<< Chapter < Page | Chapter >> Page > |
Simplify Expressions with Exponents
In the following exercises, simplify.
${10}^{4}$
${\left(\frac{2}{9}\right)}^{2}$
${\left(\mathrm{-2}\right)}^{6}$
Simplify Expressions Using the Product Property for Exponents
In the following exercises, simplify each expression.
${x}^{4}\xb7{x}^{3}$
${4}^{10}\xb7{4}^{6}$
$n\xb7{n}^{2}\xb7{n}^{4}$
Simplify Expressions Using the Power Property for Exponents
In the following exercises, simplify each expression.
${({m}^{3})}^{5}$
${\left({y}^{4}\right)}^{x}$
Simplify Expressions Using the Product to a Power Property
In the following exercises, simplify each expression.
${(4a)}^{2}$
${(2mn)}^{5}$
Simplify Expressions by Applying Several Properties
In the following exercises, simplify each expression.
${\left({p}^{2}\right)}^{5}\xb7{\left({p}^{3}\right)}^{6}$
${(5x)}^{2}(7x)$
${\left(\frac{1}{3}{x}^{2}\right)}^{2}{\left(\frac{1}{2}x\right)}^{3}$
${\left(\frac{2}{5}{m}^{2}n\right)}^{3}$
$\frac{8}{125}{m}^{6}{n}^{3}$
Multiply Monomials
In the following exercises 8, multiply the monomials.
$\left(\mathrm{-15}{x}^{2}\right)\left(6{x}^{4}\right)$
$(7{p}^{5}{q}^{3})(8p{q}^{9})$
$\left(\frac{5}{9}a{b}^{2}\right)\left(27a{b}^{3}\right)$
$15{a}^{2}{b}^{5}$
Multiply a Polynomial by a Monomial
In the following exercises, multiply.
$\mathrm{-5}(r-2)$
$\text{\u2212}m(m+15)$
$9({b}^{2}+6b+8)$
$(5z-1)z$
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.
$(x+3)(x+9)$
$(p-7)(p+4)$
$(5m-8)(12m+1)$
$\left(9x-y\right)\left(6x-5\right)$
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.
$(y-2)({y}^{2}-8y+9)$
Square a Binomial Using the Binomial Squares Pattern
In the following exercises, square each binomial using the Binomial Squares Pattern.
${(c+11)}^{2}$
${\left(x+\frac{1}{3}\right)}^{2}$
${(3{n}^{3}-2)}^{2}$
Multiply Conjugates Using the Product of Conjugates Pattern
In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.
$(s-7)(s+7)$
$\left(y+\frac{2}{5}\right)\left(y-\frac{2}{5}\right)$
${y}^{2}-\frac{4}{25}$
$(12c+13)(12c-13)$
$\left(u+\frac{3}{4}v\right)\left(u-\frac{3}{4}v\right)$
$(5{p}^{4}-4{q}^{3})(5{p}^{4}+4{q}^{3})$
$25{p}^{8}-16{q}^{6}$
Recognize and Use the Appropriate Special Product Pattern
In the following exercises, find each product.
${(3m+10)}^{2}$
$(5x+y)(x-5y)$
$({p}^{5}+{q}^{5})({p}^{5}-{q}^{5})$
Simplify Expressions Using the Quotient Property for Exponents
In the following exercises, simplify.
$\frac{{u}^{24}}{{u}^{6}}$
$\frac{{3}^{4}}{{3}^{6}}$
$\frac{x}{{x}^{5}}$
Simplify Expressions with Zero Exponents
In the following exercises, simplify.
${75}^{0}$
$\text{\u2212}{12}^{0}$
$\left(\text{\u2212}{12}^{0}\right)$ ${\left(\mathrm{-12}\right)}^{0}$
1
$25{x}^{0}$
$19{n}^{0}-25{m}^{0}$
Simplify Expressions Using the Quotient to a Power Property
In the following exercises, simplify.
${\left(\frac{2}{5}\right)}^{3}$
${\left(\frac{r}{s}\right)}^{8}$
${\left(\frac{x}{2y}\right)}^{6}$
$\frac{{x}^{6}}{64{y}^{6}}$
Simplify Expressions by Applying Several Properties
In the following exercises, simplify.
$\frac{{\left({x}^{3}\right)}^{5}}{{x}^{9}}$
${\left(\frac{{q}^{6}}{{q}^{8}}\right)}^{3}$
${\left(\frac{{c}^{2}}{{d}^{5}}\right)}^{9}$
${\left(\frac{3{x}^{4}}{2{y}^{2}}\right)}^{5}$
$\frac{343{x}^{20}}{32{y}^{10}}$
${\left(\frac{{v}^{3}{v}^{9}}{{v}^{6}}\right)}^{4}$
$\frac{{\left(3{n}^{2}\right)}^{4}{\left(\mathrm{-5}{n}^{4}\right)}^{3}}{{\left(\mathrm{-2}{n}^{5}\right)}^{2}}$
$-\frac{\mathrm{10,125}{n}^{10}}{4}$
Divide Monomials
In the following exercises, divide the monomials.
$\mathrm{-65}{y}^{14}\xf75{y}^{2}$
$\frac{64{a}^{5}{b}^{9}}{\mathrm{-16}{a}^{10}{b}^{3}}$
$-\frac{4{b}^{6}}{{a}^{5}}$
$\frac{144{x}^{15}{y}^{8}{z}^{3}}{18{x}^{10}{y}^{2}{z}^{12}}$
$\frac{\left(8{p}^{6}{q}^{2}\right)\left(9{p}^{3}{q}^{5}\right)}{16{p}^{8}{q}^{7}}$
$\frac{9p}{2}$
Divide a Polynomial by a Monomial
In the following exercises, divide each polynomial by the monomial.
$\frac{42{z}^{2}-18z}{6}$
$\frac{81{n}^{4}+105{n}^{2}}{\mathrm{-3}}$
$\left(63x{y}^{3}+56{x}^{2}{y}^{4}\right)\xf7\left(7xy\right)$
$\frac{96{a}^{5}{b}^{2}-48{a}^{4}{b}^{3}-56{a}^{2}{b}^{4}}{8a{b}^{2}}$
$12{a}^{4}-6{a}^{3}b-7a{b}^{2}$
$\frac{57{m}^{2}-12m+1}{\mathrm{-3}m}$
$\frac{105{y}^{5}+50{y}^{3}-5y}{5{y}^{3}}$
$21{y}^{2}+10-\frac{1}{{y}^{2}}$
Divide a Polynomial by a Binomial
In the following exercises, divide each polynomial by the binomial.
$\left({k}^{2}-2k-99\right)\xf7\left(k+9\right)$
$\left(3{x}^{2}-8x-35\right)\xf7\left(x-5\right)$
$\left({n}^{2}-3n-14\right)\xf7\left(n+3\right)$
$n-6+\frac{4}{n+3}$
$\left(4{m}^{3}+m-5\right)\xf7\left(m-1\right)$
Use the Definition of a Negative Exponent
In the following exercises, simplify.
${9}^{\mathrm{-2}}$
$3\xb7{4}^{\mathrm{-3}}$
${\left(\frac{2}{5}\right)}^{\mathrm{-1}}$
Simplify Expressions with Integer Exponents
In the following exercises, simplify.
${p}^{\mathrm{-2}}\xb7{p}^{8}$
${q}^{\mathrm{-6}}\xb7{q}^{\mathrm{-5}}$
$\frac{1}{{q}^{11}}$
$\left({c}^{\mathrm{-2}}d\right)\left({c}^{\mathrm{-3}}{d}^{\mathrm{-2}}\right)$
${\left({q}^{\mathrm{-4}}\right)}^{\mathrm{-3}}$
$\frac{{n}^{5}}{{n}^{\mathrm{-4}}}$
Convert from Decimal Notation to Scientific Notation
In the following exercises, write each number in scientific notation.
8,500,000
0.00429
$4.29\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-3}}$
The thickness of a dime is about 0.053 inches.
In 2015, the population of the world was about 7,200,000,000 people.
$7.2\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{9}$
Convert Scientific Notation to Decimal Form
In the following exercises, convert each number to decimal form.
$3.8\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{5}$
$1.5\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{10}$
$\mathrm{15,000,000,000}$
$9.1\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-7}}$
$5.5\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-1}}$
$0.55$
Multiply and Divide Using Scientific Notation
In the following exercises, multiply and write your answer in decimal form.
$\left(2\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{5}\right)\left(4\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-3}}\right)$
$\left(3.5\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-2}}\right)\left(6.2\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-1}}\right)$
$0.0217$
In the following exercises, divide and write your answer in decimal form.
$\frac{8\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{5}}{4\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-1}}}$
$\frac{9\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-5}}}{3\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{2}}$
$0.0000003$
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.
$\left(12{a}^{2}-7a+4\right)+\left(3{a}^{2}+8a-10\right)$
$15{a}^{2}+a-6$
$\left(9{p}^{2}-5p+1\right)-\left(2{p}^{2}-6\right)$
$u\xb7{u}^{4}$
$\left(\mathrm{-9}{r}^{4}{s}^{5}\right)\left(4r{s}^{7}\right)$
$\left(m+6\right)\left(m+12\right)$
$\left(4c-11\right)\left(3c-8\right)$
$\left(n-6\right)\left({n}^{2}-5n+4\right)$
${n}^{3}-11{n}^{2}+34n-24$
$\left(2x-15y\right)\left(5x+7y\right)$
${\left(9v-2\right)}^{2}$
${\left(\frac{{m}^{4}\xb7m}{{m}^{3}}\right)}^{6}$
$\frac{80{c}^{8}{d}^{2}}{16c{d}^{10}}$
$\left(70x{y}^{4}+95{x}^{3}y\right)\xf75xy$
$\left({y}^{2}-5y-18\right)\xf7(y+3)$
${\left(4m\right)}^{\mathrm{-3}}$
${q}^{\mathrm{-4}}\xb7{q}^{\mathrm{-5}}$
$\frac{1}{{q}^{9}}$
$\frac{{n}^{\mathrm{-2}}}{{n}^{\mathrm{-10}}}$
Convert 83,000,000 to scientific notation.
$8.3\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{7}$
Convert $6.91\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-5}}$ to decimal form.
In the following exercises, simplify, and write your answer in decimal form.
$\left(3.4\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{9}\right)\left(2.2\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-5}}\right)$
74,800
$\frac{8.4\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-3}}}{4\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{3}}$
A helicopter flying at an altitude of 1000 feet drops a rescue package. The polynomial $\mathrm{-16}{t}^{2}+1000$ gives the height of the package $t$ seconds a after it was dropped. Find the height when $t=6$ seconds.
424 feet
Notification Switch
Would you like to follow the 'Elementary algebra' conversation and receive update notifications?