# 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 3. When Jenna spent 10 minutes on the elliptical trainer and then did circuit training for20 minutes, her fitness app says she burned 278 calories. When she spent 20 minutes onthe elliptical trainer and 30 minutes circuit training she burned 473 calories. How manycalories does she burn for each minute on the elliptical trainer? How many calories doesshe burn for each minute of circuit training? Edwin Reply John left his house in Irvine at 8:35 am to drive to a meeting in Los Angeles, 45 miles away. He arrived at the meeting at 9:50. At 3:30 pm, he left the meeting and drove home. He arrived home at 5:18. DaYoungan Reply p-2/3=5/6 how do I solve it with explanation pls Adedamola Reply P=3/2 Vanarith 1/2p2-2/3p=5p/6 James don't understand answer Cindy 4.5 Ruth is y=7/5 a solution of 5y+3=10y-4 Adedamola Reply yes James don't understand answer Cindy Lucinda has a pocketful of dimes and quarters with a value of$6.20. The number of dimes is 18 more than 3 times the number of quarters. How many dimes and how many quarters does Lucinda have?
Find an equation for the line that passes through the point P ( 0 , − 4 ) and has a slope 8/9 .
is that a negative 4 or positive 4?
Felix
y = mx + b
Felix
if negative -4, then -4=8/9(0) + b
Felix
-4=b
Felix
if positive 4, then 4=b
Felix
then plug in y=8/9x - 4 or y=8/9x+4
Felix
Macario is making 12 pounds of nut mixture with macadamia nuts and almonds. macadamia nuts cost $9 per pound and almonds cost$5.25 per pound. how many pounds of macadamia nuts and how many pounds of almonds should macario use for the mixture to cost $6.50 per pound to make? Cherry Reply Nga and Lauren bought a chest at a flea market for$50. They re-finished it and then added a 350 % mark - up
$1750 Cindy the sum of two Numbers is 19 and their difference is 15 Abdulai Reply 2, 17 Jose interesting saw 4,2 Cindy Felecia left her home to visit her daughter, driving 45mph. Her husband waited for the dog sitter to arrive and left home 20 minutes, or 13 hour later. He drove 55mph to catch up to Felecia. How long before he reaches her? Rafi Reply integer greater than 2 and less than 12 Emily Reply 2 < x < 12 Felix I'm guessing you are doing inequalities... Felix Actually, translating words into algebraic expressions / equations... Felix hi Darianna hello Mister Eric here Eric 6 Cindy He charges$125 per job. His monthly expenses are $1,600. How many jobs must he work in order to make a profit of at least$2,400?
at least 20
Ayla
what are the steps?
Alicia
6.4 jobs
Grahame
32
Grahame
1600+2400= total amount with expenses. 4000/125= number of jobs needed to make that min profit of 2400. answer is 32
Orlando
He must work 32 jobs to make a profit
POP
32
Cindy
what is algebra
repeated addition and subtraction of the order of operations. i love algebra I'm obsessed.
Shemiah
hi
Krekar
Eric here. I'm a parent. 53 years old. I have never taken algebra. I want to learn.
Eric
I am 63 and never learned algebra
Cindy
One-fourth of the candies in a bag of M&M’s are red. If there are 23 red candies, how many candies are in the bag?
they are 92 candies in the bag
POP
92
Cindy
rectangular field solutions
What is this?
Donna
t
muqtaar