# 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

#### Questions & Answers

Priam has pennies and dimes in a cup holder in his car. The total value of the coins is $4.21 . The number of dimes is three less than four times the number of pennies. How many pennies and how many dimes are in the cup? Cecilia Reply Arnold invested$64,000 some at 5.5% interest and the rest at 9% interest how much did he invest at each rate if he received $4500 in interest in one year Heidi Reply List five positive thoughts you can say to yourself that will help youapproachwordproblemswith a positive attitude. You may want to copy them on a sheet of paper and put it in the front of your notebook, where you can read them often. Elbert Reply Avery and Caden have saved$27,000 towards a down payment on a house. They want to keep some of the money in a bank account that pays 2.4% annual interest and the rest in a stock fund that pays 7.2% annual interest. How much should they put into each account so that they earn 6% interest per year?
324.00
Irene
1.2% of 27.000
Irene
i did 2.4%-7.2% i got 1.2%
Irene
I have 6% of 27000 = 1620 so we need to solve 2.4x +7.2y =1620
Catherine
I think Catherine is on the right track. Solve for x and y.
Scott
next bit : x=(1620-7.2y)/2.4 y=(1620-2.4x)/7.2 I think we can then put the expression on the right hand side of the "x=" into the second equation. 2.4x in the second equation can be rewritten as 2.4(rhs of first equation) I write this out tidy and get back to you...
Catherine
Darrin is hanging 200 feet of Christmas garland on the three sides of fencing that enclose his rectangular front yard. The length is five feet less than five times the width. Find the length and width of the fencing.
Mario invested $475 in$45 and $25 stock shares. The number of$25 shares was five less than three times the number of $45 shares. How many of each type of share did he buy? Jawad Reply let # of$25 shares be (x) and # of $45 shares be (y) we start with$25x + $45y=475, right? we are told the number of$25 shares is 3y-5) so plug in this for x. $25(3y-5)+$45y=$475 75y-125+45y=475 75y+45y=600 120y=600 y=5 so if #$25 shares is (3y-5) plug in y.
Joshua
will every polynomial have finite number of multiples?
a=# of 10's. b=# of 20's; a+b=54; 10a + 20b=$910; a=54 -b; 10(54-b) + 20b=$910; 540-10b+20b=$910; 540+10b=$910; 10b=910-540; 10b=370; b=37; so there are 37 20's and since a+b=54, a+37=54; a=54-37=17; a=17, so 17 10's. So lets check. $740+$170=$910. David Reply . A cashier has 54 bills, all of which are$10 or $20 bills. The total value of the money is$910. How many of each type of bill does the cashier have?
whats the coefficient of 17x
the solution says it 14 but how i thought it would be 17 im i right or wrong is the exercise wrong
Dwayne
17
Melissa
wow the exercise told me 17x solution is 14x lmao
Dwayne
thank you
Dwayne
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
Washing his dad’s car alone, eight-year-old Levi takes 2.5 hours. If his dad helps him, then it takes 1 hour. How long does it take the Levi’s dad to wash the car by himself?
Ethan and Leo start riding their bikes at the opposite ends of a 65-mile bike path. After Ethan has ridden 1.5 hours and Leo has ridden 2 hours, they meet on the path. Ethan’s speed is 6 miles per hour faster than Leo’s speed. Find the speed of the two bikers.
Nathan walked on an asphalt pathway for 12 miles. He walked the 12 miles back to his car on a gravel road through the forest. On the asphalt he walked 2 miles per hour faster than on the gravel. The walk on the gravel took one hour longer than the walk on the asphalt. How fast did he walk on the gravel?
Mckenzie
Nancy took a 3 hour drive. She went 50 miles before she got caught in a storm. Then she drove 68 miles at 9 mph less than she had driven when the weather was good. What was her speed driving in the storm?
Mr Hernaez runs his car at a regular speed of 50 kph and Mr Ranola at 36 kph. They started at the same place at 5:30 am and took opposite directions. At what time were they 129 km apart?
90 minutes