Exponents and Scientific Notation

Elizabeth Pople

26 Exponents and Scientific Notation

Topics Covered:

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:

Rounding Decimals
Simplify Expressions Using the Properties of Exponents
Using the Definition of Negative Exponent
Summary of Exponent Properties
Using Scientific Notation
Key Concepts

Rounding Decimals

Decimals are another way of writing fractions whose denominators are powers of ten.

Decimals

Pronunciation

0.1 = $\displaystyle\frac{1}{10}$

is “one tenth

0.01 = $\displaystyle\frac{1}{100}$

is “one hundredth”

0.001 = $\displaystyle\frac{1}{1000}$

is “one thousandth”

0.0001 = $\displaystyle\frac{1}{10,000}$

is “one ten-thousandth”

Just as in whole numbers, each digit of a decimal corresponds to the place value based on the powers of ten. The table shows the names of the place values to the left and right of the decimal point.

This table is labeled place value and has 12 columns. The seventh column is blank. Starting from here and going left the columns are labeled: ones, tens, hundreds, thousands, ten thousands, hundred thousands. Starting from the blank column and going right the columns are labeled: tenths, hundredths, thousandths, ten thousandths hundred thousandths. There is a dot under the blank column.

When we work with decimals, it is often necessary to round the number to the nearest required place value. We summarize the steps for rounding a decimal here.

How to round decimals.

Locate the given place value and mark it with an arrow.
Underline the digit to the right of the place value.
Is the underlined digit greater than or equal to 5?
- Yes: add 1 to the digit in the given place value.
- No: do not change the digit in the given place value
Rewrite the number, deleting all digits to the right of the rounding digit.

Try it!

1. Round 18.379 to the nearest a. hundredth b. tenth c. whole number.

Solution:

Round 18.379.

a. to the nearest hundredth

Steps

Algebraic

Locate the hundredths place with an arrow.

Underline the digit to the right of the given
place value.

Because 9 is greater than or equal to 5, add 1 to the 7.

Rewrite the number, deleting all digits to the right of the rounding digit.

18.38

Notice that the deleted digits were NOT replaced with zeros.

So 18.379 rounded to the nearest hundredth is 18.38.

b. to the nearest tenth

Steps

Algebraic

Locate the tenths place with an arrow.

Underline the digit to the right of the given place value.

Because 7 is greater than or equal to 5, add 1 to the 3.

Rewrite the number, deleting all digits to the right of the rounding digit.

18.4

Notice that the deleted digits were NOT replaced with zeros.

So 18.379 rounded to the nearest hundredth is 18.4.

c. to the nearest whole number

Steps

Algebraic

Locate the ones place with an arrow.

Underline the digit to the right of the given place value.

Since 3 is not greater than or equal to 5, do not add 1 to the 8.

Rewrite the number, deleting all digits to the right of the rounding digit.

18

Solution

So 18.379 rounded to the nearest whole number is 18.

2. Round 6.582 to the nearest a. hundredth b. tenth c. whole number.

Solution:

a. 6.58 b. 6.6 c. 7

Simplify Expressions Using the Properties of Exponents

Remember that an exponent indicates repeated multiplication of the same quantity. For example, in the expression $\displaystyle a^m$ , the exponent m tells us how many times we use the base a as a factor. First example: a raised to the power of m equals a times a times a times a and so on until you have multiplied m different factors of a together. Second example: the quantity negative 9 raised to the power of 5 equals negative 9 times negative 9 times negative 9 times negative 9 times negative 9, a total of 5 factors of negative 9.

Let’s review the vocabulary for expressions with exponents.

Exponential Notation

The figure shows the letter a in a normal font with the label base and the letter m in a superscript font with the label exponent. This means we multiply the number a with itself, m times.

This is read a to the m^th power.

In the expression a^m, the exponent m tells us how many times we use the base a as a factor.

When we combine like terms by adding and subtracting, we need to have the same base with the same exponent. But when you multiply and divide, the exponents may be different, and sometimes the bases may be different, too.

First, we will look at an example that leads to the Product Property.

Product Property

Example

What does this mean?

Figure shows we are multiplying 2 factors of x with 3 factors of x for a total of 5 factors of x .

Simplify

Notice that 5 is the sum of the exponents, 2 and 3. We see x²· x³ is x²⁺³ or x⁵.

The base stayed the same and we added the exponents. This leads to the Product Property for Exponents.

Product Property for Exponents

If a is a real number and m and n are integers, then

a^m· aⁿ= a^m+n

To multiply with like bases, add the exponents.

Try it!

1. Simplify each expression: a. y⁵· y⁶ b. 2a⁷· 3a.

Solution:

a.

Steps

Algebraic

Expression

Use the Product Property, a^m· aⁿ= a^m+n

Simplify

b.

Steps

Algebraic

Expression

Rewrite, a = a¹.

Use the Commutative Property and use the Product Property, $\displaystyle a^m \cdot a^n =$ a^m+n.

Simplify

2. Simplify each expression:

a. x¹²· x⁴b. 10 · 10^xc. 2z · 6z⁷d. b⁵ · b⁹ · b⁵

Solution:

a. x¹⁶b. 10^x+1c. 12z⁸d. b¹⁹

Now we will look at the exponent property for division. As before, we’ll try to discover a property by looking at some examples.

Exponent Property for Division

Consider

$\displaystyle\frac{x^5}{x^2}$

and

$\displaystyle\frac{x^2}{x^3}$

What do they mean?

$\displaystyle\frac{x\cdot x\cdot x\cdot x\cdot x}{x\cdot x}$

$\displaystyle\frac{x\cdot x}{x\cdot x\cdot x}$

Use the Equivalent Fractions Property.

$\displaystyle\frac{\cancel{x}\cdot \cancel{x}\cdot x\cdot x\cdot x}{\cancel{x}\cdot \cancel{x}}$

$\displaystyle\frac{\cancel{x}\cdot \cancel{x}\cdot 1}{\cancel{x}\cdot \cancel{x}\cdot x}$

Simplify.

$\displaystyle x^3$

$\displaystyle\frac{1}{x}$

Notice, in each case the bases were the same and we subtracted exponents. We see $\displaystyle \frac{x^5}{x^2}$ is x^{5 – 2}or x³. We see $\displaystyle \frac{x^2}{x^3}$ is x^2-3 or $\displaystyle \frac{1}{x}$ . When the larger exponent was in the numerator, we were left with factors in the numerator. When the larger exponent was in the denominator, we were left with factors in the denominator — notice the numerator of 1. When all the factors in the numerator have been removed, remember this is really dividing the factors to one, and so we need a 1 in the numerator. $\displaystyle\frac{x}{x} = 1$ . This leads to the Quotient Property for Exponents.

Quotient Property for Exponents

If a is a real number, a ≠ 0, and m and n are integers, then

$\displaystyle \frac{a^m}{a^n} = a^{m - n},\quad m > n \qquad$ and $\qquad\displaystyle \frac{a^m}{a^n} = \frac{1}{a^{n - m}},\quad n > m$

Try it!

Simplify each expression: a. $\displaystyle\frac{x^9}{x^7}$ b. $\displaystyle \frac{b^8}{b^{12}}$

Solution:

To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.

a.

Steps

Algebraic

Since 9 > 7, there are more factors of x in the numerator.

Use Quotient Property, $\displaystyle \frac{a^m}{a^n} = a^{m - n}$

Simplify.

Notice that when the larger exponent is in the numerator, we are left with factors in the numerator.

b.

Steps

Algebraic

Since 12 > 8, there are more factors of b in the denominator.

Use Quotient Property, $\displaystyle \frac{a^m}{a^n} = \frac{1}{a^{n-m}}$

Simplify

Notice that when the larger exponent is in the denominator, we are left with factors in the denominator.

A special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as an expression like $\displaystyle \frac{a^m}{a^m}$ . We know, $\displaystyle \frac{x}{x} = 1$ , for any x (x ≠ 0) since any number divided by itself is 1.

The Quotient Property for Exponents shows us how to simplify $\displaystyle \frac{a^m}{a^m}$ when m > n and when n < m by subtracting exponents. What if m = n? We will simplify $\displaystyle \frac{a^m}{a^m}$ in two ways to lead us to the definition of the Zero Exponent Property.

In general, for a ≠ 0,
$In the first way we write a to the power of m divided by a to the power of m as a to the power of the quantity m minus m. This is equal to a to the power of 0. In the second way we write a to the power of m divided by a to the power of m as a fraction with m factors of a in the numerator and a factors of m in the denominator. Simplifying this we can cross of all the factors and are left with the number 1. This shows that a to the power of 0 is equal to 1.$ We see $\displaystyle \frac{a^m}{a^m}$ simplifies to a⁰and to 1. So a⁰= 1. Any non-zero base raised to the power of zero equals 1.

Zero Exponent Property

If a is a non-zero number, then a⁰= 1.

If a is a non-zero number, then a to the power of zero equals 1.

Any non-zero number raised to the zero power is 1.

In this text, we assume any variable that we raise to the zero power is not zero.

Try it!

Simplify each expression: a. 9⁰ b. n⁰.

Solution:

The definition says any non-zero number raised to the zero power is 1.

a.

Steps

Algebraic

Expression

9⁰

Use the definition of the zero exponent.

1

b.

Steps

Algebraic

Expression

n⁰

Use the definition of the zero exponent.

1

To simplify the expression n raised to the zero power we just use the definition of the zero exponent. The result is 1.

Using the Definition of Negative Exponent

We saw that the Quotient Property for Exponents has two forms depending on whether the exponent is larger in the numerator or the denominator. What if we just subtract exponents regardless of which is larger?

Let’s consider $\displaystyle\frac{x^2}{x^5}$ . We subtract the exponent in the denominator from the exponent in the numerator. We see $\displaystyle\frac{x^2}{x^5}$ is x^{2 − 5} or x⁻³.

We can also simplify $\displaystyle\frac{x^2}{x^5}$ by dividing out common factors:

$In the figure the expression x raised to the power of 2 divided by x raised to the power of 5 is written as a fraction with 2 factors of x in the numerator divided by 5 factors of x in the denominator. Two factors are crossed off in both the numerator and denominator. This only leaves 3 factors of x in the denominator. The simplified fraction is 1 divided by x to the power of 3.$ This implies that $\displaystyle x^{-3} = \frac{1}{x^3}$ and it leads us to the definition of a negative exponent. If n is an integer and a ≠ 0, then $\displaystyle a^{-n} = \frac{1}{a^n}$ .

Let’s now look at what happens to a fraction whose numerator is one and whose denominator is an integer raised to a negative exponent.

Steps

Algebraic

Expression

$\displaystyle \frac{1}{a^{-n}}$

Use the definition of a negative exponent, $\displaystyle a^{-n} = \frac{1}{a^n}$

$\displaystyle \frac{1}{\frac{1}{a^n}}$

Simplify the complex fraction

$\displaystyle 1 \cdot \frac{a^n}{1}$

Multiply

$\displaystyle a^n$

This implies $\displaystyle \frac{1}{a^{-n}} = a^n$ and is another form of the definition of Properties of Negative Exponents.

Properties of Negative Exponents

If n is an integer and a ≠ 0, then $\displaystyle a^{-n} = \frac{1}{a^{n}}$ or $\displaystyle \frac{1}{a^{-n}} = a^n$ .

The negative exponent tells us we can rewrite the expression by taking the reciprocal of the base and then changing the sign of the exponent.

Any expression that has negative exponents is not considered to be in simplest form. We will use the definition of a negative exponent and other properties of exponents to write the expression with only positive exponents.

For example, if after simplifying an expression we end up with the expression x⁻³, we will take one more step and write $\displaystyle \frac{1}{x^{3}}$ . The answer is considered to be in simplest form when it has only positive exponents.

Try it!

Simplify each expression: a. $\displaystyle x^{-5}$ b. $\displaystyle 10^{-3}$ c. $\displaystyle \frac{1}{y^{-4}}$ d. $\displaystyle \frac{1}{3^{-2}}$ .

Solution:

a.

Steps

Algebraic

Expression

$\displaystyle x^{-5}$

Use the definition of a negative exponent, $\displaystyle a^{-n} = \frac{1}{a^n}$ .

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

b.

Steps

Algebraic

Expression

$\displaystyle 10^{-3}$

Use the definition of a negative exponent, $\displaystyle a^{-n} = \frac{1}{a^n}$ .

$\displaystyle \frac{1}{10^3}$

Simplify.

$\displaystyle \frac{1}{1000}$

c.

Steps

Algebraic

Expression

$\displaystyle \frac{1}{y^{-4}}$

Use the definition of a negative exponent, $\displaystyle \frac{1}{ a^{-n}} = a^n$ .

$\displaystyle y^4$

d.

Steps

Algebraic

Expression

$\displaystyle \frac{1}{3^{-2}}$

Use the definition of a negative exponent, $\displaystyle \frac{1}{ a^{-n}} = a^n$ .

$\displaystyle 3^2$

Simplify.

$\displaystyle 9$

Suppose now we have a fraction raised to a negative exponent. Let’s use our definition of negative exponents to lead us to a new property.

Steps

Algebraic

Expression

$\displaystyle \left(\frac{3}{4}\right)^{-2}$

Use the definition of a negative exponent, $\displaystyle a^{-n} = \frac{1}{a^n}$

$\displaystyle \frac{1}{\left(\frac{3}{4}\right)^2}$

Simplify the denominator.

$\displaystyle \frac{1}{\frac{9}{16}}$

Simplify the complex fraction.

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

But we know that $\displaystyle \frac{16}{9}$ is $\displaystyle \left(\frac{4}{3}\right)^2$

This tells us

$\displaystyle \left(\frac{3}{4}\right)^{-2}$ = $\displaystyle \left(\frac{4}{3}\right)^2$

To get from the original fraction raised to a negative exponent to the final result, we took the reciprocal of the base—the fraction—and changed the sign of the exponent.

This leads us to the Quotient to a Negative Power Property.

Quotient to a Negative Power Property

If a and b are real numbers, a ≠ 0, b ≠ 0 and n is an integer, then

$\displaystyle \left(\frac{a}{b}\right)^{-n}$ = $\displaystyle \left(\frac{b}{a}\right)^n$

Try it!

Simplify each expression: a. $\displaystyle \left(\frac{5}{7}\right)^{-2}$ b. $\displaystyle \left(-\frac{x}{y}\right)^{-3}$

Solution:

a.

Steps

Algebraic

Expression

$\displaystyle \left(\frac{5}{7}\right)^{-2}$

Use the Quotient to a Negative Exponent Property, $\displaystyle \left(\frac{a}{b}\right)^{-n}$ = $\displaystyle \left(\frac{b}{a}\right)^n$

Take the reciprocal of the fraction and change the sign of the exponent.

$\displaystyle \left(\frac{7}{5}\right)^{2}$

Simplify.

$\displaystyle \frac{49}{25}$

b.

Steps

Algebraic

Expression

$\displaystyle \left(-\frac{x}{y}\right)^{-3}$

Use the Quotient to a Negative Exponent Property, $\displaystyle \left(\frac{a}{b}\right)^{-n}$ = $\displaystyle \left(\frac{b}{a}\right)^n$

Take the reciprocal of the fraction and change the sign of the exponent.

$\displaystyle \left(-\frac{y}{x}\right)^{3}$

Simplify.

$\displaystyle -\frac{y^3}{x^3}$

Now that we have negative exponents, we will use the Product Property with expressions that have negative exponents.

Try it!

Simplify each expression: a. $\displaystyle z^{-5}\cdot z^{-3}$ b. $\displaystyle \left(m^4 n^{-3}\right) \left(m^{-5} n^{-2}\right)$ c. $\displaystyle \left(2x^{-6} y^{8}\right) \left(-5x^{5} y^{-3}\right)$ .

Solution:

a.

Steps

Algebraic

Expression

$\displaystyle z^{-5}\cdot z^{-3}$

Add the exponents, since the bases are the same.

$\displaystyle z^{-5-3}$

Simplify.

$\displaystyle z^{-8}$

Use the definition of a negative exponent.

$\displaystyle \frac{1}{z^{8}}$

b.

Steps

Algebraic

Expression

$\displaystyle \left(m^4 n^{-3}\right) \left(m^{-5} n^{-2}\right)$

Use the Commutative Property to get like bases together.

$\displaystyle m^4 m^{-5}\cdot n^{-2}n^{-3}$

Add the exponents for each base.

$\displaystyle m^{-1}\cdot n^{-5}$

Take reciprocals and change the signs of the exponents.

$\displaystyle \frac{1}{m^1}\cdot \frac{1}{n^{5}}$

Simplify.

$\displaystyle \frac{1}{mn^{5}}$

c.

Steps

Algebraic

Expression

$\displaystyle \left(2x^{-6} y^{8}\right) \left(-5x^{5} y^{-3}\right)$

Rewrite with the like bases together.

$\displaystyle 2(-5) \cdot \left(x^{-6} x^5\right)\cdot \left(y^8 y^{-3}\right)$

Multiply the coefficients and add the exponents of each variable.

$\displaystyle -10 \cdot x^{-1}\cdot y^5$

Use the definition of a negative exponent, $\displaystyle a^{-n}= \frac{1}{ a^n}$ .

$\displaystyle -10 \cdot \frac{1}{x} \cdot y^5$

Simplify.

$\displaystyle \frac{-10y^5}{x}$

Now let’s look at an exponential expression that contains a power raised to a power. See if you can discover a general property.

Steps

Algebraic

Expression

(x²)³

What does this mean?

x² · x² · x²

How many factors altogether?

The quantity x raised to the power of 2 raised to the power of 3 is written as x to the power of 2 times x to the power of 2 times x to the power of 2. Figure shows x times x with a bracket underneath saying two factors. The same is repeat two more times. The set of 3 figures is also bracketed underneath stating 6 factors.

So we have

x⁶

Notice the 6 is the product of the exponents, 2 and 3. We see that (x²)³is x^{2 · 3} or x⁶.

We multiplied the exponents. This leads to the Power Property for Exponents.

Power Property for Exponents

If a is a real number and m and n are integers, then

(a^m)ⁿ= a^{m · n}

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

Try it!

Simplify each expression: a. (y⁵)⁹ b. (y³)⁶(y⁵)⁴

Solution:

a.

Steps

Algebraic

Expression

Use the Power Property, (a^m)ⁿ= a^{m · n}

Simplify.

b.

Steps

Algebraic

Expression

(y³)⁶(y⁵)⁴

Use the Power Property.

y^18 _·y²⁰

Add the exponents.

y³⁸

We will now look at an expression containing a product that is raised to a power. Can you find the pattern?

Steps

Algebraic

Expression

(2x)³

What does this mean?

2x · 2x · 2x

We group the like factors together.

2 · 2 · 2 · x · x · x

How many factors of 2 and of x

2³ · x³

Notice that each factor was raised to the power and (2x)³is 2³ · x³.

The exponent applies to each of the factors! This leads to the Product to a Power Property for Exponents.

Product to a Power Property for Exponents

If a and b are real numbers and m is a whole number, then

(ab)^m = a^mb^m

To raise a product to a power, raise each factor to that power.

Try it!

Simplify each expression: a. (−3mn)³ b. (−4a²b)⁰ c. (6k³)⁻² d. (5x⁻³)².

Solution:

a.

Steps

Algebraic

Expression

Use Power of a Product Property, (ab)^m = a^mb^m.

Simplify.

b.

Steps

Algebraic

Expression

(−4a²b)⁰

Use Power of a Product Property, (ab)^m = a^m b^m.

(−4)⁰(a²)⁰(b)⁰

Simplify.

1 · 1 · 1

Multiply.

1

c.

Steps

Algebraic

Expression

(6k³)⁻²

Use Power of a Product Property, (ab)^m = a^m b^m.

(6)⁻²(k³)⁻²

Use the Power Property, (a^m)ⁿ = a^m·n.

6⁻²k⁻⁶

Use the Definition of a negative exponent, $\displaystyle a^{−n} = \frac{1}{a^n}$ .

$\displaystyle \frac{1}{6^2}\cdot \frac{1}{k^6}$

Simplify.

$\displaystyle \frac{1}{36k^6}$

d.

Steps

Algebraic

Expression

(5x⁻³)²

Use Power of a Product Property, (ab)^m = a^m b^m.

5²(x⁻³)²

Simplify.

25 · x⁻⁶

Rewrite x⁻⁶ using, $\displaystyle a^{−n} = \frac{1}{a^n}$ .

$\displaystyle 25\cdot \frac{1}{x^6}$

Simplify.

$\displaystyle \frac{25}{x^6}$

Now we will look at an example that will lead us to the Quotient to a Power Property.

Steps

Algebraic

Example

$\displaystyle\left( \frac{x}{y}\right)^3$

This means

$\displaystyle \frac{x}{y}\cdot \frac{x}{y} \cdot \frac{x}{y}$

Multiply the fractions.

$\displaystyle \frac{x \cdot x \cdot x}{y \cdot y \cdot y }$

Write with exponents.

$\displaystyle \frac{x^3}{y^3}$

Notice that the exponent applies to both the numerator and the denominator.

We see that $\displaystyle\left( \frac{x}{y}\right)^3$ is $\displaystyle \frac{x^3}{y^3}$ .

This leads to the Quotient to a Power Property for Exponents.

Quotient to a Power Property for Exponents

If a and b are real numbers, b ≠ 0, and m is an integer, then

$\displaystyle \left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$

To raise a fraction to a power, raise the numerator and denominator to that power.

Try it!

Simplify each expression: a. $\displaystyle \left(\frac{b}{3}\right)^{4}$ b. $\displaystyle \left(\frac{k}{j}\right)^{-3}$ c. $\displaystyle \left(\frac{2xy^2}{z}\right)^3$ d. $\displaystyle \left(\frac{4p^{-3}}{q^2}\right)^2$ .

Solution:

a.

Steps

Algebraic

Expression

Use Quotient to a Power Property, $\displaystyle \left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$ .

Simplify.

b.

Steps

Algebraic

Expression

Raise the numerator and denominator to the power.

Use the definition of negative exponent.

Multiply.

c.

Steps

Algebraic

Expression

$\displaystyle \left(\frac{2xy^2}{z}\right)^3$

Use Quotient to a Power Property, $\displaystyle \left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$ .

$\displaystyle \frac{\left(2xy^2\right)^3}{z^3}$

Use the Product to a Power Property, (ab)^m = a^m b^m.

$\displaystyle \frac{8x^3 y^6}{z^3}$

d.

Steps

Algebraic

Expression

$\displaystyle \left(\frac{4p^{-3}}{q^2}\right)^2$

Use Quotient to a Power Property, $\displaystyle \left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$ .

$\displaystyle \frac{\left(4p^{-3}\right)^2}{\left(q^2\right)^2}$

Use the Product to a Power Property, (ab)^m = a^m b^m.

$\displaystyle \frac{4^2\left(p^{-3}\right)^2}{\left(q^2\right)^2}$

Simplify using the Power Property, (a^m)ⁿ = a^m·n.

$\displaystyle \frac{16p^{-6}}{q^4}$

Use the definition of negative exponent.

$\displaystyle \frac{16}{q^4} \cdot \frac{1}{p^6}$

Simplify.

$\displaystyle \frac{16}{p^6 q^4}$

Summary of Exponent Properties

We now have several properties for exponents. Let’s summarize them and then we’ll do some more examples that use more than one of the properties.

Summary of Exponent Properties

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

Property

Description

Product Property

a^m · aⁿ= a^m+n

Power Property

(a^m)ⁿ = a^m.n

Product to a Power

(ab)^m = a^mb^m

Quotient Property

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

Zero Exponent Property

a⁰= 1, a ≠ 0

Quotient to a Power Property

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

Properties of Negative Exponents

$\displaystyle a^{-n} = \frac{1}{a^n}$ and $\displaystyle \frac{1}{a^{-n}}= a^n$

Quotient to a Negative Exponent

$\displaystyle \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n}$

Try it!

1. Simplify each expression by applying several properties:

a. $\displaystyle (3x^2y)^4(2xy^2)^3$ b. $\displaystyle \frac{(x^3)^4(x^{-2})^5}{(x^6)^5}$ c. $\displaystyle \left(\frac{2xy^2}{x^3y^{-2}}\right)^{2}\left(\frac{12xy^3}{x^3y^{-1}}\right)^{-1}$ .

Solution

a.

Steps

Algebraic

Expression

(3x²y)⁴(2xy²)³

Use the Product to a Power Property, (ab)^m= a^m b^m.

(3⁴x⁸y⁴)(2³x³y⁶)

Simplify.

(81x⁸y⁴)(8x³y⁶)

Use the Commutative Property.

81 · 8 · x⁸· x³· y⁴· y⁶

Multiply the constants and add the exponents.

648x¹¹y¹⁰

b.

Steps

Algebraic

Expression

$\displaystyle \frac{(x^3)^4(x^{-2})^5}{(x^6)^5}$

Use the Power Property, (a^m)ⁿ= a^m·n.

$\displaystyle \frac{(x^{12})(x^{-10})}{x^{30}}$

Add the exponents in the numerator.

$\displaystyle \frac{x^{2}}{x^{30}}$

Use the Quotient Property, $\displaystyle \frac{a^m}{a^n} = \frac{1}{a^{n-m}}$ .

$\displaystyle \frac{1}{x^{28}}$

c.

Steps

Algebraic

Expression

$\displaystyle \left(\frac{2xy^2}{x^3y^{-2}}\right)^{2}\left(\frac{12xy^3}{x^3y^{-1}}\right)^{-1}$

Simplify inside the parentheses first.

$\displaystyle \left(\frac{2y^4}{x^2}\right)^{2}\left(\frac{12y^4}{x^2}\right)^{-1}$

Use the Quotient to a Power Property, $\displaystyle \left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$ .

$\displaystyle \frac{(2y^4)^2(12y^4)^{-1}}{(x^2)^2 (x^2)^{-1}}$

Use the Product to a Power Property, (ab)^m= a^mb^m.

$\displaystyle \frac{ 4y^8}{x^4}\cdot \frac{12^{-1} y^{-4}}{x^{-2}}$

Simplify.

$\displaystyle \frac{ 4y^4}{12x^2}$

Simplify.

$\displaystyle \frac{ y^4}{3x^2}$

2. Simplify each expression: a. (c⁴d²)⁵(3cd⁵)⁴ b. $\displaystyle \frac{(a^{-2})^3(a^2)^4}{(a^4)^5}$ c. $\displaystyle \left(\frac{3xy^2}{x^2y^{-3}}\right)^{2}\left(\frac{9xy^{-3}}{x^3y^{2}}\right)^{-1}$ .

Solution

a. 81c²⁴d³⁰ b. $\displaystyle \frac{1}{a^{18}}$ c. $\displaystyle y^{15}$

Using Scientific Notation

Working with very large or very small numbers can be awkward. Since our number system is base ten we can use powers of ten to rewrite very large or very small numbers to make them easier to work with. Consider the numbers 4,000 and 0.004.

Using place value, we can rewrite the numbers 4,000 and 0.004. We know that 4,000 means 4×1,000 and 0.004 means $\displaystyle 4\times\frac{1}{1,000}$ .

If we write the 1,000 as a power of ten in exponential form, we can rewrite these numbers in this way:

Examples

4,000

$\displaystyle 4 \times 1,000$

$\displaystyle 4 \times 10^3$

0.004

$\displaystyle 4 \times \frac{1}{1,000}$

$\displastyle 4 \times \frac{1}{10^3}$

$\displaystyle 4 \times 10^{-3}$

When a number is written as a product of two numbers, where the first factor is a number greater than or equal to one but less than ten, and the second factor is a power of 10 written in exponential form, it is said to be in scientific notation.

Scientific Notation

A number is expressed in scientific notation when it is of the form

a × 10ⁿ where 1 ≤ a < 10 and n is an integer.

It is customary in scientific notation to use the × multiplication sign, even though we avoid using this sign elsewhere in algebra.

If we look at what happened to the decimal point, we can see a method to easily convert from decimal notation to scientific notation.

In both cases, the decimal was moved 3 places to get the first factor between 1 and 10.

The power of 10 is positive when the number is larger than 1: 4,000 = 4 × 10³

The power of 10 is negative when the number is between 0 and 1: 0.004 = 4 × 10⁻³

How to convert a decimal to scientific notation.

Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
Count the number of decimal places, n, that the decimal point was moved.
Write the number as a product with a power of 10. If the original number is.
- greater than 1, the power of 10 will be 10ⁿ.
- between 0 and 1, the power of 10 will be 10⁻ⁿ.
Check.

Try it!

Write in scientific notation: a. 37,000 b. 0.0052.

Solution

a.

Steps

Algebraic

The original number, 37,000, is greater than 1 so we will have a positive power of 10.

37,000

Move the decimal point to get 3.7, a number between 1 and 10.

Count the number of decimal places the point was moved.

4 places

Write as a product with a power of 10.

$3.7 \times 10^4$

Check:

$\begin{align*} &3.7\times 10^4 \\ & 3.7 \times 10,000\\ &37,000\\ \end{align*}$

$37,000 = 3.7 \times 10^4$

b.

Steps

Algebraic

The original number, 0.0052, is between 0 and 1 so we will have a negative power of 10.

0.0052

Move the decimal point to get 5.2, a number between 1 and 10.

Count the number of decimal places the point was moved.

3 places

Write as a product with a power of 10.

$5.2 \times 10^{-3}$

Check:

$\begin{align*} &5.2\times 10^{-3} \\ & 5.2 \times \frac{1}{10^3}\\ &5.2 \times\frac{1}{1,000}\\ &5.2 \times 0.001\\ &0.0052\\ \end{align*}$

$0.0052 = 5.2 \times 10^{-3}$

2. Write in scientific notation: a. 96,000 b. 0.0078.

Solution

a. 9.6 × 10⁴ b. 7.8 × 10⁻³

How can we convert from scientific notation to decimal form? Let’s look at two numbers written in scientific notation and see.

Ex. 1

Ex. 2

$9.12 \times 10^4$

$9.12 \times 10^{-4}$

$9.12 \times 10,000$

$9.12 \times 0.0001$

$91,200$

$0.000912$

If we look at the location of the decimal point, we can see an easy method to convert a number from scientific notation to decimal form.

In both cases the decimal point moved 4 places. When the exponent was positive, the decimal moved to the right. When the exponent was negative, the decimal point moved to the left.

How to convert scientific notation to decimal form.

Determine the exponent, n, on the factor 10.
Move the decimal n places, adding zeros if needed.
- If the exponent is positive, move the decimal point n places to the right.
- If the exponent is negative, move the decimal point |n| places to the left.
Check.

Try it!

Convert to decimal form: a. 6.2 × 10³b. –8.9 × 10⁻².

Solution

a.

Steps

Algebraic

Number

6.2 × 10³

Determine the exponent, n, on the factor 10.

The exponent is 3.

Since the exponent is positive, move the decimal point 3 places to the right.

Add zeros as needed for placeholders.

6,200

Solution

6.2 × 10³= 6,200

b.

Steps

Algebraic

Number

–8.9 × 10⁻²

Determine the exponent, n, on the factor 10.

The exponent is −2.

Since the exponent is negative, move the decimal point 2 places to the left.

Add zeros as needed for placeholders.

-0.089

Solution

–8.9 × 10⁻²= -0.089

When scientists perform calculations with very large or very small numbers, they use scientific notation. Scientific notation provides a way for the calculations to be done without writing a lot of zeros. We will see how the Properties of Exponents are used to multiply and divide numbers in scientific notation.

Try it!

Multiply or divide as indicated. Write answers in decimal form:

a. $\displaystyle(-4\times 10^5) (2\times 10^{-7})$ b. $\displaystyle\frac{9\times 10^3}{3\times 10^{-2}}$ .

Solution

a.

Steps

Algebraic

Example

$\displaystyle(-4\times 10^5) (2\times 10^{-7})$

Use the Commutative Property to rearrange the factors.

$\displaystyle -4\cdot 2\cdot 10^5\cdot 10^{-7}$

Multiply.

$\displaystyle -8\times 10^{-2}$

Change to decimal form by moving the decimal two places left.

−0.08

b.

Steps

Algebraic

Example

$\displaystyle\frac{9\times 10^3}{3\times 10^{-2}}$

Separate the factors, rewriting as the product of two fractions.

$\displaystyle\frac{9}{3}\times\frac{10^3}{10^{-2}}$

Divide.

$\displaystyle 3\times 10^5$

Change to decimal form by moving the decimal five places right.

300,000

Additional Online Resources

Key Concepts

Exponential Notation

This is read a to the m^th power.
In the expression a^m, the exponent m tells us how many times we use the base a as a factor.
Product Property for Exponents
If a is a real number and m and n are integers, then

a^m · aⁿ = a^{m + n}

To multiply with like bases, add the exponents.
Quotient Property for Exponents
If a is a real number, a ≠ 0, and m and n are integers, then

$\displaystyle \frac{a^m}{a^n} = a^{m-n} ,\ m > n$ and $\displaystyle \frac{a^m}{a^n} = \frac{1}{a^{n-m}},\ n>m$
Zero Exponent
- If a is a non-zero number, then a⁰ = 1.
- If a is a non-zero number, then a to the power of zero equals 1.
- Any non-zero number raised to the zero power is 1.
Negative Exponent
- If n is an integer and $a \ne 0$ , then $\displaystyle a^{-n} =\frac{1}{a^n}$ or $\displaystyle \frac{1}{a^{-n}} = a^n$ .
Quotient to a Negative Exponent Property
If a, b are real numbers, a ≠ 0, b ≠ 0 and n is an integer, then

$\displaystyle \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n}$
Power Property for Exponents
If a is a real number and m, n are integers, then

(a^m)ⁿ= a^{m · n}

To raise a power to a power, multiply the exponents.
Product to a Power Property for Exponents
If a and b are real numbers and m is a whole number, then

(ab)^m = a^m b^m

To raise a product to a power, raise each factor to that power.
Quotient to a Power Property for Exponents
If a and b are real numbers, b ≠ 0, and m is an integer, then

$\displaystyle\left (\frac{a}{b}\right)^m=\frac{a^m}{b^m}$

To raise a fraction to a power, raise the numerator and denominator to that power.

Summary of Exponent Properties
If a and b are real numbers, and m and n are integers, then

Property

Description

Product Property

a^m · aⁿ= a^m+n

Power Property

(a^m)ⁿ = a^m

Product to a Power

(ab)^m = a^m b^m

Quotient Property

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

Zero Exponent Property

a⁰= 1, a ≠ 0

Quotient to a Power Property

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

Properties of Negative Exponents

$\displaystyle a^{-n} = \frac{1}{a^n}$ and $\displaystyle \frac{1}{a^{-n}}= a^n$

Quotient to a Negative Exponent

$\displaystyle \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n}$

Scientific Notation
A number is expressed in scientific notation when it is of the form

$a\times 10^n$ where $1\le a < 10$ and n is an integer.
How to convert a decimal to scientific notation.
1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
2. Count the number of decimal places, n, that the decimal point was moved.
3. Write the number as a product with a power of 10. If the original number is.
  - greater than 1, the power of 10 will be 10ⁿ.
  - between 0 and 1, the power of 10 will be 10⁻ⁿ.
4. Check.
How to convert scientific notation to decimal form.
1. Determine the exponent, n, on the factor 10.
2. Move the decimal n places, adding zeros if needed.
  - If the exponent is positive, move the decimal point n places to the right.
  - If the exponent is negative, move the decimal point |n| places to the left.
3. Check.