# 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

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.
p-2/3=5/6 how do I solve it with explanation pls
P=3/2
Vanarith
1/2p2-2/3p=5p/6
James
Cindy
is y=7/5 a solution of 5y+3=10y-4
yes
James
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? Rhonda Reply Find an equation for the line that passes through the point P ( 0 , − 4 ) and has a slope 8/9 . Gabriel Reply 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?
Nga and Lauren bought a chest at a flea market for $50. They re-finished it and then added a 350 % mark - up Makaila Reply$1750
Cindy
the sum of two Numbers is 19 and their difference is 15
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?
integer greater than 2 and less than 12
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
the proudact of 3x^3-5×^2+3 and 2x^2+5x-4 in z7[x]/ is
?
Choli
a rock is thrown directly upward with an initial velocity of 96feet per second from a cliff 190 feet above a beach. The hight of tha rock above the beach after t second is given by the equation h=_16t^2+96t+190
Usman