# 6.7 Integer exponents and scientific notation  (Page 8/10)

 Page 8 / 10

## Section 6.2 Use Multiplication Properties of Exponents

Simplify Expressions with Exponents

In the following exercises, simplify.

${10}^{4}$

${17}^{1}$

17

${\left(\frac{2}{9}\right)}^{2}$

${\left(0.5\right)}^{3}$

0.125

${\left(-2\right)}^{6}$

$\text{−}{2}^{6}$

$-64$

Simplify Expressions Using the Product Property for Exponents

In the following exercises, simplify each expression.

${x}^{4}·{x}^{3}$

${p}^{15}·{p}^{16}$

${p}^{31}$

${4}^{10}·{4}^{6}$

$8·{8}^{5}$

${8}^{6}$

$n·{n}^{2}·{n}^{4}$

${y}^{c}·{y}^{3}$

${y}^{c+3}$

Simplify Expressions Using the Power Property for Exponents

In the following exercises, simplify each expression.

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

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

${5}^{6}$

${\left({y}^{4}\right)}^{x}$

${\left({3}^{r}\right)}^{s}$

${3}^{rs}$

Simplify Expressions Using the Product to a Power Property

In the following exercises, simplify each expression.

${\left(4a\right)}^{2}$

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

$-125{y}^{3}$

${\left(2mn\right)}^{5}$

${\left(10xyz\right)}^{3}$

$1000{x}^{3}{y}^{3}{z}^{3}$

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

${\left({p}^{2}\right)}^{5}·{\left({p}^{3}\right)}^{6}$

${\left(4{a}^{3}{b}^{2}\right)}^{3}$

$64{a}^{9}{b}^{6}$

${\left(5x\right)}^{2}\left(7x\right)$

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

$48{q}^{14}$

${\left(\frac{1}{3}{x}^{2}\right)}^{2}{\left(\frac{1}{2}x\right)}^{3}$

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

$\frac{8}{125}{m}^{6}{n}^{3}$

Multiply Monomials

In the following exercises 8, multiply the monomials.

$\left(-15{x}^{2}\right)\left(6{x}^{4}\right)$

$\left(-9{n}^{7}\right)\left(-16n\right)$

$144{n}^{8}$

$\left(7{p}^{5}{q}^{3}\right)\left(8p{q}^{9}\right)$

$\left(\frac{5}{9}a{b}^{2}\right)\left(27a{b}^{3}\right)$

$15{a}^{2}{b}^{5}$

## Section 6.3 Multiply Polynomials

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$7\left(a+9\right)$

$-4\left(y+13\right)$

$-4y-52$

$-5\left(r-2\right)$

$p\left(p+3\right)$

${p}^{2}+3p$

$\text{−}m\left(m+15\right)$

$-6u\left(2u+7\right)$

$-12{u}^{2}-42u$

$9\left({b}^{2}+6b+8\right)$

$3{q}^{2}\left({q}^{2}-7q+6\right)$ 3

$3{q}^{4}-21{q}^{3}+18{q}^{2}$

$\left(5z-1\right)z$

$\left(b-4\right)·11$

$11b-44$

Multiply a Binomial by a Binomial

In the following exercises, multiply the binomials using: the Distributive Property, the FOIL method, the Vertical Method.

$\left(x-4\right)\left(x+10\right)$

$\left(6y-7\right)\left(2y-5\right)$

$12{y}^{2}-44y+35$ $12{y}^{2}-44y+35$ $12{y}^{2}-44y+35$

In the following exercises, multiply the binomials. Use any method.

$\left(x+3\right)\left(x+9\right)$

$\left(y-4\right)\left(y-8\right)$

${y}^{2}-12y+32$

$\left(p-7\right)\left(p+4\right)$

$\left(q+16\right)\left(q-3\right)$

${q}^{2}+13q-48$

$\left(5m-8\right)\left(12m+1\right)$

$\left({u}^{2}+6\right)\left({u}^{2}-5\right)$

${u}^{4}+{u}^{2}-30$

$\left(9x-y\right)\left(6x-5\right)$

$\left(8mn+3\right)\left(2mn-1\right)$

$16{m}^{2}{n}^{2}-2mn-3$

Multiply a Trinomial by a Binomial

In the following exercises, multiply using the Distributive Property, the Vertical Method.

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

$\left(3x-4\right)\left(6{x}^{2}+x-10\right)$

$18{x}^{3}-21{x}^{2}-34x+40$ $18{x}^{3}-21{x}^{2}-34x+40$

In the following exercises, multiply. Use either method.

$\left(y-2\right)\left({y}^{2}-8y+9\right)$

$\left(7m+1\right)\left({m}^{2}-10m-3\right)$

$7{m}^{3}-69{m}^{2}-31m-3$

## Section 6.4 Special Products

Square a Binomial Using the Binomial Squares Pattern

In the following exercises, square each binomial using the Binomial Squares Pattern.

${\left(c+11\right)}^{2}$

${\left(q-15\right)}^{2}$

${q}^{2}-30q+225$

${\left(x+\frac{1}{3}\right)}^{2}$

${\left(8u+1\right)}^{2}$

$64{u}^{2}+16u+1$

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

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

$16{a}^{2}-24ab+9{b}^{2}$

Multiply Conjugates Using the Product of Conjugates Pattern

In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.

$\left(s-7\right)\left(s+7\right)$

$\left(y+\frac{2}{5}\right)\left(y-\frac{2}{5}\right)$

${y}^{2}-\frac{4}{25}$

$\left(12c+13\right)\left(12c-13\right)$

$\left(6-r\right)\left(6+r\right)$

$36-{r}^{2}$

$\left(u+\frac{3}{4}v\right)\left(u-\frac{3}{4}v\right)$

$\left(5{p}^{4}-4{q}^{3}\right)\left(5{p}^{4}+4{q}^{3}\right)$

$25{p}^{8}-16{q}^{6}$

Recognize and Use the Appropriate Special Product Pattern

In the following exercises, find each product.

${\left(3m+10\right)}^{2}$

$\left(6a+11\right)\left(6a-11\right)$

$36{a}^{2}-121$

$\left(5x+y\right)\left(x-5y\right)$

${\left({c}^{4}+9d\right)}^{2}$

${c}^{8}+18{c}^{4}d+81{d}^{2}$

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

$\left({a}^{2}+4b\right)\left(4a-{b}^{2}\right)$

$4{a}^{3}+3{a}^{2}b-4{b}^{3}$

## Section 6.5 Divide Monomials

Simplify Expressions Using the Quotient Property for Exponents

In the following exercises, simplify.

$\frac{{u}^{24}}{{u}^{6}}$

$\frac{{10}^{25}}{{10}^{5}}$

${10}^{20}$

$\frac{{3}^{4}}{{3}^{6}}$

$\frac{{v}^{12}}{{v}^{48}}$

$\frac{1}{{v}^{36}}$

$\frac{x}{{x}^{5}}$

$\frac{5}{{5}^{8}}$

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

Simplify Expressions with Zero Exponents

In the following exercises, simplify.

${75}^{0}$

${x}^{0}$

1

$\text{−}{12}^{0}$

$\left(\text{−}{12}^{0}\right)$ ${\left(-12\right)}^{0}$

1

$25{x}^{0}$

${\left(25x\right)}^{0}$

1

$19{n}^{0}-25{m}^{0}$

${\left(19n\right)}^{0}-{\left(25m\right)}^{0}$

0

Simplify Expressions Using the Quotient to a Power Property

In the following exercises, simplify.

${\left(\frac{2}{5}\right)}^{3}$

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

$\frac{{m}^{4}}{81}$

${\left(\frac{r}{s}\right)}^{8}$

${\left(\frac{x}{2y}\right)}^{6}$

$\frac{{x}^{6}}{64{y}^{6}}$

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

$\frac{{\left({x}^{3}\right)}^{5}}{{x}^{9}}$

$\frac{{n}^{10}}{{\left({n}^{5}\right)}^{2}}$

1

${\left(\frac{{q}^{6}}{{q}^{8}}\right)}^{3}$

${\left(\frac{{r}^{8}}{{r}^{3}}\right)}^{4}$

${r}^{20}$

${\left(\frac{{c}^{2}}{{d}^{5}}\right)}^{9}$

${\left(\frac{3{x}^{4}}{2{y}^{2}}\right)}^{5}$

$\frac{343{x}^{20}}{32{y}^{10}}$

${\left(\frac{{v}^{3}{v}^{9}}{{v}^{6}}\right)}^{4}$

$\frac{{\left(3{n}^{2}\right)}^{4}{\left(-5{n}^{4}\right)}^{3}}{{\left(-2{n}^{5}\right)}^{2}}$

$-\frac{10,125{n}^{10}}{4}$

Divide Monomials

In the following exercises, divide the monomials.

$-65{y}^{14}÷5{y}^{2}$

$\frac{64{a}^{5}{b}^{9}}{-16{a}^{10}{b}^{3}}$

$-\frac{4{b}^{6}}{{a}^{5}}$

$\frac{144{x}^{15}{y}^{8}{z}^{3}}{18{x}^{10}{y}^{2}{z}^{12}}$

$\frac{\left(8{p}^{6}{q}^{2}\right)\left(9{p}^{3}{q}^{5}\right)}{16{p}^{8}{q}^{7}}$

$\frac{9p}{2}$

## Section 6.6 Divide Polynomials

Divide a Polynomial by a Monomial

In the following exercises, divide each polynomial by the monomial.

$\frac{42{z}^{2}-18z}{6}$

$\left(35{x}^{2}-75x\right)÷5x$

$7x-15$

$\frac{81{n}^{4}+105{n}^{2}}{-3}$

$\frac{550{p}^{6}-300{p}^{4}}{10{p}^{3}}$

$55{p}^{3}-30p$

$\left(63x{y}^{3}+56{x}^{2}{y}^{4}\right)÷\left(7xy\right)$

$\frac{96{a}^{5}{b}^{2}-48{a}^{4}{b}^{3}-56{a}^{2}{b}^{4}}{8a{b}^{2}}$

$12{a}^{4}-6{a}^{3}b-7a{b}^{2}$

$\frac{57{m}^{2}-12m+1}{-3m}$

$\frac{105{y}^{5}+50{y}^{3}-5y}{5{y}^{3}}$

$21{y}^{2}+10-\frac{1}{{y}^{2}}$

Divide a Polynomial by a Binomial

In the following exercises, divide each polynomial by the binomial.

$\left({k}^{2}-2k-99\right)÷\left(k+9\right)$

$\left({v}^{2}-16v+64\right)÷\left(v-8\right)$

$v-8$

$\left(3{x}^{2}-8x-35\right)÷\left(x-5\right)$

$\left({n}^{2}-3n-14\right)÷\left(n+3\right)$

$n-6+\frac{4}{n+3}$

$\left(4{m}^{3}+m-5\right)÷\left(m-1\right)$

$\left({u}^{3}-8\right)÷\left(u-2\right)$

${u}^{2}+2u+4$

## Section 6.7 Integer Exponents and Scientific Notation

Use the Definition of a Negative Exponent

In the following exercises, simplify.

${9}^{-2}$

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

$-\frac{1}{125}$

$3·{4}^{-3}$

${\left(6u\right)}^{-3}$

$\frac{1}{216{u}^{3}}$

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

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

$\frac{16}{9}$

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

${p}^{-2}·{p}^{8}$

${q}^{-6}·{q}^{-5}$

$\frac{1}{{q}^{11}}$

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

${\left({y}^{8}\right)}^{-1}$

$\frac{1}{{y}^{8}}$

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

$\frac{{a}^{8}}{{a}^{12}}$

$\frac{1}{{a}^{4}}$

$\frac{{n}^{5}}{{n}^{-4}}$

$\frac{{r}^{-2}}{{r}^{-3}}$

$r$

Convert from Decimal Notation to Scientific Notation

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

8,500,000

0.00429

$4.29\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-3}$

The thickness of a dime is about 0.053 inches.

In 2015, the population of the world was about 7,200,000,000 people.

$7.2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{9}$

Convert Scientific Notation to Decimal Form

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

$3.8\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{5}$

$1.5\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{10}$

$15,000,000,000$

$9.1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-7}$

$5.5\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-1}$

$0.55$

Multiply and Divide Using Scientific Notation

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

$\left(2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{5}\right)\left(4\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-3}\right)$

$\left(3.5\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-2}\right)\left(6.2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-1}\right)$

$0.0217$

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

$\frac{8\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{5}}{4\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-1}}$

$\frac{9\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-5}}{3\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{2}}$

$0.0000003$

## Chapter practice test

For the polynomial $10{x}^{4}+9{y}^{2}-1$
Is it a monomial, binomial, or trinomial?
What is its degree?

In the following exercises, simplify each expression.

$\left(12{a}^{2}-7a+4\right)+\left(3{a}^{2}+8a-10\right)$

$15{a}^{2}+a-6$

$\left(9{p}^{2}-5p+1\right)-\left(2{p}^{2}-6\right)$

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

$-\frac{8}{125}$

$u·{u}^{4}$

${\left(4{a}^{3}{b}^{5}\right)}^{2}$

$16{a}^{6}{b}^{10}$

$\left(-9{r}^{4}{s}^{5}\right)\left(4r{s}^{7}\right)$

$3k\left({k}^{2}-7k+13\right)$

$3{k}^{3}-21{k}^{2}+39k$

$\left(m+6\right)\left(m+12\right)$

$\left(v-9\right)\left(9v-5\right)$

$9{v}^{2}-86v+45$

$\left(4c-11\right)\left(3c-8\right)$

$\left(n-6\right)\left({n}^{2}-5n+4\right)$

${n}^{3}-11{n}^{2}+34n-24$

$\left(2x-15y\right)\left(5x+7y\right)$

$\left(7p-5\right)\left(7p+5\right)$

$49{p}^{2}-25$

${\left(9v-2\right)}^{2}$

$\frac{{3}^{8}}{{3}^{10}}$

$\frac{1}{9}$

${\left(\frac{{m}^{4}·m}{{m}^{3}}\right)}^{6}$

${\left(87{x}^{15}{y}^{3}{z}^{22}\right)}^{0}$

$1$

$\frac{80{c}^{8}{d}^{2}}{16c{d}^{10}}$

$\frac{12{x}^{2}+42x-6}{2x}$

$6x+21-\frac{3}{x}$

$\left(70x{y}^{4}+95{x}^{3}y\right)÷5xy$

$\frac{64{x}^{3}-1}{4x-1}$

$16{x}^{2}+4x+1$

$\left({y}^{2}-5y-18\right)÷\left(y+3\right)$

${5}^{-2}$

$\frac{1}{25}$

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

${q}^{-4}·{q}^{-5}$

$\frac{1}{{q}^{9}}$

$\frac{{n}^{-2}}{{n}^{-10}}$

Convert 83,000,000 to scientific notation.

$8.3\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{7}$

Convert $6.91\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-5}$ to decimal form.

In the following exercises, simplify, and write your answer in decimal form.

$\left(3.4\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{9}\right)\left(2.2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-5}\right)$

74,800

$\frac{8.4\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-3}}{4\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3}}$

A helicopter flying at an altitude of 1000 feet drops a rescue package. The polynomial $-16{t}^{2}+1000$ gives the height of the package $t$ seconds a after it was dropped. Find the height when $t=6$ seconds.

424 feet

Stella bought a dinette set on sale for $725. The original price was$1,299. To the nearest tenth of a percent, what was the rate of discount?
44.19%
Scott
40.22%
Terence
44.2%
Orlando
p mulripied-5 and add 30 to it
Tausif
Tausif
Can you explain further
p mulripied-5 and add to 30
Tausif
How do you find divisible numbers without a calculator?
TAKE OFF THE LAST DIGIT AND MULTIPLY IT 9. SUBTRACT IT THE DIGITS YOU HAVE LEFT. IF THE ANSWER DIVIDES BY 13(OR IS ZERO), THEN YOUR ORIGINAL NUMBER WILL ALSO DIVIDE BY 13!IS DIVISIBLE BY 13
BAINAMA
When she graduates college, Linda will owe $43,000 in student loans. The interest rate on the federal loans is 4.5% and the rate on the private bank loans is 2%. The total interest she owes for one year was$1,585. What is the amount of each loan?
Sean took the bus from Seattle to Boise, a distance of 506 miles. If the trip took 7 2/3 hours, what was the speed of the bus?
66miles/hour
snigdha
How did you work it out?
Esther
s=mi/hr 2/3~0.67 s=506mi/7.67hr = ~66 mi/hr
Orlando
hello, I have algebra phobia. Subtracting negative numbers always seem to get me confused.
what do you need help in?
Felix
Heather
look at the numbers if they have different signs, it's like subtracting....but you keep the sign of the largest number...
Felix
for example.... -19 + 7.... different signs...subtract.... 12 keep the sign of the "largest" number 19 is bigger than 7.... 19 has the negative sign... Therefore, -12 is your answer...
Felix
—12
Thanks Felix.l also get confused with signs.
Esther
Thank you for this
Shatey
ty
Graham
think about it like you lost $19 (-19), then found$7(+7). Totally you lost just $12 (-12) Annushka I used to struggle a lot with negative numbers and math in general what I typically do is look at it in terms of money I have -$5 in my account I then take out 5 more dollars how much do I have in my account well-\$10 ... I also for a long time would draw it out on a number line to visualize it
Meg
practicing with smaller numbers to understand then working with larger numbers helps too and the song/rhyme same sign add and keep opposite signs subtract keep the sign of the bigger # then you'll be exact
Meg
Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles
168.50=R
Heather
john is 5years older than wanjiru.the sum of their years is27years.what is the age of each
46
mustee
j 17 w 11
Joseph
john is 16. wanjiru is 11.
Felix
27-5=22 22÷2=11 11+5=16
Joyce
I don't see where the answers are.
Ed
Cindy and Richard leave their dorm in Charleston at the same time. Cindy rides her bicycle north at a speed of 18 miles per hour. Richard rides his bicycle south at a speed of 14 miles per hour. How long will it take them to be 96 miles apart?
3
Christopher
18t+14t=96 32t=96 32/96 3
Christopher
show that a^n-b^2n is divisible by a-b
What does 3 times your weight right now
Use algebra to combine 39×5 and the half sum of travel of 59+30
Cherokee
What is the segment of 13? Explain
Cherokee
my weight is 49. So 3 times is 147
Cherokee
kg to lbs you goin to convert 2.2 or one if the same unit your going to time your body weight by 3. example if my body weight is 210lb. what would be my weight if I was 3 times as much in kg. that's you do 210 x3 = 630lb. then 630 x 2.2= .... hope this helps
tyler
How to convert grams to pounds?
paul
What is the lcm of 340
Yes
Cherokee
How many numbers each equal to y must be taken to make 15xy
15x
Martin
15x
Asamoah
15x
Hugo
1y
Tom
1y x 15y
Tom