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

Tickets for a show are $70 for adults and$50 for children. For one evening performance, a total of 300 tickets were sold and the receipts totaled $17,200. How many adult tickets and how many child tickets were sold? Mum Reply A 50% antifreeze solution is to be mixed with a 90% antifreeze solution to get 200 liters of a 80% solution. How many liters of the 50% solution and how many liters of the 90% solution will be used? Edi Reply June needs 45 gallons of punch for a party and has 2 different coolers to carry it in. The bigger cooler is 5 times as large as the smaller cooler. How many gallons can each cooler hold? Jesus Reply 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? Ronald Reply 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. Dojzae Reply LeBron needs 150 milliliters of a 30% solution of sulfuric acid for a lab experiment but only has access to a 25% and a 50% solution. How much of the 25% and how much of the 50% solution should he mix to make the 30% solution? Xona Reply 5% Michael hey everyone how to do algebra The Reply Felecia answer 1.5 hours before he reaches her Adriana Reply I would like to solve the problem -6/2x rachel Reply 12x Andrew how Christian Does the x represent a number or does it need to be graphed ? latonya -3/x Venugopal -3x is correct Atul Arnold invested$64,000, some at 5.5% interest and the rest at 9%. How much did he invest at each rate if he received $4,500 in interest in one year? Stephanie Reply Tickets for the community fair cost$12 for adults and $5 for children. On the first day of the fair, 312 tickets were sold for a total of$2204. How many adult tickets and how many child tickets were sold?
220
gayla
Three-fourths of the people at a concert are children. If there are 87 children, what is the total number of people at the concert?
Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was$1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college?
Tsimmuaj
Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was$1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college?
Tsimmuaj
? Is there anything wrong with this passage I found the total sum for 2 jobs, but found why elaborate on extra If I total one week from the store *4 would = the month than the total is = x than x can't calculate 10 month of a year
candido
what would be wong
candido
87 divided by 3 then multiply that by 4. 116 people total.
Melissa
the actual number that has 3 out of 4 of a whole pie
candido
was having a hard time finding
Teddy
use Matrices for the 2nd question
Daniel
One number is 11 less than the other number. If their sum is increased by 8, the result is 71. Find the numbers.
26 + 37 = 63 + 8 = 71
gayla
26+37=63+8=71
11+52=63+8=71
Thisha
how do we know the answer is correct?
Thisha
23 is 11 less than 37. 23+37=63. 63+8=71. that is what the question asked for.
gayla
23 +11 = 37. 23+37=63 63+8=71
Gayla
by following the question. one number is 11 less than the other number 26+11=37 so 26+37=63+8=71
Gayla
your answer did not fit the guidelines of the question 11 is 41 less than 52.
gayla
71-8-11 =52 is this correct?
Ruel
let the number is 'x' and the other number is "x-11". if their sum is increased means: x+(x-11)+8 result will be 71. so x+(x-11)+8=71 2x-11+8=71 2x-3=71 2x=71+3 2x=74 1/2(2x=74)1/2 x=37 final answer
tesfu
just new
Muwanga
Amara currently sells televisions for company A at a salary of $17,000 plus a$100 commission for each television she sells. Company B offers her a position with a salary of $29,000 plus a$20 commission for each television she sells. How televisions would Amara need to sell for the options to be equal?
yes math
Kenneth
company A 13 company b 5. A 17,000+13×100=29,100 B 29,000+5×20=29,100
gayla
need help with math to do tsi test
Toocute
me too
Christian
have you tried the TSI practice test ***tsipracticetest.com
gayla
DaMarcus and Fabian live 23 miles apart and play soccer at a park between their homes. DaMarcus rode his bike for 34 of an hour and Fabian rode his bike for 12 of an hour to get to the park. Fabian’s speed was 6 miles per hour faster than DaMarcus’s speed. Find the speed of both soccer players.
?
Ann
DaMarcus: 16 mi/hr Fabian: 22 mi/hr
Sherman