# 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 -65r to the 4th power-50r cubed-15r squared+8r+23 ÷ 5r WENDY Reply write in this form a/b answer should be in the simplest form 5% August Reply convert to decimal 9/11 August Equation in the form of a pending point y+2=1/6(×-4) Jose Reply write in simplest form 3 4/2 August definition of quadratic formula Ahmed Reply From Google: The quadratic formula, , is used in algebra to solve quadratic equations (polynomial equations of the second degree). The general form of a quadratic equation is , where x represents a variable, and a, b, and c are constants, with . A quadratic equation has two solutions, called roots. Melissa what is the answer of w-2.6=7.55 What Reply 10.15 Michael w = 10.15 You add 2.6 to both sides and then solve for w (-2.6 zeros out on the left and leaves you with w= 7.55 + 2.6) Korin Nataly is considering two job offers. The first job would pay her$83,000 per year. The second would pay her $66,500 plus 15% of her total sales. What would her total sales need to be for her salary on the second offer be higher than the first? Mckenzie Reply x >$110,000
bruce
greater than $110,000 Michael Estelle is making 30 pounds of fruit salad from strawberries and blueberries. Strawberries cost$1.80 per pound, and blueberries cost $4.50 per pound. If Estelle wants the fruit salad to cost her$2.52 per pound, how many pounds of each berry should she use?
$1.38 worth of strawberries +$1.14 worth of blueberries which= $2.52 Leitha how Zaione is it right😊 Leitha lol maybe Robinson 8 pound of blueberries and 22 pounds of strawberries Melissa 8 pounds x 4.5 = 36 22 pounds x 1.80 = 39.60 36 + 39.60 = 75.60 75.60 / 30 = average 2.52 per pound Melissa 8 pounds x 4.5 equal 36 22 pounds x 1.80 equal 39.60 36 + 39.60 equal 75.60 75.60 / 30 equal average 2.52 per pound Melissa hmmmm...... ? Robinson 8 pounds x 4.5 = 36 22 pounds x 1.80 = 39.60 36 + 39.60 = 75.60 75.60 / 30 = average 2.52 per pound Melissa The question asks how many pounds of each in order for her to have an average cost of$2.52. She needs 30 lb in all so 30 pounds times $2.52 equals$75.60. that's how much money she is spending on the fruit. That means she would need 8 pounds of blueberries and 22 lbs of strawberries to equal 75.60
Melissa
good
Robinson
👍
Leitha
thanks Melissa.
Leitha
nawal let's do another😊
Leitha
we can't use emojis...I see now
Leitha
Sorry for the multi post. My phone glitches.
Melissa
Vina has $4.70 in quarters, dimes and nickels in her purse. She has eight more dimes than quarters and six more nickels than quarters. How many of each coin does she have? Mckenzie Reply 10 quarters 16 dimes 12 nickels Leitha 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. Crispy Reply wtf. is a tail wind or headwind? Robert 48 miles per hour with headwind and 68 miles per hour with tailwind Leitha average speed is 58 mph Leitha Into the wind (headwind), 125 mph; with wind (tailwind), 175 mph. Use time (t) = distance (d) ÷ rate (r). since t is equal both problems, then 1210/(x-25) = 1694/(×+25). solve for x gives x=150. bruce the jet will fly 9.68 hours to cover either distance bruce Riley is planning to plant a lawn in his yard. He will need 9 pounds of grass seed. He wants to mix Bermuda seed that costs$4.80 per pound with Fescue seed that costs $3.50 per pound. How much of each seed should he buy so that the overall cost will be$4.02 per pound?
33.336
Robinson
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? Imaan Reply Ivan has$8.75 in nickels and quarters in his desk drawer. The number of nickels is twice the number of quarters. How many coins of each type does he have?
2q=n ((2q).05) + ((q).25) = 8.75 .1q + .25q = 8.75 .35q = 8.75 q = 25 quarters 2(q) 2 (25) = 50 nickles Answer check 25 x .25 = 6.25 50 x .05 = 2.50 6.25 + 2.50 = 8.75
Melissa
John has $175 in$5 and $10 bills in his drawer. The number of$5 bills is three times the number of $10 bills. How many of each are in the drawer? mikayla Reply 7-$10 21-$5 Robert 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? (Round your answer to the nearest tenth of a percent.)
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hour longer than Tamara to go 80 miles, how fast can Samantha ride her bike?
8mph
michele
16mph
Robert
3.8 mph
Ped
16 goes into 80 5times while 20 goes into 80 4times and is 4mph faster
Robert