# 6.7 Integer exponents and scientific notation

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By the end of this section, you will be able to:
• 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

Before you get started, take this readiness quiz.

1. What is the place value of the $6$ in the number $64,891$ ?
If you missed this problem, review [link] .
2. Name the decimal: $0.0012.$
If you missed this problem, review [link] .
3. Subtract: $5-\left(-3\right).$
If you missed this problem, review [link] .

## 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.

## Quotient property for exponents

If $a$ is a real number, $a\ne 0$ , and $m\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}n$ are whole numbers, then

$\begin{array}{c}\frac{{a}^{m}}{{a}^{n}}={a}^{m-n},m>n\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}},n>m\hfill \end{array}$

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.

$\begin{array}{c}\hfill \frac{{x}^{2}}{{x}^{5}}\hfill \\ \hfill {x}^{2-5}\hfill \\ \hfill {x}^{-3}\hfill \end{array}$

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 .

## Negative exponent

If $n$ is an integer and $a\ne 0$ , then ${a}^{\text{−}n}=\frac{1}{{a}^{n}}$ .

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.

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

## Solution

1. $\begin{array}{cccc}& & & \phantom{\rule{4em}{0ex}}{4}^{-2}\hfill \\ \text{Use the definition of a negative exponent,}\phantom{\rule{0.2em}{0ex}}{a}^{\text{−}n}=\frac{1}{{a}^{n}}.\hfill & & & \phantom{\rule{4em}{0ex}}\frac{1}{{4}^{2}}\hfill \\ \text{Simplify.}\hfill & & & \phantom{\rule{4em}{0ex}}\frac{1}{16}\hfill \end{array}$

2. $\begin{array}{cccc}& & & \phantom{\rule{4em}{0ex}}{10}^{-3}\hfill \\ \text{Use the definition of a negative exponent,}\phantom{\rule{0.2em}{0ex}}{a}^{\text{−}n}=\frac{1}{{a}^{n}}.\hfill & & & \phantom{\rule{4em}{0ex}}\frac{1}{{10}^{3}}\hfill \\ \text{Simplify.}\hfill & & & \phantom{\rule{4em}{0ex}}\frac{1}{1000}\hfill \end{array}$

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

$\frac{1}{8}$ $\frac{1}{{10}^{7}}$

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

$\frac{1}{9}$ $\frac{1}{10,000}$

In [link] 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.

$\begin{array}{cccc}& & & \phantom{\rule{4em}{0ex}}\frac{1}{{a}^{\text{−}n}}\hfill \\ \text{Use the definition of a negative exponent,}\phantom{\rule{0.2em}{0ex}}{a}^{\text{−}n}=\frac{1}{{a}^{n}}.\hfill & & & \phantom{\rule{4em}{0ex}}\frac{1}{\frac{1}{{a}^{n}}}\hfill \\ \text{Simplify the complex fraction.}\hfill & & & \phantom{\rule{4em}{0ex}}1·\frac{{a}^{n}}{1}\hfill \\ \text{Multiply.}\hfill & & & \phantom{\rule{4em}{0ex}}{a}^{n}\hfill \end{array}$

This leads to the Property of Negative Exponents.

## Property of negative exponents

If $n$ is an integer and $a\ne 0$ , then $\frac{1}{{a}^{\text{−}n}}={a}^{n}$ .

Simplify: $\frac{1}{{y}^{-4}}$ $\frac{1}{{3}^{-2}}.$

## Solution

1. $\begin{array}{cccc}& & & \phantom{\rule{4em}{0ex}}\frac{1}{{y}^{-4}}\hfill \\ \text{Use the property of a negative exponent,}\phantom{\rule{0.2em}{0ex}}\frac{1}{{a}^{\text{−}n}}={a}^{n}.\hfill & & & \phantom{\rule{4em}{0ex}}{y}^{4}\hfill \end{array}$

2. $\begin{array}{cccc}& & & \phantom{\rule{4em}{0ex}}\frac{1}{{3}^{-2}}\hfill \\ \text{Use the property of a negative exponent,}\phantom{\rule{0.2em}{0ex}}\frac{1}{{a}^{\text{−}n}}={a}^{n}.\hfill & & & \phantom{\rule{4em}{0ex}}{3}^{2}\hfill \\ \text{Simplify.}\hfill & & & \phantom{\rule{4em}{0ex}}9\hfill \end{array}$

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

${p}^{8}$ $64$

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

${q}^{7}$ $16$

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.

$\begin{array}{cccc}& & & \phantom{\rule{4em}{0ex}}{\left(\frac{3}{4}\right)}^{-2}\hfill \\ \text{Use the definition of a negative exponent,}\phantom{\rule{0.2em}{0ex}}{a}^{\text{−}n}=\frac{1}{{a}^{n}}.\hfill & & & \phantom{\rule{4em}{0ex}}\frac{1}{{\left(\frac{3}{4}\right)}^{2}}\hfill \\ \text{Simplify the denominator.}\hfill & & & \phantom{\rule{4em}{0ex}}\frac{1}{\frac{9}{16}}\hfill \\ \text{Simplify the complex fraction.}\hfill & & & \phantom{\rule{4em}{0ex}}\frac{16}{9}\hfill \\ \text{But we know that}\phantom{\rule{0.2em}{0ex}}\frac{16}{9}\phantom{\rule{0.2em}{0ex}}\text{is}\phantom{\rule{0.2em}{0ex}}{\left(\frac{4}{3}\right)}^{2}.\hfill & & & \\ \text{This tells us that:}\hfill & & & \phantom{\rule{4em}{0ex}}{\left(\frac{3}{4}\right)}^{-2}={\left(\frac{4}{3}\right)}^{2}\hfill \end{array}$

Mario invested $475 in$45 and $25 stock shares. The number of$25 shares was five less than three times the number of $45 shares. How many of each type of share did he buy? Jawad Reply will every polynomial have finite number of multiples? cricket Reply a=# of 10's. b=# of 20's; a+b=54; 10a + 20b=$910; a=54 -b; 10(54-b) + 20b=$910; 540-10b+20b=$910; 540+10b=$910; 10b=910-540; 10b=370; b=37; so there are 37 20's and since a+b=54, a+37=54; a=54-37=17; a=17, so 17 10's. So lets check.$740+$170=$910.
. A cashier has 54 bills, all of which are $10 or$20 bills. The total value of the money is $910. How many of each type of bill does the cashier have? jojo Reply whats the coefficient of 17x Dwayne Reply the solution says it 14 but how i thought it would be 17 im i right or wrong is the exercise wrong Dwayne 17 Melissa wow the exercise told me 17x solution is 14x lmao Dwayne thank you Dwayne A private jet can fly 1,210 miles against a 25 mph headwind in the same amount of time it can fly 1,694 miles with a 25 mph tailwind. Find the speed of the jet Mikaela Reply Washing his dad’s car alone, eight-year-old Levi takes 2.5 hours. If his dad helps him, then it takes 1 hour. How long does it take the Levi’s dad to wash the car by himself? Sam Reply Ethan and Leo start riding their bikes at the opposite ends of a 65-mile bike path. After Ethan has ridden 1.5 hours and Leo has ridden 2 hours, they meet on the path. Ethan’s speed is 6 miles per hour faster than Leo’s speed. Find the speed of the two bikers. Mckenzie Reply Nathan walked on an asphalt pathway for 12 miles. He walked the 12 miles back to his car on a gravel road through the forest. On the asphalt he walked 2 miles per hour faster than on the gravel. The walk on the gravel took one hour longer than the walk on the asphalt. How fast did he walk on the gravel? Mckenzie Nancy took a 3 hour drive. She went 50 miles before she got caught in a storm. Then she drove 68 miles at 9 mph less than she had driven when the weather was good. What was her speed driving in the storm? Reiley Reply Mr Hernaez runs his car at a regular speed of 50 kph and Mr Ranola at 36 kph. They started at the same place at 5:30 am and took opposite directions. At what time were they 129 km apart? hamzzi Reply 90 minutes muhammad Melody wants to sell bags of mixed candy at her lemonade stand. She will mix chocolate pieces that cost$4.89 per bag with peanut butter pieces that cost $3.79 per bag to get a total of twenty-five bags of mixed candy. Melody wants the bags of mixed candy to cost her$4.23 a bag to make. How many bags of chocolate pieces and how many bags of peanut butter pieces should she use?
enrique borrowed $23,500 to buy a car he pays his uncle 2% interest on the$4,500 he borrowed from him and he pays the bank 11.5% interest on the rest. what average interest rate does he pay on the total $23,500 Nakiya Reply 13.5 Pervaiz Amber wants to put tiles on the backsplash of her kitchen counters. She will need 36 square feet of tiles. She will use basic tiles that cost$8 per square foot and decorator tiles that cost $20 per square foot. How many square feet of each tile should she use so that the overall cost of the backsplash will be$10 per square foot?
The equation P=28+2.54w models the relation between the amount of Randy’s monthly water bill payment, P, in dollars, and the number of units of water, w, used. Find the payment for a month when Randy used 15 units of water.
Bridget
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bruce
function f(x) to find each value
Marlene
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Marlene
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Melissa
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Melissa
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Marlene
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Marlene
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Marlene
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Melissa
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Melissa
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Marlene
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Melissa
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Marlene
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Melissa
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Marlene
-65r to the 4th power-50r cubed-15r squared+8r+23 ÷ 5r
Rich