6.7 Integer Exponents and Scientific Notation

Learning objectives

Learning objectives

• Use the definition of a negative exponent

• Simplify expressions with integer exponents

• Convert from decimal notation to scientific notation

• Convert scientific notation to decimal form

• Multiply and divide using scientific notation

Use the Definition of a Negative Exponent

We saw that the Quotient Property for Exponents introduced earlier in this chapter, 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 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.

The negative exponent tells us we can re-write 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.

Example 6.89

Solution

In Example 6.89 we raised an integer to a negative exponent. What happens when we raise a fraction to a negative exponent? We’ll start by looking at what happens to a fraction whose numerator is one and whose denominator is an integer raised to a negative exponent.

This leads to the Property of Negative Exponents.

Example 6.90

Solution

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.

Table

6.4

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.

Example 6.91

Solution

When simplifying an expression with exponents, we must be careful to correctly identify the base.

Example 6.92

Solution

We must be careful to follow the Order of Operations. In the next example, parts (a) and (b) look similar, but the results are different.

Example 6.93

Solution

When a variable is raised to a negative exponent, we apply the definition the same way we did with numbers. We will assume all variables are non-zero.

Example 6.94

Solution

1. ⓐ $\begin{array}{cccl} & & & \,{x}^{-6} \\ \mathrm{Use the definition of a negative exponent,}\,{a}^{-n}=\frac{1}{{a}^{n}}. & & & \,\frac{1}{{x}^{6}} \\ \end{array}$

2. ⓑ $\begin{array}{cccl} & & & \,{({u}^{4})}^{-3} \\ \mathrm{Use the definition of a negative exponent,}\,{a}^{-n}=\frac{1}{{a}^{n}}. & & & \,\frac{1}{{({u}^{4})}^{3}} \\ \mathrm{Simplify.} & & & \,\frac{1}{{u}^{12}} \\ \end{array}$

When there is a product and an exponent we have to be careful to apply the exponent to the correct quantity. According to the Order of Operations, we simplify expressions in parentheses before applying exponents. We’ll see how this works in the next example.

Example 6.95

Solution

ⓐ

ⓑ

ⓒ

With negative exponents, the Quotient Rule needs only one form $\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}$, for $a\ne{}0$. When the exponent in the denominator is larger than the exponent in the numerator, the exponent of the quotient will be negative.

Simplify Expressions with Integer Exponents

All of the exponent properties we developed earlier in the chapter with whole number exponents apply to integer exponents, too. We restate them here for reference.

Example 6.96

Solution

ⓐ

ⓑ

ⓒ

In the next two examples, we’ll start by using the Commutative Property to group the same variables together. This makes it easier to identify the like bases before using the Product Property.

Example 6.97

Solution

In the next two examples, we’ll use the Power Property and the Product to a Power Property.

Example 6.98

Solution

Example 6.99

Solution

To simplify a fraction, we use the Quotient Property and subtract the exponents.

Example 6.100

Solution

Convert from Decimal Notation to Scientific Notation

Remember working with place value for whole numbers and decimals? Our number system is based on powers of 10. We use tens, hundreds, thousands, and so on. Our decimal numbers are also based on powers of tens—tenths, hundredths, thousandths, and so on. Consider 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 1000 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 10, 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.

$\begin{array}{lccl} \mathrm{The power of 10 is positive when the number is larger than 1:} & & & 4,000=4\,\times\,{10}^{3} \\ \mathrm{The power of 10 is negative when the number is between 0 and 1:} & & & 0.004=4\,\times\,{10}^{-3} \\ \end{array}$

Example 6.101

Example 6.102

Solution

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

Convert Scientific Notation to Decimal Form

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

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 6.103

The steps are summarized below.

Example 6.104

Solution

Multiply and Divide Using Scientific Notation

Astronomers use very large numbers to describe distances in the universe and ages of stars and planets. Chemists use very small numbers to describe the size of an atom or the charge on an electron. 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.

Example 6.105

Solution

$\begin{array}{cccl} & & & (4\,\times\,{10}^{5})(2\,\times\,{10}^{-7}) \\ \mathrm{Use the Commutative Property to rearrange the factors.} & & & 4\cdot{}2\cdot{10}^{5}\cdot{10}^{-7} \\ \mathrm{Multiply.} & & & 8\,\times\,{10}^{-2} \\ \mathrm{Change to decimal form by moving the decimal two places left.} & & & 0.08 \\ \end{array}$

Example 6.106

Solution

Section 6.7 Exercises

Section 6.7 Exercises

Use the Definition of a Negative Exponent

In the following exercises, simplify.

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

Convert from Decimal Notation to Scientific Notation

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

Convert Scientific Notation to Decimal Form

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

Multiply and Divide Using Scientific Notation

In the following exercises, multiply. Write your answer in decimal form.

In the following exercises, divide. Write your answer in decimal form.

Everyday Math

Writing Exercises

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

ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

1. Simplify: ⓐ ${2}^{-3}$ ⓑ ${10}^{-7}.$

2. Simplify: ⓐ ${3}^{-2}$ ⓑ ${10}^{-4}.$

3. Simplify: ⓐ $\frac{1}{{p}^{-8}}$ ⓑ $\frac{1}{{4}^{-3}}.$

4. Simplify: ⓐ $\frac{1}{{q}^{-7}}$ ⓑ $\frac{1}{{2}^{-4}}.$

5. Simplify: ⓐ ${(\frac{2}{3})}^{-4}$ ⓑ ${(-\frac{6m}{n})}^{-2}.$

1. Simplify: ⓐ ${(\frac{3}{5})}^{-3}$ ⓑ ${(-\frac{a}{2b})}^{-4}.$

2. Simplify: ⓐ ${(-5)}^{-2}$ ⓑ $-{5}^{-2}$ ⓒ ${(-\frac{1}{5})}^{-2}$ ⓓ $-{(\frac{1}{5})}^{-2}.$

3. Simplify: ⓐ ${(-7)}^{-2}$ ⓑ $-{7}^{-2}$, ⓒ ${(-\frac{1}{7})}^{-2}$ ⓓ $-{(\frac{1}{7})}^{-2}.$

4. Simplify: ⓐ $6\cdot{3}^{-1}$ ⓑ ${(6\cdot{}3)}^{-1}.$

5. Simplify: ⓐ $8\cdot{2}^{-2}$ ⓑ ${(8\cdot{}2)}^{-2}.$

1. Simplify: ⓐ ${y}^{-7}$ ⓑ ${({z}^{3})}^{-5}.$

2. Simplify: ⓐ ${p}^{-9}$ ⓑ ${({q}^{4})}^{-6}.$

3. Simplify: ⓐ $8{p}^{-1}$ ⓑ ${(8p)}^{-1}$ ⓒ ${(-8p)}^{-1}.$

4. Simplify: ⓐ ${11q}^{-1}$ ⓑ ${(11q)}^{-1}$$-{(11q)}^{-1}$ ⓒ $-\frac{1}{11q}$

5. Simplify: ⓐ ${x}^{-3}\cdot{x}^{7}$ ⓑ ${y}^{-7}\cdot{y}^{2}$ ⓒ ${z}^{-4}\cdot{z}^{-5}.$

1. Simplify: ⓐ ${a}^{-1}\cdot{a}^{6}$ ⓑ ${b}^{-8}\cdot{b}^{4}$ ⓒ ${c}^{-8}\cdot{c}^{-7}.$

2. Simplify: $({p}^{6}{q}^{-2})({p}^{-9}{q}^{-1}).$

3. Simplify: $({r}^{5}{s}^{-3})({r}^{-7}{s}^{-5}).$

4. Simplify: $(-6{c}^{-6}{d}^{4})(-5{c}^{-2}{d}^{-1}).$

5. Simplify: ${(-4{x}^{4})}^{-2}.$

1. Simplify: ${(2{b}^{3})}^{-4}.$

2. Simplify: ${(8{a}^{-4})}^{2}.$

3. Simplify: ${(2{c}^{-4})}^{3}.$

4. Simplify: $\frac{{x}^{8}}{{x}^{-3}}.$

5. Simplify: $\frac{{y}^{8}}{{y}^{-6}}.$

1. Write in scientific notation: $96,000.$

2. Write in scientific notation: $48,300.$

3. Write in scientific notation: $0.0078.$

4. Write in scientific notation: $0.0129.$

5. Convert to decimal form: $1.3\,\times\,{10}^{3}.$

1. Convert to decimal form: $9.25\,\times\,{10}^{4}.$

2. Divide $\frac{8\,\times\,{10}^{4}}{2\,\times\,{10}^{-1}}$. Write answers in decimal form.

3. Divide $\frac{8\,\times\,{10}^{2}}{4\,\times\,{10}^{-2}}$. Write answers in decimal form.

4. ⓐ ${3}^{-4}$ ⓑ ${10}^{-2}$

5. ⓐ ${2}^{-8}$ ⓑ ${10}^{-2}$

1. ⓐ $\frac{1}{{c}^{-5}}$ ⓑ $\frac{1}{{5}^{-2}}$

2. ⓐ $\frac{1}{{t}^{-9}}$ ⓑ $\frac{1}{{10}^{-4}}$

3. ⓐ ${(\frac{3}{10})}^{-2}$ ⓑ ${(-\frac{2}{cd})}^{-3}$

4. ⓐ ${(\frac{7}{2})}^{-3}$ ⓑ ${(-\frac{3}{x{y}^{2}})}^{-3}$

5. ⓐ ${(-7)}^{-2}$ ⓑ $-{7}^{-2}$ ⓒ ${(-\frac{1}{7})}^{-2}$ ⓓ $-{(\frac{1}{7})}^{-2}$

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

2. ⓐ $2\cdot{5}^{-1}$ ⓑ ${(2\cdot{}5)}^{-1}$

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

4. ⓐ ${b}^{-5}$ ⓑ ${({k}^{2})}^{-5}$

5. ⓐ ${s}^{-8}$ ⓑ ${({a}^{9})}^{-10}$

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

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

3. ⓐ ${s}^{3}\cdot{s}^{-7}$ ⓑ ${q}^{-8}\cdot{q}^{3}$ ⓒ ${y}^{-2}\cdot{y}^{-5}$

4. ⓐ ${y}^{5}\cdot{y}^{-5}$ ⓑ $y\cdot{y}^{5}$ ⓒ $y\cdot{y}^{-5}$

5. ${x}^{4}\cdot{x}^{-2}\cdot{x}^{-3}$

1. $({m}^{3}{n}^{-3})({m}^{-5}{n}^{-1})$

2. $(p{q}^{-4})({p}^{-6}{q}^{-3})$

3. $(-2{j}^{-5}{k}^{8})(7{j}^{2}{k}^{-3})$

4. $(-5{m}^{4}{n}^{6})(8{m}^{-5}{n}^{-3})$

5. ${(4{y}^{3})}^{-3}$

1. ${(2{p}^{-5})}^{2}$

2. $\frac{{n}^{5}}{{n}^{-2}}$

3. $\frac{{y}^{-5}}{{y}^{-10}}$

4. 340,000

5. 1,290,000

1. $1.6\,\times\,{10}^{10}$

2. $(3.5\,\times\,{10}^{-4})(1.6\,\times\,{10}^{-2})$

3. $\frac{5\,\times\,{10}^{-2}}{1\,\times\,{10}^{-10}}$

4. $\frac{8\,\times\,{10}^{6}}{4\,\times\,{10}^{-1}}$

5. The population of the world on July 4, 2010 was more than 6,850,000,000. Write the number in scientific notation

1. The probability of winning the 2010 Megamillions lottery was about 0.0000000057. Write the number in scientific notation.

2. At the start of 2012, the US federal budget had a deficit of more than $$1.5\,\times\,{10}^{13}$. Convert this number to decimal form.

3. Coin production In 1942, the U.S. Mint produced 154,500,000 nickels. Write 154,500,000 in scientific notation.

4. Debt At the end of fiscal year 2015 the gross United States federal government debt was estimated to be approximately $18,600,000,000,000 ($18.6 trillion), according to the Federal Budget. The population of the United States was approximately 300,000,000 people at the end of fiscal year 2015.

1. ⓐ Write the debt in scientific notation.

2. ⓑ Write the population in scientific notation.

3. ⓒ Find the amount of debt per person by using scientific notation to divide the debt by the population. Write the answer in scientific notation.

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