt[n]{a}$

The denominator of the rational exponent is the index of the radical.

There will be times when working with expressions will be easier if you use rational exponents and times when it will be easier if you use radicals. In the first few examples, you’ll practice converting expressions between these two notations.

Try it!

Write as a radical expression: a. $\displaystyle x^{\frac{1}{2}}$ b. $\displaystyle y^{\frac{1}{3}}$ c. $\displaystyle z^{\frac{1}{4}}$

Solution A (click to reveal)

We want to write each expression in the form $\displaystyle \sqrt[n]{a}$

a.

Steps

Algebraic

Expression

$\boldsymbol{\displaystyle x^{\frac{1}{2}}}$

The denominator of the rational exponent is 2, sothe index of the radical is 2. We do not show theindex when it is 2.

$\boldsymbol{\displaystyle \sqrt{x}}$

Solution B (click to reveal)

b.

Steps

Algebraic

Expression

$\boldsymbol{\displaystyle y^{\frac{1}{3}}}$

The denominator of the exponent is 3, so the index is 3.

$\boldsymbol{\displaystyle \sqrt[3]{y}}$

Solution C (click to reveal)

c.

Steps

Algebraic

Expression

$\boldsymbol{\displaystyle z^{\frac{1}{4}}}$

The denominator of the exponent is 4, so the index is 4.

$\boldsymbol{\displaystyle \sqrt[4]{z}}$

Simplify: a. $\displaystyle 25^{\frac{1}{2}}$ b. $\displaystyle 64^{\frac{1}{3}}$ c. $\displaystyle 256^{\frac{1}{4}}$

Solution A (click to reveal)

a.

Steps

Algebraic

Expression

$\boldsymbol{\displaystyle 25^{\frac{1}{2}}}$

Rewrite as square root.

$\boldsymbol{\displaystyle \sqrt{25}}$

Simplify.

$\boldsymbol{\displaystyle 5}$

Solution B (click to reveal)

b.

Steps

Algebraic

Expression

$\boldsymbol{\displaystyle 64^{\frac{1}{3}}}$

Rewrite as a cube root.

$\boldsymbol{\displaystyle \sqrt[3]{64}}$

Recognize 64 is a perfect cube.

$\boldsymbol{\displaystyle \sqrt[3]{4^3}}$

Simplify.

$\boldsymbol{\displaystyle 4}$

Solution C (click to reveal)

c.

Steps

Algebraic

Expression

$\boldsymbol{\displaystyle 256^{\frac{1}{4}}}$

Rewrite as a fourth root.

$\boldsymbol{\displaystyle \sqrt[4]{256}}$

Recognize 256 is a perfect fourth power.

$\boldsymbol{\displaystyle \sqrt[4]{4^4}}$

Simplify.

$\boldsymbol{\displaystyle 4}$

Be careful of the placement of the negative signs in the next example. We will need to use the property $\displaystyle a^{-n} = \frac{1}{a^n}$ in one case.

Try it!

Simplify: a. $\displaystyle (-16)^{\frac{1}{4}}$ b. $\displaystyle -16^{\frac{1}{4}}$ c. $\displaystyle (16)^{-\frac{1}{4}}$

Solution A (click to reveal)

a.

Steps

Algebraic

Expression

$\boldsymbol{\displaystyle (-16)^{\frac{1}{4}}}$

Rewrite as a fourth root.

$\boldsymbol{\displaystyle \sqrt[4]{-16}}$

Simplify

$\boldsymbol{\displaystyle \sqrt[4]{-2^4}}$

Simplify.

No real solution.

Solution B (click to reveal)

b.

Steps

Algebraic

Expression

$\boldsymbol{\displaystyle -16^{\frac{1}{4}}}$

The exponent only applies to the 16.

Rewrite as a fourth root.

$\boldsymbol{\displaystyle -\sqrt[4]{16}}$

Rewrite 16 as 2⁴.

$\boldsymbol{\displaystyle -\sqrt[4]{2^4}}$

Simplify.

$\boldsymbol{\displaystyle -2}$

Solution C (click to reveal)

c.

Steps

Algebraic

Expression

$\boldsymbol{\displaystyle 16^{-\frac{1}{4}}}$

Rewrite using the property $\displaystyle \boldsymbol{a^{-n}=\frac{1}{a^n}}$

$\boldsymbol{\displaystyle \frac{1}{16^{\frac{1}{4}}}}$

Rewrite as a fourth root.

$\boldsymbol{\displaystyle \frac{1}{\sqrt[4]{16}}}$

Rewrite 16 as 2⁴.

$\boldsymbol{\displaystyle \frac{1}{\sqrt[4]{2^4}}}$

Simplify.

$\boldsymbol{\displaystyle \frac{1}{2}}$

Simplify Expressions with $\displaystyle \boldsymbol{a^{\frac{m}{n}}}$

We can look at $\displaystyle a^{\frac{m}{n}}$ in two ways. Remember, the Power Property tells us to multiply the exponents and so $\displaystyle \left(a^{\frac{1}{n}}\right)^{m}$ and $\displaystyle (a^m)^{\frac{1}{n}}$ both equal $\displaystyle a^{\frac{m}{n}}$ . If we write these expressions in radical form, we get

$\displaystyle a^{\frac{m}{n}} = \left(a^{\frac{1}{n}\right)^m = \left(\sqrt[n]{a}\right)^m\quad \text{ and }\quad a^{\frac{m}{n}} = (a^m)^{\frac{1}{n}} = \sqrt[n]{a^m}$

This leads us to the following definition.

Rational Exponent $\displaystyle {\color{white}\boldsymbol{a^{\frac{m}{n}}}}$

For any positive integers m and n,

$\displaystyle a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m\quad \text{ and }\quad a^{\frac{m}{n}} = \sqrt[n]{a^m}$

Try it!

Simplify: a. $\displaystyle 125^{\frac{2}{3}}$ b. $\displaystyle 16^{-\frac{3}{2}}$ c. $\displaystyle 32^{-\frac{2}{5}}$

Solution A (click to reveal)

We will rewrite the expression as a radical first using the definition, $\displaystyle a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^{m}$ . This form lets us take the root first and so we keep the numbers in the radicand smaller than if we used the other form.

a.

Steps

Algebraic

Expression

$\boldsymbol{\displaystyle 125^{\frac{2}{3}}}$

The power of the radical is the numerator of the exponent, 2.

The index of the radical is the denominator of the exponent, 3.

$\boldsymbol{\displaystyle \left(\sqrt[3]{125}\right)^{2}}$

Simplify.

$\boldsymbol{\displaystyle (5)^2}$

Solution

$\boldsymbol{\displaystyle 25}$

Solution B (click to reveal)

b. We will rewrite each expression first using $\displaystyle a^{-n} = \frac{1}{a^n}$ and then change to radical form.

Steps

Algebraic

Example

$\boldsymbol{\displaystyle 16^{-\frac{3}{2}}}$

Rewrite using $\displaystyle \boldsymbol{a^{-n}=\frac{1}{a^n}}$

$\boldsymbol{\displaystyle \frac{1}{16^{\frac{3}{2}}}}$

Change to radical form. The power of the radical is the numerator of the exponent, 3. The index is the denominator of the exponent, 2.

$\boldsymbol{\displaystyle \frac{1}{\left(\sqrt{16}\right)^3}}$

Simplify.

$\boldsymbol{\displaystyle \frac{1}{4^3}}$

Solution

$\boldsymbol{\displaystyle \frac{1}{64}}$

Solution C (click to reveal)

c.

Steps

Algebraic

Example

$\boldsymbol{\displaystyle 32^{-\frac{2}{5}}}$

Rewrite using $\displaystyle \boldsymbol{a^{-n}=\frac{1}{a^n}}$

$\boldsymbol{\displaystyle \frac{1}{32^{\frac{2}{5}}}}$

Change to radical form.

$\boldsymbol{\displaystyle \frac{1}{\left(\sqrt[5]{32}\right)^2}}$

Rewrite the radicand as a power

$\boldsymbol{\displaystyle \frac{1}{\left(\sqrt[5]{2^5}\right)^2}}$

Simplify.

$\boldsymbol{\displaystyle \frac{1}{2^2}}$

Solution

$\boldsymbol{\displaystyle \frac{1}{4}}$

Use the Properties of Exponents to Simplify Expressions with Rational Exponents

The same properties of exponents that we have already used also apply to rational exponents. We will list the Properties of Exponents here to have them for reference as we simplify expressions.

If a and b are real numbers and m and n are rational numbers, then

*Properties of Exponents*
Properties	Definitions
Product Property	$\displaystyle a^m \cdot a^n = a^{m + n}$
Power Property	$\displaystyle (a^m)^n = a^{m \cdot n}$
Product to a Power Property	$\displaystyle (ab)^m = a^{m}b^{m}$
Quotient Property	$\displaystyle \frac{a^m}{a^n} = a^{m - n},\quad a \ne 0$
Zero Exponent Property	$\displaystyle a^0 = 1, \quad a\ne 0$
Quotient to a Power Property	$\displaystyle \left(\frac{a}{b}\right)^m = \frac{a^m}{b^m},\quad b \ne 0$
Negative Exponent Property	$\displaystyle a^{-n} =\frac{1}{a^n},\quad a \ne 0$

We will apply these properties in the next example.

Try it!

Simplify: a. $\displaystyle x^{\frac{1}{2}} \cdot x^{\frac{5}{6}}$ b. $\displaystyle (z^9)^{\frac{2}{3}}$ c. $\displaystyle \frac{x^{\frac{1}{3}}}{x^{\frac{5}{3}}}$

Solution A (click to reveal)

a. The Product Property tells us that when we multiply the same base, we add the exponents.

Steps

Algebraic

Example

$\boldsymbol{\displaystyle x^{\frac{1}{2}} \cdot x^{\frac{5}{6}}}$

The bases are the same, so we add the exponents.

$\boldsymbol{\displaystyle x^{\frac{1}{2} + \frac{5}{6}}}$

Add the fractions.

$\boldsymbol{\displaystyle x^{\frac{8}{6}}}$

Simplify the exponent.

$\boldsymbol{\displaystyle x^{\frac{4}{3}}}$

Solution B (click to reveal)

b. The Power Property tells us that when we raise a power to a power, we multiply the exponents.

Steps

Algebraic

Example

$\boldsymbol{\displaystyle (z^9)^{\frac{2}{3}}}$

To raise a power to a power, we multiply the exponents.

$\boldsymbol{\displaystyle z^{9\cdot \frac{2}{3}}}$

Simplify.

$\boldsymbol{\displaystyle z^6}$

Solution C (click to reveal)

c. The Quotient Property tells us that when we divide with the same base, we subtract the exponents.

Steps

Algebraic

Example

$\boldsymbol{\displaystyle \frac{x^{\frac{1}{3}}}{x^{\frac{5}{3}}}}$

To divide with the same base, we subtract the exponents.

$\boldsymbol{\displaystyle \frac{1}{x^{\frac{5}{3} - \frac{1}{3}}}}$

Simplify.

$\boldsymbol{\displaystyle \frac{1}{x^{\frac{4}{3}}}}$

Access these online resources for additional instruction and practice with simplifying rational exponents.

Key Concepts

Square Root Notation
- $\displaystyle \sqrt{m}$ is read ‘the square root of m’
- If $\displaystyle n^2 = m$ , then $\displaystyle n =\sqrt{m}$ , for n ≥ 0.
  $\displaystyle\textcolor{mypurple1}{\boldsymbo{\text{radical sign }}} \textcolor{mypurple1}{\boldsymbol{\longrightarrow}} \boldsymbol{\sqrt{m}}\textcolor{mypurple1}{\boldsymbol{\longleftarrow}} \textcolor{mypurple1}{\boldsymbol{\text{ radicand}}}$
- The square root of m, $\displaystyle \sqrt{m}$ , is a positive number whose square is m.
- Simplified Square Root $\displaystyle \sqrt{a}$ is considered simplified if a has no perfect-square factors.
Product Property of Square Roots If a, b are non-negative real numbers, then

$\displaystyle \sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$
Simplify a Square Root Using the Product Property To simplify a square root using the Product Property:
1. Find the largest perfect square factor of the radicand. Rewrite the radicand as a product using the perfect square factor.
2. Use the product rule to rewrite the radical as the product of two radicals.
3. Simplify the square root of the perfect square.
Quotient Property of Square Roots If a, b are non-negative real numbers and b ≠ 0, then

$\displaystyle \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$
Simplify a Square Root Using the Quotient Property To simplify a square root using the Quotient Property:
1. Simplify the fraction in the radicand, if possible.
2. Use the Quotient Rule to rewrite the radical as the quotient of two radicals.
3. Simplify the radicals in the numerator and the denominator.
n^th Root of a Number
- If bⁿ = a, then b is an n^th root of a.
- The principal n^th root of a is written $\displaystyle \sqrt[n]{a}$ .
- n is called the index of the radical.
Properties of $\displaystyle \boldsymbol{\sqrt[n]{a}}$
- When n is an even number and
  - a ≥ 0, then $\displaystyle \sqrt[n]{a}$ is a real number
  - a < 0, then $\displaystyle \sqrt[n]{a}$ is not a real number
- When n is an odd number, $\displaystyle \sqrt[n]{a}$ is a real number for all values of a.
Simplifying Odd and Even Roots
- For any integer n ≥ 2,
  - when n is odd $\displaystyle \sqrt[n]{a} = a$
  - when n is even $\displaystyle \sqrt[n]{a} = |a|$
- We must use the absolute value signs when we take an even root of an expression with a variable in the radical.

Simplified Radical Expression
- For real numbers a, m and n ≥ 2
  $\displaystyle \sqrt[n]{a}$ is considered simplified if a has no factors of mⁿ
Product Property of n^th Roots
- For any real numbers, $\displaystyle \sqrt[n]{a}$ and $\displaystyle \sqrt[n]{b}$ and for any integer n ≥ 2
  $\displaystyle \sqrt[n]{ab} = \sqrt[n]{a}\cdot \sqrt[n]{b}$ and $\displaystyle \sqrt[n]{a}\cdot \sqrt[n]{b} = \sqrt[n]{ab}$
How to simplify a radical expression using the Product Property
1. Find the largest factor in the radicand that is a perfect power of the index.
  Rewrite the radicand as a product of two factors, using that factor.
2. Use the product rule to rewrite the radical as the product of two radicals.
3. Simplify the root of the perfect power.
Quotient Property of Radical Expressions
- If $\displaystyle \sqrt[n]{a}$ and $\displaystyle \sqrt[n]{b}$ are real numbers, b ≠ 0, and for any integer n ≥ 2 then,
  $\displaystyle \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$ and $\displaystyle \frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}$
How to simplify a radical expression using the Quotient Property.
1. Simplify the fraction in the radicand, if possible.
2. Use the Quotient Property to rewrite the radical as the quotient of two radicals.
3. Simplify the radicals in the numerator and the denominator.

Rational Exponent $\displaystyle \boldsymbol{a^{\frac{1}{n}}}$
- If $\displaystyle \sqrt[n]{a}$ is a real number and n ≥ 2, then $\displaystyle a^{\frac{1}{n}} = \sqrt[n]{a}$ .
Rational Exponent $\displaystyle \boldsymbol{a^{\frac{m}{n}}}$
- For any positive integers m and n,
  $\displaystyle a^{\frac{m}{n}} = (\sqrt[n]{a})^m$ and $\displaystyle a^{\frac{m}{n}} = \sqrt[n]{a^m}$
Properties of Exponents
- If a, b are real numbers and m, n are rational numbers, then
  - Product Property: $\displaystyle a^m \cdot a^n = a^{m + n}$
  - Power Property: $\displaystyle (a^m)^n = a^{m \cdot n}$
  - Product to a Power Property: $\displaystyle (ab)^m = a^{m}b^{m}$
  - Quotient Property: $\displaystyle \frac{a^m}{a^n} = a^{m - n},\quad a \ne 0$
  - Zero Exponent Definition: $\displaystyle a^0 = 1, \quad a\ne 0$
  - Quotient to a Power Property: $\displaystyle \left(\frac{a}{b}\right)^m = \frac{a^m}{b^m},\quad b \ne 0$
  - Negative Exponent Property: $\displaystyle a^{-n} =\frac{1}{a^n},\quad a \ne 0$

Derived from Openstax Elementary Algebra; Access for free at https://openstax.org/books/intermediate-algebra-2e/pages/1-introduction. Intermediate Algebra;Access for free at https://openstax.org/books/intermediate-algebra-2e/pages/1-introduction ↵

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Simplify Expressions with Roots

Square Root Notation

Simplified Square Root

Use Product Property to Simplify Square Roots

Product Property of Square Roots

Simplify a square root using the product property.

Quotient Property with Roots

Quotient Property of Square Roots

Add, Subtract Square Roots

Rationalize Denominators

Simplify Variable Expressions with Roots

Use Product and Quotient Rule to Simplify

Simplified Radical Expression

Product Property of nth Roots

Simplify a radical expression using the Product Property.

Using Quotient Property to Simplify

Quotient Property of Radical Expressions

Simplify Expressions with

Rational Exponent

Simplify Expressions with

Use the Properties of Exponents to Simplify Expressions with Rational Exponents

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Product Property of n^th Roots

Simplify Expressions with $\displaystyle \boldsymbol{a^{\frac{1}{n}}}$

Rational Exponent $\displaystyle \boldsymbol{a^{\frac{1}{n}}}$

Simplify Expressions with $\displaystyle \boldsymbol{a^{\frac{m}{n}}}$