# 6.7 Integer exponents and scientific notation  (Page 6/10)

 Page 6 / 10

## Practice makes perfect

Use the Definition of a Negative Exponent

In the following exercises, simplify.

${4}^{-2}$
${10}^{-3}$

${3}^{-4}$
${10}^{-2}$

$\frac{1}{81}$ $\frac{1}{100}$

${5}^{-3}$
${10}^{-5}$

${2}^{-8}$
${10}^{-2}$

$\frac{1}{256}$ $\frac{1}{100}$

$\frac{1}{{c}^{-5}}$
$\frac{1}{{3}^{-2}}$

$\frac{1}{{c}^{-5}}$
$\frac{1}{{5}^{-2}}$

${c}^{5}$ 25

$\frac{1}{{q}^{-10}}$
$\frac{1}{{10}^{-3}}$

$\frac{1}{{t}^{-9}}$
$\frac{1}{{10}^{-4}}$

${t}^{9}$ 10000

${\left(\frac{5}{8}\right)}^{-2}$
${\left(-\frac{3m}{n}\right)}^{-2}$

${\left(\frac{3}{10}\right)}^{-2}$
${\left(-\frac{2}{cd}\right)}^{-3}$

$\frac{100}{9}$ $-\frac{{c}^{3}{d}^{3}}{8}$

${\left(\frac{4}{9}\right)}^{-3}$
${\left(-\frac{{u}^{2}}{2v}\right)}^{-5}$

${\left(\frac{7}{2}\right)}^{-3}$
${\left(-\frac{3}{x{y}^{2}}\right)}^{-3}$

$\frac{8}{343}$ $-\frac{{x}^{3}{y}^{6}}{27}$

${\left(-5\right)}^{-2}$
$\text{−}{5}^{-2}$
${\left(-\frac{1}{5}\right)}^{-2}$
$\text{−}{\left(\frac{1}{5}\right)}^{-2}$

${\left(-7\right)}^{-2}$
$-{7}^{-2}$
${\left(-\frac{1}{7}\right)}^{-2}$
$\text{−}{\left(\frac{1}{7}\right)}^{-2}$

$\frac{1}{49}$ $-\frac{1}{49}$ 49 $-49$

$\text{−}{3}^{-3}$
${\left(-\frac{1}{3}\right)}^{-3}$
$\text{−}{\left(\frac{1}{3}\right)}^{-3}$
${\left(-3\right)}^{-3}$

$\text{−}{5}^{-3}$
${\left(-\frac{1}{5}\right)}^{-3}$
$\text{−}{\left(\frac{1}{5}\right)}^{-3}$
${\left(-5\right)}^{-3}$

$-\frac{1}{125}$ $-125$ $-125$ $-\frac{1}{125}$

$3·{5}^{-1}$
${\left(3·5\right)}^{-1}$

$2·{5}^{-1}$
${\left(2·5\right)}^{-1}$

$\frac{2}{5}$ $\frac{1}{10}$

$4·{5}^{-2}$
${\left(4·5\right)}^{-2}$

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

$\frac{3}{16}$ $\frac{1}{144}$

${m}^{-4}$
${\left({x}^{3}\right)}^{-4}$

${b}^{-5}$
${\left({k}^{2}\right)}^{-5}$

$\frac{1}{{b}^{5}}$ $\frac{1}{{k}^{10}}$

${p}^{-10}$
${\left({q}^{6}\right)}^{-8}$

${s}^{-8}$
${\left({a}^{9}\right)}^{-10}$

$\frac{1}{{s}^{8}}$ $\frac{1}{{a}^{90}}$

$7{n}^{-1}$
${\left(7n\right)}^{-1}$
${\left(-7n\right)}^{-1}$

$6{r}^{-1}$
${\left(6r\right)}^{-1}$
${\left(-6r\right)}^{-1}$

$\frac{6}{r}$ $\frac{1}{6r}$ $-\frac{1}{6r}$

${\left(3p\right)}^{-2}$
$3{p}^{-2}$
$-3{p}^{-2}$

${\left(2q\right)}^{-4}$
$2{q}^{-4}$
$-2{q}^{-4}$

$\frac{1}{16{q}^{4}}$ $\frac{2}{{q}^{4}}$ $-\frac{2}{{q}^{4}}$

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

${b}^{4}{b}^{-8}$
${r}^{-2}{r}^{5}$
${x}^{-7}{x}^{-3}$

${s}^{3}·{s}^{-7}$
${q}^{-8}·{q}^{3}$
${y}^{-2}·{y}^{-5}$

$\frac{1}{{s}^{4}}$ $\frac{1}{{q}^{5}}$ $\frac{1}{{y}^{7}}$

${a}^{3}·{a}^{-3}$
$a·{a}^{3}$
$a·{a}^{-3}$

${y}^{5}·{y}^{-5}$
$y·{y}^{5}$
$y·{y}^{-5}$

1 ${y}^{6}$ $\frac{1}{{y}^{4}}$

${p}^{5}·{p}^{-2}·{p}^{-4}$

${x}^{4}·{x}^{-2}·{x}^{-3}$

$\frac{1}{x}$

$\left({w}^{4}{x}^{-5}\right)\left({w}^{-2}{x}^{-4}\right)$

$\left({m}^{3}{n}^{-3}\right)\left({m}^{-5}{n}^{-1}\right)$

$\frac{1}{{m}^{2}{n}^{4}}$

$\left(u{v}^{-2}\right)\left({u}^{-5}{v}^{-3}\right)$

$\left(p{q}^{-4}\right)\left({p}^{-6}{q}^{-3}\right)$

$\frac{1}{{p}^{5}{q}^{7}}$

$\left(-6{c}^{-3}{d}^{9}\right)\left(2{c}^{4}{d}^{-5}\right)$

$\left(-2{j}^{-5}{k}^{8}\right)\left(7{j}^{2}{k}^{-3}\right)$

$-\frac{14{k}^{5}}{{j}^{3}}$

$\left(-4{r}^{-2}{s}^{-8}\right)\left(9{r}^{4}{s}^{3}\right)$

$\left(-5{m}^{4}{n}^{6}\right)\left(8{m}^{-5}{n}^{-3}\right)$

$-\frac{40{n}^{3}}{m}$

${\left(5{x}^{2}\right)}^{-2}$

${\left(4{y}^{3}\right)}^{-3}$

$\frac{1}{64{y}^{9}}$

${\left(3{z}^{-3}\right)}^{2}$

${\left(2{p}^{-5}\right)}^{2}$

$\frac{4}{{p}^{10}}$

$\frac{{t}^{9}}{{t}^{-3}}$

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

${n}^{7}$

$\frac{{x}^{-7}}{{x}^{-3}}$

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

${y}^{5}$

Convert from Decimal Notation to Scientific Notation

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

57,000

340,000

$3.4\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{5}$

8,750,000

1,290,000

$1.29\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{6}$

0.026

0.041

$4.1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-2}$

0.00000871

0.00000103

$1.03\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-6}$

Convert Scientific Notation to Decimal Form

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

$5.2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{2}$

$8.3\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{2}$

830

$7.5\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{6}$

$1.6\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{10}$

16,000,000,000

$2.5\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-2}$

$3.8\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-2}$

0.038

$4.13\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-5}$

$1.93\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-5}$

0.0000193

Multiply and Divide Using Scientific Notation

$\left(3\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-5}\right)\left(3\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{9}\right)$

$\left(2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{2}\right)\left(1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-4}\right)$

0.02

$\left(7.1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-2}\right)\left(2.4\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-4}\right)$

$\left(3.5\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-4}\right)\left(1.6\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-2}\right)$

$5.6\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-6}$

$\frac{7\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-3}}{1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-7}}$

$\frac{5\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-2}}{1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-10}}$

500,000,000

$\frac{6\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{4}}{3\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-2}}$

$\frac{8\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{6}}{4\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-1}}$

20,000,000

## Everyday math

The population of the United States on July 4, 2010 was almost 310,000,000. Write the number in scientific notation.

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

$6.85\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{9}$ .

The average width of a human hair is 0.0018 centimeters. Write the number in scientific notation.

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

$5.7\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-10}$

In 2010, the number of Facebook users each day who changed their status to ‘engaged’ was $2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{4}$ . Convert this number to decimal form.

At the start of 2012, the US federal budget had a deficit of more than $\text{}1.5\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{13}$ . Convert this number to decimal form.

15,000,000,000,000

The concentration of carbon dioxide in the atmosphere is $3.9\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-4}$ . Convert this number to decimal form.

The width of a proton is $1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-5}$ of the width of an atom. Convert this number to decimal form.

0.00001

Health care costs The Centers for Medicare and Medicaid projects that consumers will spend more than $4 trillion on health care by 2017. 1. Write 4 trillion in decimal notation. 2. Write 4 trillion in scientific notation. #### Questions & Answers how did you get the value of 2000N.What calculations are needed to arrive at it Smarajit Reply Privacy Information Security Software Version 1.1a Good Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike? Got questions? Get instant answers now! Seera Reply how do u solve that question Seera Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike? Seera Speed=distance ÷ time Tremayne x-3y =1; 3x-2y+4=0 graph Juned Reply Brandon has a cup of quarters and dimes with a total of 5.55$. The number of quarters is five less than three times the number of dimes
app is wrong how can 350 be divisible by 3.
June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold?
Susanna if the first cooler holds five times the gallons then the other cooler. The big cooler holda 40 gallons and the 2nd will hold 8 gallons is that correct?
Georgie
@Susanna that person is correct if you divide 40 by 8 you can see it's 5 it's simple
Ashley
@Geogie my bad that was meant for u
Ashley
Hi everyone, I'm glad to be connected with you all. from France.
I'm getting "math processing error" on math problems. Anyone know why?
Can you all help me I don't get any of this
4^×=9
Did anyone else have trouble getting in quiz link for linear inequalities?
operation of trinomial
y=2×+9
Keshad gets paid $2,400 per month plus 6% of his sales. His brother earns$3,300 per month. For what amount of total sales will Keshad’s monthly pay be higher than his brother’s monthly pay?
Mayra has $124 in her checking account. She writes a check for$152. What is the New Balance in her checking account?
-28$ashley -$28
Stephanie