Section 5.2Properties of Exponents and Scientific Notation

Simplify Expressions Using the Properties for Exponents

Remember that an exponent indicates repeated multiplication of the same quantity. For example, in the expression \({a}^{m},\) the exponent m tells us how many times we use the base a as a factor.

Let’s review the vocabulary for expressions with exponents.

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.

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.

Table 1

		\(\,\)
What does this mean?	\(\,\)
		\(\,\)

Notice that 5 is the sum of the exponents, 2 and 3. We see \({x}^{2}·{x}^{3}\) is \({x}^{2+3}\) or \({x}^{5}.\)

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

Example 1

Simplify each expression: ⓐ \({y}^{5}·{y}^{6}\) ⓑ \({2}^{x}·{2}^{3x}\) ⓒ \(2{a}^{7}·3a.\) ⓓ \({d}^{4}\cdot {d}^{5}\cdot {d}^{2}\)

Check that each pair shares the same base, then add the exponents.

Show solution

ⓐ

Table 2

Use the Product Property, \({a}^{m}·{a}^{n}={a}^{m+n}.\)
Simplify.

ⓑ

Table 3

Use the Product Property, \({a}^{m}·{a}^{n}={a}^{m+n}.\)
Simplify.

ⓒ

Table 4

Rewrite, \(a={a}^{1}.\)
Use the Commutative Property and use the Product Property, \({a}^{m}·{a}^{n}={a}^{m+n}.\)
Simplify.

ⓓ

Table 5

Add the exponents, since bases are the same.
Simplify.

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

Notice, in each case the bases were the same and we subtracted exponents. We see \(\frac{{x}^{5}}{{x}^{2}}\) is \({x}^{5-2}\) or \({x}^{3}\) . We see \(\frac{{x}^{2}}{{x}^{3}}\) is or \(\frac{1}{x}.\) 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. \(\frac{x}{x}=1\) . This leads to the Quotient Property for Exponents.

Example 2

Simplify each expression: ⓐ \(\frac{{x}^{9}}{{x}^{7}}\) ⓑ \(\frac{{3}^{10}}{{3}^{2}}\) ⓒ \(\frac{{b}^{8}}{{b}^{12}}\) ⓓ \(\frac{{7}^{3}}{{7}^{5}}.\)

Compare the exponents in the numerator and denominator to decide which form of the Quotient Property to use.

Show solution

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

ⓐ

Table 7

Since \(9>7,\) there are more factors of \(x\) in the numerator. \(\,\)
Use Quotient Property, \(\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}.\)
Simplify.

ⓑ

Table 8

Since \(10>2,\) there are more factors of \(3\) in the numerator. \(\,\)
Use Quotient Property, \(\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}.\)
Simplify.

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

ⓒ

Table 9

Since \(12>8,\) there are more factors of \(b\) in the denominator.
Use Quotient Property, \(\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}.\)
Simplify.

ⓓ

Table 10

Since \(5>3,\) there are more factors of \(3\) in the denominator. \(\,\)
Use Quotient Property, \(\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}.\)
Simplify.
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 \(\frac{{a}^{m}}{{a}^{m}}.\) We know \(,\frac{x}{x}=1,\) for any \(x\,(x\ne 0)\) since any number divided by itself is 1.

The Quotient Property for Exponents shows us how to simplify \(\frac{{a}^{m}}{{a}^{m}}.\) when \(m>n\) and when \(n<m\) by subtracting exponents. What if \(m=n?\) We will simplify \(\frac{{a}^{m}}{{a}^{m}}\) in two ways to lead us to the definition of the Zero Exponent Property. In general, for \(a\ne 0:\)

We see \(\frac{{a}^{m}}{{a}^{m}}\) simplifies to \({a}^{0}\) and to 1. So \({a}^{0}=1.\) Any non-zero base raised to the power of zero equals 1.

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

Use the Definition of a 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 \(\frac{{x}^{2}}{{x}^{5}}.\) We subtract the exponent in the denominator from the exponent in the numerator. We see \(\frac{{x}^{2}}{{x}^{5}}\) is \({x}^{2-5}\) or \({x}^{-3}.\)

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

This implies that \({x}^{-3}=\frac{1}{{x}^{3}}\) and it leads us to the definition of a negative exponent. If n is an integer and \(a\ne 0,\) then \({a}^{\text{-}n}=\frac{1}{{a}^{n}}.\)

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.

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

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 \(\frac{1}{{x}^{3}}.\) The answer is considered to be in simplest form when it has only positive exponents.

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.

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.

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

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

\(\begin{array}{llllll} & & & & & \,{({x}^{2})}^{3} \\ \text{What does this mean?} & & & & & \,{x}^{2}·{x}^{2}·{x}^{2}\end{array}\)

Table 22

How many factors altogether?
So we have

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

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

Example 7

Simplify each expression: ⓐ \({({y}^{5})}^{9}\) ⓑ \({({4}^{4})}^{7}\) ⓒ \({({y}^{3})}^{6}{({y}^{5})}^{4}.\)

Apply the Power Property — multiply the exponents — to each factor.

Show solution

ⓐ

Table 23

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

ⓑ

Table 24

Use the Power Property.
Simplify.

ⓒ

Table 25

	\(\,{({y}^{3})}^{6}{({y}^{5})}^{4}\)
Use the Power Property.	\(\,{y}^{18}·{y}^{20}\)
Add the exponents.	\(\,{y}^{38}\)

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

Notice that each factor was raised to the power and \({(2x)}^{3}\) is \({2}^{3}·{x}^{3}.\)

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

Example 8

Simplify each expression: ⓐ \({(-3mn)}^{3}\) ⓑ \({(-4{a}^{2}b)}^{0}\) ⓒ \({(6{k}^{3})}^{-2}\) ⓓ \({(5{x}^{-3})}^{2}.\)

Distribute the outer exponent to every factor inside the parentheses first.

Show solution

ⓐ

Table 27

Use Power of a Product Property, \({(ab)}^{m}={a}^{m}{b}^{m}.\)
Simplify.

ⓑ

Table 28

	\(\,{(-4{a}^{2}b)}^{0}\)
Use Power of a Product Property, \({(ab)}^{m}={a}^{m}{b}^{m}.\)	\(\,{(-4)}^{0}{({a}^{2})}^{0}{(b)}^{0}\)
Simplify.	\(\,1·1·1\)
Multiply.	\(\,1\)

ⓒ

Table 29

	\({(6{k}^{3})}^{-2}\)
Use the Product to a Power Property, \({(ab)}^{m}={a}^{m}{b}^{m}.\)	\({(6)}^{-2}{({k}^{3})}^{-2}\)
Use the Power Property, \({({a}^{m})}^{n}={a}^{m·n}.\)	\({6}^{-2}{k}^{-6}\)
Use the Definition of a negative exponent, \({a}^{\text{-}n}=\frac{1}{{a}^{n}}.\)	\(\frac{1}{{6}^{2}}·\frac{1}{{k}^{6}}\)
Simplify.	\(\frac{1}{36{k}^{6}}\)

ⓓ

Table 30

	\({(5{x}^{-3})}^{2}\)
Use the Product to a Power Property, \({(ab)}^{m}={a}^{m}{b}^{m}.\)	\({5}^{2}{({x}^{-3})}^{2}\)
Simplify.	\(25·{x}^{-6}\)
Rewrite \({x}^{-6}\) using, \({a}^{\text{-}n}=\frac{1}{{a}^{n}}.\)	\(25·\frac{1}{{x}^{6}}\)
Simplify.	\(\frac{25}{{x}^{6}}\)

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

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

We see that \({(\frac{x}{y})}^{3}\) is \(\frac{{x}^{3}}{{y}^{3}}.\)

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

Example 9

Simplify each expression:

ⓐ \({(\frac{b}{3})}^{4}\) ⓑ \({(\frac{k}{j})}^{-3}\) ⓒ \({(\frac{2x{y}^{2}}{z})}^{3}\) ⓓ \({(\frac{4{p}^{-3}}{{q}^{2}})}^{2}.\)

Raise both the numerator and denominator to the given power.

Show solution

ⓐ

Table 32

Use Quotient to a Power Property, \({(\frac{a}{b})}^{m}=\frac{{a}^{m}}{{b}^{m}}.\)
Simplify.

ⓑ

Table 33

Raise the numerator and denominator to the power.
Use the definition of negative exponent.
Multiply.

ⓒ

Table 34

	\({(\frac{2x{y}^{2}}{z})}^{3}\)
Use Quotient to a Power Property, \({(\frac{a}{b})}^{m}=\frac{{a}^{m}}{{b}^{m}}.\)	\(\frac{{(2x{y}^{2})}^{3}}{{z}^{3}}\)
Use the Product to a Power Property, \({(ab)}^{m}={a}^{m}{b}^{m}.\)	\(\frac{8{x}^{3}{y}^{6}}{{z}^{3}}\)

ⓓ

Table 35

	\({(\frac{4{p}^{-3}}{{q}^{2}})}^{2}\)
Use Quotient to a Power Property, \({(\frac{a}{b})}^{m}=\frac{{a}^{m}}{{b}^{m}}.\)	\(\frac{{(4{p}^{-3})}^{2}}{{({q}^{2})}^{2}}\)
Use the Product to a Power Property, \({(ab)}^{m}={a}^{m}{b}^{m}.\)	\(\frac{{4}^{2}{({p}^{-3})}^{2}}{{({q}^{2})}^{2}}\)
Simplify using the Power Property, \({({a}^{m})}^{n}={a}^{m·n}.\)	\(\frac{16{p}^{-6}}{{q}^{4}}\)
Use the definition of negative exponent.	\(\frac{16}{{q}^{4}}·\frac{1}{{p}^{6}}\)
Simplify.	\(\frac{16}{{p}^{6}{q}^{4}}\)

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.

Use 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 \(4\,\times \,1,000\) and 0.004 means \(4\,\times \,\frac{1}{1,000}.\)

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

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.

It is customary in scientific notation to use as the \(\,\times \,\) 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: \(4,000=4\,\times \,{10}^{3}\)

The power of 10 is negative when the number is between 0 and 1: \(0.004=4\,\times \,{10}^{-3}\)

Example 11

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

Move the decimal point until only one nonzero digit remains to its left, then count the places moved.

Show solution

ⓐ

Table 41

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.
Write as a product with a power of 10.
Check: \(\begin{array}{l} \\ 3.7\,\times \,{10}^{4} \\ 3.7\,\times \,10,000 \\ 37,000\end{array}\)

ⓑ

Table 42

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.
Write as a product with a power of 10.
\(\begin{array}{llll} \\ \\ \text{Check:} & & & 5.2\,\times \,{10}^{-3} \\ & & & 5.2\,\times \,\frac{1}{{10}^{3}} \\ & & & 5.2\,\times \,\frac{1}{1000} \\ & & & 5.2\,\times \,0.001 \\ & & & 0.0052\end{array}\)

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

\[\begin{array}{llllll}9.12\,\times \,{10}^{4} & & & & & 9.12\,\times \,{10}^{-4} \\ 9.12\,\times \,10,000 & & & & & 9.12\,\times \,0.0001 \\ 91,200 & & & & & 0.000912\end{array}\]

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.

Example 12

Convert to decimal form: ⓐ \(6.2\,\times \,{10}^{3}\) ⓑ \(-8.9\,\times \,{10}^{-2}.\)

Look at the sign of the exponent to decide which direction to move the decimal point.

Show solution

ⓐ

Table 43

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.

ⓑ

Table 44

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.

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.

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
  
  To multiply with like bases, add the exponents. (See Example 1.)

  \[{a}^{m}·{a}^{n}={a}^{m+n}\]

• Quotient Property for Exponents
  If \(a\) is a real number, \(a\ne 0,\) and m and n are integers, then
  

  \[\frac{{a}^{m}}{{a}^{n}}={a}^{m-n},\,\,\,m>n\,\text{and}\,\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}},\,n>m\]

  (See Example 2.)
• Zero Exponent
  • If a is a non-zero number, then \({a}^{0}=1.\)
  • 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. (See Example 3.)
• Negative Exponent
  • If n is an integer and \(a\ne 0,\) then \({a}^{\text{-}n}=\frac{1}{{a}^{n}}\) or \(\frac{1}{{a}^{\text{-}n}}={a}^{n}.\) (See Example 4 and Example 6.)
• Quotient to a Negative Exponent Property
  If \(a,b\) are real numbers, \(a\ne 0,b\ne 0\) and \(n\) is an integer, then
  

  \[{(\frac{a}{b})}^{\text{-}n}={(\frac{b}{a})}^{n}\]

  (See Example 5.)
• Power Property for Exponents
  If \(a\) is a real number and \(m,n\) are integers, then
  
  To raise a power to a power, multiply the exponents. (See Example 7.)

  \[{({a}^{m})}^{n}={a}^{m·n}\]

• Product to a Power Property for Exponents
  If a and b are real numbers and m is a whole number, then
  
  To raise a product to a power, raise each factor to that power. (See Example 8.)

  \[{(ab)}^{m}={a}^{m}{b}^{m}\]

• Quotient to a Power Property for Exponents
  If \(a\) and are real numbers, \(b\ne 0,\) and \(m\) is an integer, then
  
  To raise a fraction to a power, raise the numerator and denominator to that power. (See Example 9.)

  \[{(\frac{a}{b})}^{m}=\frac{{a}^{m}}{{b}^{m}}\]

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

  Table 47

  Property	Description
  Product Property	\({a}^{m}·{a}^{n}={a}^{m+n}\)
  Power Property	\({({a}^{m})}^{n}={a}^{m·n}\)
  Product to a Power	\({(ab)}^{n}={a}^{n}{b}^{n}\)
  Quotient Property	\(\frac{{a}^{m}}{{a}^{n}}={a}^{m-n},a\ne 0\)
  Zero Exponent Property	\({a}^{0}=1,a\ne 0\)
  Quotient to a Power Property:	\({(\frac{a}{b})}^{m}=\frac{{a}^{m}}{{b}^{m}},b\ne 0\)
  Properties of Negative Exponents	\({a}^{\text{-}n}=\frac{1}{{a}^{n}}\) and \(\frac{1}{{a}^{\text{-}n}}={a}^{n}\)
  Quotient to a Negative Exponent	\({(\frac{a}{b})}^{\text{-}n}={(\frac{b}{a})}^{n}\)

  (See Example 10.)
• Scientific Notation
  A number is expressed in scientific notation when it is of the form
  

  \[a\,\times \,{10}^{n}\,\text{where}\,1\le a<10\,\text{and}\,n\,\text{is an integer.}\]

• 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 \({10}^{n}.\)
    • between 0 and 1, the power of 10 will be \({10}^{\text{-}n}.\)
  • Check. (See Example 11.)
• 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. (See Example 12 and Example 13.)

Section Exercises

Practice Makes Perfect

Simplify Expressions Using the Properties for Exponents

In the following exercises, simplify each expression using the properties for exponents.

4

ⓐ \({d}^{3}·{d}^{6}\) ⓑ \({4}^{5x}·{4}^{9x}\) ⓒ \(2y·4{y}^{3}\) ⓓ \(w·{w}^{2}·{w}^{3}\)

Show answer

ⓐ \({d}^{9}\) ⓑ \({4}^{14x}\) ⓒ \(8{y}^{4}\) ⓓ \({w}^{6}\)

5

ⓐ \({x}^{4}·{x}^{2}\) ⓑ \({8}^{9x}·{8}^{3}\) ⓒ \(3{z}^{25}·5{z}^{8}\) ⓓ \(y·{y}^{3}·{y}^{5}\)

6

ⓐ \({n}^{19}·{n}^{12}\) ⓑ \({3}^{x}·{3}^{6}\) ⓒ \(7{w}^{5}·8w\) ⓓ \({a}^{4}·{a}^{3}·{a}^{9}\)

Show answer

ⓐ \({n}^{31}\) ⓑ \({3}^{x+6}\) ⓒ \(56{w}^{6}\)
ⓓ \({a}^{16}\)

7

ⓐ \({q}^{27}·{q}^{15}\) ⓑ \({5}^{x}·{5}^{4x}\) ⓒ \(9{u}^{41}·7{u}^{53}\)
ⓓ \({c}^{5}·{c}^{11}·{c}^{2}\)

8

\({m}^{x}·{m}^{3}\)

Show answer

\({m}^{x+3}\)

9

\({n}^{y}·{n}^{2}\)

10

\({y}^{a}·{y}^{b}\)

Show answer

\({y}^{a+b}\)

11

\({x}^{p}·{x}^{q}\)

12

ⓐ \(\frac{{x}^{18}}{{x}^{3}}\) ⓑ \(\frac{{5}^{12}}{{5}^{3}}\) ⓒ \(\frac{{q}^{18}}{{q}^{36}}\) ⓓ \(\frac{{10}^{2}}{{10}^{3}}\)

Show answer

ⓐ \({x}^{15}\) ⓑ \({5}^{9}\) ⓒ \(\frac{1}{{q}^{18}}\) ⓓ \(\frac{1}{10}\)

13

ⓐ \(\frac{{y}^{20}}{{y}^{10}}\) ⓑ \(\frac{{7}^{16}}{{7}^{2}}\) ⓒ \(\frac{{t}^{10}}{{t}^{40}}\) ⓓ \(\frac{{8}^{3}}{{8}^{5}}\)

14

ⓐ \(\frac{{p}^{21}}{{p}^{7}}\) ⓑ \(\frac{{4}^{16}}{{4}^{4}}\) ⓒ \(\frac{b}{{b}^{9}}\) ⓓ \(\frac{4}{{4}^{6}}\)

Show answer

ⓐ \({p}^{14}\) ⓑ \({4}^{12}\) ⓒ \(\frac{1}{{b}^{8}}\) ⓓ \(\frac{1}{{4}^{5}}\)

15

ⓐ \(\frac{{u}^{24}}{{u}^{3}}\) ⓑ \(\frac{{9}^{15}}{{9}^{5}}\) ⓒ \(\frac{x}{{x}^{7}}\) ⓓ \(\frac{10}{{10}^{3}}\)

16

ⓐ \({20}^{0}\) ⓑ \({b}^{0}\)

Show answer

ⓐ 1 ⓑ 1

17

ⓐ \({13}^{0}\) ⓑ \({k}^{0}\)

18

ⓐ \(\text{-}{27}^{0}\) ⓑ \(\text{-}({27}^{0})\)

Show answer

ⓐ \(-1\) ⓑ \(-1\)

19

ⓐ \(\text{-}{15}^{0}\) ⓑ \(\text{-}({15}^{0})\)

Use the Definition of a Negative Exponent

In the following exercises, simplify each expression.

20

ⓐ \({a}^{-2}\) ⓑ \({10}^{-3}\) ⓒ \(\frac{1}{{c}^{-5}}\) ⓓ \(\frac{1}{{3}^{-2}}\)

Show answer

ⓐ \(\frac{1}{{a}^{2}}\) ⓑ \(\frac{1}{1000}\) ⓒ \({c}^{5}\) ⓓ \(9\)

21

ⓐ \({b}^{-4}\) ⓑ \({10}^{-2}\) ⓒ \(\frac{1}{{b}^{-3}}\) ⓓ \(\frac{1}{{5}^{-2}}\)

22

ⓐ \({r}^{-3}\) ⓑ \({10}^{-5}\) ⓒ \(\frac{1}{{q}^{-10}}\) ⓓ \(\frac{1}{{10}^{-3}}\)

Show answer

ⓐ \(\frac{1}{{r}^{3}}\) ⓑ \(\frac{1}{100,000}\) ⓒ \({q}^{10}\)
ⓓ \(1,000\)

23

ⓐ \({s}^{-8}\) ⓑ \({10}^{-2}\) ⓒ \(\frac{1}{{t}^{-9}}\) ⓓ \(\frac{1}{{10}^{-4}}\)

24

ⓐ \({(\frac{5}{8})}^{-2}\) ⓑ \({(-\frac{b}{a})}^{-2}\)

Show answer

ⓐ \(\frac{64}{25}\) ⓑ \(\frac{{a}^{2}}{{b}^{2}}\)

25

ⓐ \({(\frac{3}{10})}^{-2}\) ⓑ \({(-\frac{2}{z})}^{-3}\)

26

ⓐ \({(\frac{4}{9})}^{-3}\) ⓑ \({(-\frac{u}{v})}^{-5}\)

Show answer

ⓐ \(\frac{729}{64}\) ⓑ \(-\frac{{v}^{5}}{{u}^{5}}\)

27

ⓐ \({(\frac{7}{2})}^{-3}\) ⓑ \({(-\frac{3}{x})}^{-3}\)

28

ⓐ \({(-5)}^{-2}\) ⓑ \(\text{-}{5}^{-2}\) ⓒ \({(-\frac{1}{5})}^{-2}\) ⓓ \(\text{-}{(\frac{1}{5})}^{-2}\)

Show answer

ⓐ \(\frac{1}{25}\) ⓑ \(-\frac{1}{25}\) ⓒ \(25\) ⓓ \(-25\)

29

ⓐ \(\text{-}{5}^{-3}\) ⓑ \({(-\frac{1}{5})}^{-3}\) ⓒ \(\text{-}{(\frac{1}{5})}^{-3}\) ⓓ \({(-5)}^{-3}\)

30

ⓐ \(3·{5}^{-1}\) ⓑ \({(3·5)}^{-1}\)

Show answer

ⓐ \(\frac{3}{5}\) ⓑ \(\frac{1}{15}\)

31

ⓐ \(3·{4}^{-2}\) ⓑ \({(3·4)}^{-2}\)

In the following exercises, simplify each expression using the Product Property.

32

ⓐ \({b}^{4}{b}^{-8}\) ⓑ \(({w}^{4}{x}^{-5})({w}^{-2}{x}^{-4})\) ⓒ \((-6{c}^{-3}{d}^{9})(2{c}^{4}{d}^{-5})\)

Show answer

ⓐ \(\frac{1}{{b}^{4}}\) ⓑ \(\frac{{w}^{2}}{{x}^{9}}\) ⓒ \(-12c{d}^{4}\)

33

ⓐ \({s}^{3}·{s}^{-7}\) ⓑ \(({m}^{3}{n}^{-3})({m}^{-5}{n}^{-1})\) ⓒ \((-2{j}^{-5}{k}^{8})(7{j}^{2}{k}^{-3})\)

34

ⓐ \({a}^{3}·{a}^{-3}\) ⓑ \((u{v}^{-2})({u}^{-5}{v}^{-3})\) ⓒ \((-4{r}^{-2}{s}^{-8})(9{r}^{4}{s}^{3})\)

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ⓐ 1 ⓑ \(\frac{1}{{u}^{4}{v}^{5}}\) ⓒ \(\frac{-36{r}^{2}}{{s}^{5}}\)

35

ⓐ \({y}^{5}·{y}^{-5}\) ⓑ \((p{q}^{-4})({p}^{-6}{q}^{-3})\) ⓒ \((-5{m}^{4}{n}^{6})(8{m}^{-5}{n}^{-3})\)

36

\({p}^{5}·{p}^{-2}·{p}^{-4}\)

Show answer

\(\frac{1}{p}\)

37

\({x}^{4}·{x}^{-2}·{x}^{-3}\)

In the following exercises, simplify each expression using the Power Property.

38

ⓐ \({({m}^{4})}^{2}\) ⓑ \({({10}^{3})}^{6}\) ⓒ \({({x}^{3})}^{-4}\)

Show answer

ⓐ \({m}^{8}\) ⓑ \({10}^{18}\) ⓒ \(\frac{1}{{x}^{12}}\)

39

ⓐ \({({b}^{2})}^{7}\) ⓑ \({({3}^{8})}^{2}\) ⓒ \({({k}^{2})}^{-5}\)

40

ⓐ \({({y}^{3})}^{x}\) ⓑ \({({5}^{x})}^{y}\) ⓒ \({({q}^{6})}^{-8}\)

Show answer

ⓐ \({y}^{3x}\) ⓑ \({5}^{xy}\) ⓒ \(\frac{1}{{q}^{48}}\)

41

ⓐ \({({x}^{2})}^{y}\) ⓑ \({({7}^{a})}^{b}\) ⓒ \({({a}^{9})}^{-10}\)

In the following exercises, simplify each expression using the Product to a Power Property.

42

ⓐ \({(-3xy)}^{2}\) ⓑ \({(6a)}^{0}\) ⓒ \({(5{x}^{2})}^{-2}\) ⓓ \({(-4{y}^{-3})}^{2}\)

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ⓐ \(9{x}^{2}{y}^{2}\) ⓑ 1 ⓒ \(\frac{1}{25{x}^{4}}\)
ⓓ \(\frac{16}{{y}^{6}}\)

43

ⓐ \({(-4ab)}^{2}\) ⓑ \({(5x)}^{0}\) ⓒ \({(4{y}^{3})}^{-3}\) ⓓ \({(-7{y}^{-3})}^{2}\)

44

ⓐ \({(-5ab)}^{3}\) ⓑ \({(-4pq)}^{0}\) ⓒ \({(-6{x}^{3})}^{-2}\) ⓓ \({(3{y}^{-4})}^{2}\)

Show answer

ⓐ \(-125{a}^{3}{b}^{3}\) ⓑ 1 ⓒ \(\frac{1}{36{x}^{6}}\) ⓓ \(\frac{9}{{y}^{8}}\)

45

ⓐ \({(-3xyz)}^{4}\) ⓑ \({(-7mn)}^{0}\) ⓒ \({(-3{x}^{3})}^{-2}\)
ⓓ \({(2{y}^{-5})}^{2}\)

In the following exercises, simplify each expression using the Quotient to a Power Property.

46

ⓐ \({(\frac{p}{2})}^{5}\) ⓑ \({(\frac{x}{y})}^{-6}\) ⓒ \({(\frac{2x{y}^{2}}{z})}^{3}\) ⓓ \({(\frac{4{p}^{-3}}{{q}^{2}})}^{2}\)

Show answer

ⓐ \(\frac{{p}^{5}}{32}\) ⓑ \(\frac{{y}^{6}}{{x}^{6}}\) ⓒ \(\frac{8{x}^{3}{y}^{6}}{{z}^{3}}\)
ⓓ \(\frac{16}{{p}^{6}{q}^{4}}\)

47

ⓐ \({(\frac{x}{3})}^{4}\) ⓑ \({(\frac{a}{b})}^{-5}\) ⓒ \({(\frac{2{x}^{2}{y}^{3}}{{z}^{2}})}^{2}\) ⓓ \({(\frac{{x}^{3}y}{{z}^{4}})}^{2}\)

48

ⓐ \({(\frac{a}{3b})}^{4}\) ⓑ \({(\frac{5}{4m})}^{-2}\) ⓒ \({(\frac{3{a}^{-2}{b}^{3}}{{c}^{2}})}^{-2}\) ⓓ \({(\frac{{p}^{-1}{q}^{4}}{{r}^{-4}})}^{2}\)

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ⓐ \(\frac{{a}^{4}}{81{b}^{4}}\) ⓑ \(\frac{16{m}^{2}}{25}\) ⓒ \(\frac{{a}^{4}{c}^{4}}{9{b}^{6}}\) ⓓ \(\frac{{q}^{8}{r}^{8}}{{p}^{2}}\)

49

ⓐ \({(\frac{x}{2y})}^{3}\) ⓑ \({(\frac{10}{3q})}^{-4}\) ⓒ \({(\frac{2{x}^{3}{y}^{4}}{3{z}^{2}})}^{5}\) ⓓ \({(\frac{5{a}^{3}{b}^{-1}}{2{c}^{4}})}^{-3}\)

In the following exercises, simplify each expression by applying several properties.

50

ⓐ \({(5{t}^{2})}^{3}{(3t)}^{2}\) ⓑ \(\frac{{({t}^{2})}^{5}{({t}^{-4})}^{2}}{{({t}^{3})}^{7}}\) ⓒ \({(\frac{2x{y}^{2}}{{x}^{3}{y}^{-2}})}^{2}{(\frac{12x{y}^{3}}{{x}^{3}{y}^{-1}})}^{-1}\)

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ⓐ \(1125{t}^{8}\) ⓑ \(\frac{1}{{t}^{19}}\) ⓒ \(\frac{{y}^{4}}{3{x}^{2}}\)

51

ⓐ \({(10{k}^{4})}^{3}{(5{k}^{6})}^{2}\) ⓑ \(\frac{{({q}^{3})}^{6}{({q}^{-2})}^{3}}{{({q}^{4})}^{8}}\)

52

ⓐ \({({m}^{2}n)}^{2}{(2m{n}^{5})}^{4}\) ⓑ \(\frac{{(-2{p}^{-2})}^{4}{(3{p}^{4})}^{2}}{{(-6{p}^{3})}^{2}}\)

Show answer

ⓐ \(16{m}^{8}{n}^{22}\) ⓑ \(\frac{4}{{p}^{6}}\)

53

ⓐ \({(3p{q}^{4})}^{2}{(6{p}^{6}q)}^{2}\) ⓑ \(\frac{{(-2{k}^{-3})}^{2}{(6{k}^{2})}^{4}}{{(9{k}^{4})}^{2}}\)

Mixed Practice

In the following exercises, simplify each expression.

54

ⓐ \(7{n}^{-1}\) ⓑ \({(7n)}^{-1}\) ⓒ \({(-7n)}^{-1}\)

Show answer

ⓐ \(\frac{7}{n}\) ⓑ \(\frac{1}{7n}\) ⓒ \(-\frac{1}{7n}\)

55

ⓐ \(6{r}^{-1}\) ⓑ \({(6r)}^{-1}\) ⓒ \({(-6r)}^{-1}\)

56

ⓐ \({(3p)}^{-2}\) ⓑ \(3{p}^{-2}\) ⓒ \(-3{p}^{-2}\)

Show answer

ⓐ \(\frac{1}{9{p}^{2}}\) ⓑ \(\frac{3}{{p}^{2}}\) ⓒ \(\frac{-3}{{p}^{2}}\)

57

ⓐ \({(2q)}^{-4}\) ⓑ \(2{q}^{-4}\) ⓒ \(-2{q}^{-4}\)

58

\({({x}^{2})}^{4}·{({x}^{3})}^{2}\)

Show answer

\({x}^{14}\)

59

\({({y}^{4})}^{3}·{({y}^{5})}^{2}\)

60

\({({a}^{2})}^{6}·{({a}^{3})}^{8}\)

Show answer

\({a}^{36}\)

61

\({({b}^{7})}^{5}·{({b}^{2})}^{6}\)

62

\({(2{m}^{6})}^{3}\)

Show answer

\(8{m}^{18}\)

63

\({(3{y}^{2})}^{4}\)

64

\({(10{x}^{2}y)}^{3}\)

Show answer

\(1,000{x}^{6}{y}^{3}\)

65

\({(2m{n}^{4})}^{5}\)

66

\({(-2{a}^{3}{b}^{2})}^{4}\)

Show answer

\(16{a}^{12}{b}^{8}\)

67

\({(-10{u}^{2}{v}^{4})}^{3}\)

68

\({(\frac{2}{3}{x}^{2}y)}^{3}\)

Show answer

\(\frac{8}{27}{x}^{6}{y}^{3}\)

69

\({(\frac{7}{9}p{q}^{4})}^{2}\)

70

\({(8{a}^{3})}^{2}{(2a)}^{4}\)

Show answer

\(1,024{a}^{10}\)

71

\({(5{r}^{2})}^{3}{(3r)}^{2}\)

72

\({(10{p}^{4})}^{3}{(5{p}^{6})}^{2}\)

Show answer

\(25,000{p}^{24}\)

73

\({(4{x}^{3})}^{3}{(2{x}^{5})}^{4}\)

74

\({(\frac{1}{2}{x}^{2}{y}^{3})}^{4}{(4{x}^{5}{y}^{3})}^{2}\)

Show answer

\({x}^{18}{y}^{18}\)

75

\({(\frac{1}{3}{m}^{3}{n}^{2})}^{4}{(9{m}^{8}{n}^{3})}^{2}\)

76

\({(3{m}^{2}n)}^{2}{(2m{n}^{5})}^{4}\)

Show answer

\(144{m}^{8}{n}^{22}\)

77

\({(2p{q}^{4})}^{3}{(5{p}^{6}q)}^{2}\)

78

ⓐ \({(3x)}^{2}(5x)\) ⓑ \({(2y)}^{3}(6y)\)

Show answer

ⓐ \(45{x}^{3}\) ⓑ \(48{y}^{4}\)

79

ⓐ \({(\frac{1}{2}{y}^{2})}^{3}{(\frac{2}{3}y)}^{2}\) ⓑ \({(\frac{1}{2}{j}^{2})}^{5}{(\frac{2}{5}{j}^{3})}^{2}\)

80

ⓐ \({(2{r}^{-2})}^{3}{({4}^{-1}r)}^{2}\) ⓑ \({(3{x}^{-3})}^{3}{({3}^{-1}{x}^{5})}^{4}\)

Show answer

ⓐ \(\frac{1}{2{r}^{4}}\) ⓑ \(\frac{1}{3}{x}^{11}\)

81

\({(\frac{{k}^{-2}{k}^{8}}{{k}^{3}})}^{2}\)

82

\({(\frac{{j}^{-2}{j}^{5}}{{j}^{4}})}^{3}\)

Show answer

\(\frac{1}{{j}^{3}}\)

83

\(\frac{{(-4{m}^{-3})}^{2}{(5{m}^{4})}^{3}}{{(-10{m}^{6})}^{3}}\)

84

\(\frac{{(-10{n}^{-2})}^{3}{(4{n}^{5})}^{2}}{{(2{n}^{8})}^{2}}\)

Show answer

\(-\frac{4000}{{n}^{12}}\)

Use Scientific Notation

In the following exercises, write each number in scientific notation.

85

ⓐ 57,000 ⓑ 0.026

86

ⓐ 340,000 ⓑ 0.041

Show answer

ⓐ \(3.4\,\times \,{10}^{5}\) ⓑ \(4.1\,\times \,{10}^{-2}\)

87

ⓐ 8,750,000 ⓑ 0.00000871

88

ⓐ 1,290,000 ⓑ 0.00000103

Show answer

ⓐ \(1.29\,\times \,{10}^{6}\)
ⓑ \(1.03\,\times \,{10}^{-6}\)

In the following exercises, convert each number to decimal form.

89

ⓐ \(5.2\,\times \,{10}^{2}\) ⓑ \(2.5\,\times \,{10}^{-2}\)

90

ⓐ \(-8.3\,\times \,{10}^{2}\) ⓑ \(3.8\,\times \,{10}^{-2}\)

Show answer

ⓐ \(-830\) ⓑ 0.038

91

ⓐ \(7.5\,\times \,{10}^{6}\) ⓑ \(-4.13\,\times \,{10}^{-5}\)

92

ⓐ \(1.6\,\times \,{10}^{10}\) ⓑ \(8.43\,\times \,{10}^{-6}\)

Show answer

ⓐ 16,000,000,000
ⓑ 0.00000843

In the following exercises, multiply or divide as indicated. Write your answer in decimal form.

93

ⓐ \((3\,\times \,{10}^{-5})(3\,\times \,{10}^{9})\) ⓑ \(\frac{7\,\times \,{10}^{-3}}{1\,\times \,{10}^{-7}}\)

94

ⓐ \((2\,\times \,{10}^{2})(1\,\times \,{10}^{-4})\) ⓑ \(\frac{5\,\times \,{10}^{-2}}{1\,\times \,{10}^{-10}}\)

Show answer

ⓐ 0.02 ⓑ 500,000,000

95

ⓐ \((7.1\,\times \,{10}^{-2})(2.4\,\times \,{10}^{-4})\) ⓑ \(\frac{6\,\times \,{10}^{4}}{3\,\times \,{10}^{-2}}\)

96

ⓐ \((3.5\,\times \,{10}^{-4})(1.6\,\times \,{10}^{-2})\) ⓑ \(\frac{8\,\times \,{10}^{6}}{4\,\times \,{10}^{-1}}\)

Show answer

ⓐ 0.0000056 ⓑ 20,000,000

Writing Exercises

97

Use the Product Property for Exponents to explain why \(x·x={x}^{2}.\)

98

Jennifer thinks the quotient \(\frac{{a}^{24}}{{a}^{6}}\) simplifies to \({a}^{4}.\) What is wrong with her reasoning?

Show answer

Answers will vary.

99

Explain why \(\text{-}{5}^{3}={(-5)}^{3}\) but \(\text{-}{5}^{4}\ne {(-5)}^{4}.\)

100

When you convert a number from decimal notation to scientific notation, how do you know if the exponent will be positive or negative?

Show answer

Answers will vary.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After reviewing this checklist, what will you do to become confident for all goals?

Glossary

Product Property

According to the Product Property, a to the m times a to the n equals a to the m plus n.

Power Property

According to the Power Property, a to the m to the n equals a to the m times n.

Product to a Power

According to the Product to a Power Property, a times b in parentheses to the m equals a to the m times b to the m.

Quotient Property

According to the Quotient Property, a to the m divided by a to the n equals a to the m minus n as long as a is not zero.

Zero Exponent Property

According to the Zero Exponent Property, a to the zero is 1 as long as a is not zero.

Quotient to a Power Property

According to the Quotient to a Power Property, a divided by b in parentheses to the power of m is equal to a to the m divided by b to the m as long as b is not zero.

Properties of Negative Exponents

According to the Properties of Negative Exponents, a to the negative n equals 1 divided by a to the n and 1 divided by a to the negative n equals a to the n.

Quotient to a Negative Exponent

Raising a quotient to a negative exponent occurs when a divided by b in parentheses to the power of negative n equals b divided by a in parentheses to the power of n.