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Mathematics LibreTexts

Chapter 6 Review Exercises

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    Chapter 6 Review Exercises

    Add and Subtract Polynomials

    Identify Polynomials, Monomials, Binomials and Trinomials

    In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.

    Exercise \(\PageIndex{1}\)

    1. \(11 c^{4}-23 c^{2}+1\)
    2. \(9 p^{3}+6 p^{2}-p-5\)
    3. \(\frac{3}{7} x+\frac{5}{14}\)
    4. 10
    5. 2y−12

    Exercise \(\PageIndex{2}\)

    1. \(a^{2}-b^{2}\)
    2. 24\(d^{3}\)
    3. \(x^{2}+8 x-10\)
    4. \(m^{2} n^{2}-2 m n+6\)
    5. \(7 y^{3}+y^{2}-2 y-4\)
    Answer
    1. binomial
    2. monomial
    3. trinomial
    4. trinomial
    5. other polynomial

    Determine the Degree of Polynomials

    In the following exercises, determine the degree of each polynomial.

    Exercise \(\PageIndex{3}\)

    1. \(3 x^{2}+9 x+10\)
    2. 14\(a^{2} b c\)
    3. 6y+1
    4. \(n^{3}-4 n^{2}+2 n-8\)
    5. −19

    Exercise \(\PageIndex{4}\)

    1. \(5 p^{3}-8 p^{2}+10 p-4\)
    2. \(-20 q^{4}\)
    3. \(x^{2}+6 x+12\)
    4. \(23 r^{2} s^{2}-4 r s+5\)
    5. 100
    Answer
    1. 3
    2. 4
    3. 2
    4. 4
    5. 0

    Add and Subtract Monomials

    In the following exercises, add or subtract the monomials.

    Exercise \(\PageIndex{5}\)

    \(5 y^{3}+8 y^{3}\)

    Exercise \(\PageIndex{6}\)

    \(-14 k+19 k\)

    Answer

    5k

    Exercise \(\PageIndex{7}\)

    12q−(−6q)

    Exercise \(\PageIndex{8}\)

    −9c−18c

    Answer

    −27c

    Exercise \(\PageIndex{9}\)

    12x−4y−9x

    Exercise \(\PageIndex{2}\)

    \(3 m^{2}+7 n^{2}-3 m^{2}\)

    Answer

    7\(n^{2}\)

    Exercise \(\PageIndex{3}\)

    \(6 x^{2} y-4 x+8 x y^{2}\)

    Exercise \(\PageIndex{4}\)

    13a+b

    Answer

    13a+b

    Add and Subtract Polynomials

    In the following exercises, add or subtract the polynomials.

    Exercise \(\PageIndex{5}\)

    \(\left(5 x^{2}+12 x+1\right)+\left(6 x^{2}-8 x+3\right)\)

    Exercise \(\PageIndex{6}\)

    \(\left(9 p^{2}-5 p+3\right)+\left(4 p^{2}-4\right)\)

    Answer

    \(13 p^{2}-5 p-1\)

    Exercise \(\PageIndex{7}\)

    \(\left(10 m^{2}-8 m-1\right)-\left(5 m^{2}+m-2\right)\)

    Exercise \(\PageIndex{8}\)

    \(\left(7 y^{2}-8 y\right)-(y-4)\)

    Answer

    \(7 y^{2}-9 y+4\)

    Exercise \(\PageIndex{9}\)

    Subtract
    \(\left(3 s^{2}+10\right)\) from \(\left(15 s^{2}-2 s+8\right)\)

    Exercise \(\PageIndex{10}\)

    Find the sum of \(\left(a^{2}+6 a+9\right)\) and \(\left(5 a^{3}-7\right)\)

    Answer

    \(5 a^{3}+a^{2}+6 a+2\)

    Evaluate a Polynomial for a Given Value of the Variable

    In the following exercises, evaluate each polynomial for the given value.

    Exercise \(\PageIndex{11}\)

    Evaluate \(3 y^{2}-y+1\) when:

    1. y=5
    2. y=−1
    3. y=0

    Exercise \(\PageIndex{12}\)

    Evaluate 10−12x when:

    1. x=3
    2. x=0
    3. x=−1
    Answer
    1. −26
    2. 10
    3. 22

    Exercise \(\PageIndex{13}\)

    Randee drops a stone off the 200 foot high cliff into the ocean. The polynomial \(-16 t^{2}+200\) gives the height of a stone t seconds after it is dropped from the cliff. Find the height after t=3 seconds.

    Exercise \(\PageIndex{14}\)

    A manufacturer of stereo sound speakers has found that the revenue received from selling the speakers at a cost of p dollars each is given by the polynomial \(-4 p^{2}+460 p\). Find the revenue received when p=75 dollars.

    Answer

    12,000

    Use Multiplication Properties of Exponents

    Simplify Expressions with Exponents

    In the following exercises, simplify.

    Exercise \(\PageIndex{15}\)

    \(10^{4}\)

    Exercise \(\PageIndex{16}\)

    \(17^{1}\)

    Answer

    17

    Exercise \(\PageIndex{17}\)

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

    Exercise \(\PageIndex{18}\)

    \((0.5)^{3}\)

    Answer

    0.125

    Exercise \(\PageIndex{19}\)

    \((-2)^{6}\)

    Exercise \(\PageIndex{20}\)

    \(-2^{6}\)

    Answer

    −64

    Simplify Expressions Using the Product Property for Exponents

    In the following exercises, simplify each expression.

    Exercise \(\PageIndex{21}\)

    \(x^{4} \cdot x^{3}\)

    Exercise \(\PageIndex{22}\)

    \(p^{15} \cdot p^{16}\)

    Answer

    \(p^{31}\)

    Exercise \(\PageIndex{23}\)

    \(4^{10} \cdot 4^{6}\)

    Exercise \(\PageIndex{24}\)

    8\(\cdot 8^{5}\)

    Answer

    \(8^{6}\)

    Exercise \(\PageIndex{25}\)

    \(n \cdot n^{2} \cdot n^{4}\)

    Exercise \(\PageIndex{26}\)

    \(y^{c} \cdot y^{3}\)

    Answer

    \(y^{c+3}\)

    Simplify Expressions Using the Power Property for Exponents

    In the following exercises, simplify each expression.

    Exercise \(\PageIndex{27}\)

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

    Exercise \(\PageIndex{28}\)

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

    Answer

    \(5^{6}\)

    Exercise \(\PageIndex{29}\)

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

    Exercise \(\PageIndex{30}\)

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

    Answer

    \(3^{r s}\)

    Simplify Expressions Using the Product to a Power Property

    In the following exercises, simplify each expression.

    Exercise \(\PageIndex{31}\)

    \((4 a)^{2}\)

    Exercise \(\PageIndex{32}\)

    \((-5 y)^{3}\)

    Answer

    \(-125 y^{3}\)

    Exercise \(\PageIndex{33}\)

    \((2 m n)^{5}\)

    Exercise \(\PageIndex{34}\)

    \((10 x y z)^{3}\)

    Answer

    1000\(x^{3} y^{3} z^{3}\)

    Simplify Expressions by Applying Several Properties

    In the following exercises, simplify each expression.

    Exercise \(\PageIndex{35}\)

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

    Exercise \(\PageIndex{36}\)

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

    Answer

    64\(a^{9} b^{6}\)

    Exercise \(\PageIndex{37}\)

    \((5 x)^{2}(7 x)\)

    Exercise \(\PageIndex{38}\)

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

    Answer

    48\(q^{14}\)

    Exercise \(\PageIndex{39}\)

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

    Exercise \(\PageIndex{40}\)

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

    Answer

    \(\frac{8}{125} m^{6} n^{3}\)

    Multiply Monomials

    In the following exercises 8, multiply the monomials.

    Exercise \(\PageIndex{41}\)

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

    Exercise \(\PageIndex{42}\)

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

    Answer

    144\(n^{8}\)

    Exercise \(\PageIndex{43}\)

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

    Exercise \(\PageIndex{44}\)

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

    Answer

    15\(a^{2} b^{5}\)

    Multiply Polynomials

    Multiply a Polynomial by a Monomial

    In the following exercises, multiply.

    Exercise \(\PageIndex{45}\)

    7(a+9)

    Exercise \(\PageIndex{46}\)

    −4(y+13)

    Answer

    −4y−52

    Exercise \(\PageIndex{47}\)

    −5(r−2)

    Exercise \(\PageIndex{48}\)

    p(p+3)

    Answer

    \(p^{2}+3 p\)

    Exercise \(\PageIndex{49}\)

    −m(m+15)

    Exercise \(\PageIndex{50}\)

    −6u(2u+7)

    Answer

    \(-12 u^{2}-42 u\)

    Exercise \(\PageIndex{51}\)

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

    Exercise \(\PageIndex{52}\)

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

    Answer

    \(3 q^{4}-21 q^{3}+18 q^{2}\)

    Exercise \(\PageIndex{53}\)

    \((5 z-1) z\)

    Exercise \(\PageIndex{54}\)

    \((b-4) \cdot 11\)

    Answer

    11b−44

    Multiply a Binomial by a Binomial

    In the following exercises, multiply the binomials using:

    1. the Distributive Property,
    2. the FOIL method,
    3. the Vertical Method.

    Exercise \(\PageIndex{55}\)

    (x−4)(x+10)

    Exercise \(\PageIndex{56}\)

    (6y−7)(2y−5)

    Answer
    1. \(12 y^{2}-44y+35\)
    2. \(12 y^{2}-44y+35\)
    3. \(12 y^{2}-44y+35\)

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

    Exercise \(\PageIndex{57}\)

    (x+3)(x+9)

    Exercise \(\PageIndex{58}\)

    (y−4)(y−8)

    Answer

    \(y^{2}-12 y+32\)

    Exercise \(\PageIndex{59}\)

    (p−7)(p+4)

    Exercise \(\PageIndex{60}\)

    (q+16)(q−3)

    Answer

    \(q^{2}+13 q-48\)

    Exercise \(\PageIndex{61}\)

    (5m−8)(12m+1)

    Exercise \(\PageIndex{62}\)

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

    Answer

    \(u^{4}+u^{2}-30\)

    Exercise \(\PageIndex{63}\)

    (9x−y)(6x−5)

    Exercise \(\PageIndex{64}\)

    (8mn+3)(2mn−1)

    Answer

    \(16 m^{2} n^{2}-2 m n-3\)

    Multiply a Trinomial by a Binomial

    In the following exercises, multiply using

    1. the Distributive Property,
    2. the Vertical Method.

    Exercise \(\PageIndex{65}\)

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

    Exercise \(\PageIndex{66}\)

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

    Answer
    1. \(18 x^{3}-21 x^{2}-34 x+40\)
    2. \(18 x^{3}-21 x^{2}-34 x+40\)

    In the following exercises, multiply. Use either method.

    Exercise \(\PageIndex{67}\)

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

    Exercise \(\PageIndex{68}\)

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

    Answer

    \(7 m^{3}-69 m^{2}-31 m-3\)

    Special Products

    Square a Binomial Using the Binomial Squares Pattern

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

    Exercise \(\PageIndex{69}\)

    \((c+11)^{2}\)

    Exercise \(\PageIndex{70}\)

    \((q-15)^{2}\)

    Answer

    \(q^{2}-30 q+225\)

    Exercise \(\PageIndex{71}\)

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

    Exercise \(\PageIndex{72}\)

    \((8 u+1)^{2}\)

    Answer

    \(64 u^{2}+16 u+1\)

    Exercise \(\PageIndex{73}\)

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

    Exercise \(\PageIndex{74}\)

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

    Answer

    \(16 a^{2}-24 a b+9 b^{2}\)

    Multiply Conjugates Using the Product of Conjugates Pattern

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

    Exercise \(\PageIndex{75}\)

    (s−7)(s+7)

    Exercise \(\PageIndex{76}\)

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

    Answer

    \(y^{2}-\frac{4}{25}\)

    Exercise \(\PageIndex{77}\)

    \((12 c+13)(12 c-13)\)

    Exercise \(\PageIndex{78}\)

    (6−r)(6+r)

    Answer

    \(36-r^{2}\)

    Exercise \(\PageIndex{79}\)

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

    Exercise \(\PageIndex{80}\)

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

    Answer

    \(25 p^{8}-16 q^{6}\)

    Recognize and Use the Appropriate Special Product Pattern

    In the following exercises, find each product.

    Exercise \(\PageIndex{81}\)

    \((3 m+10)^{2}\)

    Exercise \(\PageIndex{82}\)

    (6a+11)(6a−11)

    Answer

    \(36 a^{2}-121\)

    Exercise \(\PageIndex{83}\)

    (5x+y)(x−5y)

    Exercise \(\PageIndex{84}\)

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

    Answer

    \(c^{8}+18 c^{4} d+81 d^{2}\)

    Exercise \(\PageIndex{85}\)

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

    Exercise \(\PageIndex{86}\)

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

    Answer

    \(4 a^{3}+3 a^{2} b-4 b^{3}\)

    Divide Monomials

    Simplify Expressions Using the Quotient Property for Exponents

    In the following exercises, simplify.

    Exercise \(\PageIndex{87}\)

    \(\frac{u^{24}}{u^{6}}\)

    Exercise \(\PageIndex{88}\)

    \(\frac{10^{25}}{10^{5}}\)

    Answer

    \(10^{20}\)

    Exercise \(\PageIndex{89}\)

    \(\frac{3^{4}}{3^{6}}\)

    Exercise \(\PageIndex{90}\)

    \(\frac{v^{12}}{v^{48}}\)

    Answer

    \(\frac{1}{v^{36}}\)

    Exercise \(\PageIndex{91}\)

    \(\frac{x}{x^{5}}\)

    Exercise \(\PageIndex{92}\)

    \(\frac{5}{5^{8}}\)

    Answer

    \(\frac{1}{5^{7}}\)

    Simplify Expressions with Zero Exponents

    In the following exercises, simplify.

    Exercise \(\PageIndex{93}\)

    \(75^{0}\)

    Exercise \(\PageIndex{94}\)

    \(x^{0}\)

    Answer

    1

    Exercise \(\PageIndex{95}\)

    \(-12^{0}\)

    Exercise \(\PageIndex{96}\)

    \(\left(-12^{0}\right)(-12)^{0}\)

    Answer

    1

    Exercise \(\PageIndex{97}\)

    25\(x^{0}\)

    Exercise \(\PageIndex{98}\)

    \((25 x)^{0}\)

    Answer

    1

    Exercise \(\PageIndex{99}\)

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

    Exercise \(\PageIndex{100}\)

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

    Answer

    0

    Simplify Expressions Using the Quotient to a Power Property

    In the following exercises, simplify.

    Exercise \(\PageIndex{101}\)

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

    Exercise \(\PageIndex{102}\)

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

    Answer

    \(\frac{m^{4}}{81}\)

    Exercise \(\PageIndex{103}\)

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

    Exercise \(\PageIndex{104}\)

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

    Answer

    \(\frac{x^{6}}{64 y^{6}}\)

    Simplify Expressions by Applying Several Properties

    In the following exercises, simplify.

    Exercise \(\PageIndex{105}\)

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

    Exercise \(\PageIndex{106}\)

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

    Answer

    1

    Exercise \(\PageIndex{107}\)

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

    Exercise \(\PageIndex{108}\)

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

    Answer

    \(r^{20}\)

    Exercise \(\PageIndex{109}\)

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

    Exercise \(\PageIndex{110}\)

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

    Answer

    \(\frac{343 x^{20}}{32 y^{10}}\)

    Exercise \(\PageIndex{111}\)

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

    Exercise \(\PageIndex{112}\)

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

    Answer

    \(-\frac{10,125 n^{10}}{4}\)

    Divide Monomials

    In the following exercises, divide the monomials.

    Exercise \(\PageIndex{113}\)

    \(-65 y^{14} \div 5 y^{2}\)

    Exercise \(\PageIndex{114}\)

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

    Answer

    \(-\frac{4 b^{6}}{a^{5}}\)

    Exercise \(\PageIndex{115}\)

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

    Exercise \(\PageIndex{116}\)

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

    Answer

    \(\frac{9 p}{2}\)

    Divide Polynomials

    Divide a Polynomial by a Monomial

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

    Exercise \(\PageIndex{117}\)

    \(\frac{42 z^{2}-18 z}{6}\)

    Exercise \(\PageIndex{118}\)

    \(\left(35 x^{2}-75 x\right) \div 5 x\)

    Answer

    7x−15

    Exercise \(\PageIndex{119}\)

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

    Exercise \(\PageIndex{120}\)

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

    Answer

    \(55 p^{3}-30 p\)

    Exercise \(\PageIndex{121}\)

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

    Exercise \(\PageIndex{122}\)

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

    Answer

    \(12 a^{4}-6 a^{3} b-7 a b^{2}\)

    Exercise \(\PageIndex{123}\)

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

    Exercise \(\PageIndex{124}\)

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

    Answer

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

    Divide a Polynomial by a Binomial

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

    Exercise \(\PageIndex{125}\)

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

    Exercise \(\PageIndex{126}\)

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

    Answer

    v−8

    Exercise \(\PageIndex{127}\)

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

    Exercise \(\PageIndex{128}\)

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

    Answer

    \(n-6+\frac{4}{n+3}\)

    Exercise \(\PageIndex{129}\)

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

    Exercise \(\PageIndex{130}\)

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

    Answer

    \(u^{2}+2 u+4\)

    Integer Exponents and Scientific Notation

    Use the Definition of a Negative Exponent

    In the following exercises, simplify.

    Exercise \(\PageIndex{131}\)

    \(9^{-2}\)

    Exercise \(\PageIndex{132}\)

    \((-5)^{-3}\)

    Answer

    \(-\frac{1}{125}\)

    Exercise \(\PageIndex{133}\)

    3\(\cdot 4^{-3}\)

    Exercise \(\PageIndex{134}\)

    \((6 u)^{-3}\)

    Answer

    \(\frac{1}{216 u^{3}}\)

    Exercise \(\PageIndex{135}\)

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

    Exercise \(\PageIndex{136}\)

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

    Answer

    \(\frac{16}{9}\)

    Simplify Expressions with Integer Exponents

    In the following exercises, simplify.

    Exercise \(\PageIndex{137}\)

    \(p^{-2} \cdot p^{8}\)

    Exercise \(\PageIndex{138}\)

    \(q^{-6} \cdot q^{-5}\)

    Answer

    \(\frac{1}{q^{11}}\)

    Exercise \(\PageIndex{139}\)

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

    Exercise \(\PageIndex{140}\)

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

    Answer

    \(\frac{1}{y^{8}}\)

    Exercise \(\PageIndex{141}\)

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

    Exercise \(\PageIndex{142}\)

    \(\frac{a^{8}}{a^{12}}\)

    Answer

    \(\frac{1}{a^{4}}\)

    Exercise \(\PageIndex{143}\)

    \(\frac{n^{5}}{n^{-4}}\)

    Exercise \(\PageIndex{144}\)

    \(\frac{r^{-2}}{r^{-3}}\)

    Answer

    r

    Convert from Decimal Notation to Scientific Notation

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

    Exercise \(\PageIndex{145}\)

    8,500,000

    Exercise \(\PageIndex{146}\)

    0.00429

    Answer

    \(4.29 \times 10^{-3}\)

    Exercise \(\PageIndex{147}\)

    The thickness of a dime is about 0.053 inches.

    Exercise \(\PageIndex{148}\)

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

    Answer

    \(7.2 \times 10^{9}\)

    Convert Scientific Notation to Decimal Form

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

    Exercise \(\PageIndex{149}\)

    \(3.8 \times 10^{5}\)

    Exercise \(\PageIndex{150}\)

    \(1.5 \times 10^{10}\)

    Answer

    15,000,000,000

    Exercise \(\PageIndex{151}\)

    \(9.1 \times 10^{-7}\)

    Exercise \(\PageIndex{152}\)

    \(5.5 \times 10^{-1}\)

    Answer

    0.55

    Multiply and Divide Using Scientific Notation

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

    Exercise \(\PageIndex{153}\)

    \(\left(2 \times 10^{5}\right)\left(4 \times 10^{-3}\right)\)

    Exercise \(\PageIndex{154}\)

    \(\left(3.5 \times 10^{-2}\right)\left(6.2 \times 10^{-1}\right)\)

    Answer

    0.0217

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

    Exercise \(\PageIndex{155}\)

    \(\frac{8 \times 10^{5}}{4 \times 10^{-1}}\)

    Exercise \(\PageIndex{156}\)

    \(\frac{9 \times 10^{-5}}{3 \times 10^{2}}\)

    Answer

    0.0000003

    Chapter Practice Test

    Exercise \(\PageIndex{1}\)

    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.

    Exercise \(\PageIndex{2}\)

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

    Answer

    \(15 a^{2}+a-6\)

    Exercise \(\PageIndex{3}\)

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

    Exercise \(\PageIndex{4}\)

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

    Answer

    \(-\frac{8}{125}\)

    Exercise \(\PageIndex{5}\)

    \(u \cdot u^{4}\)

    Exercise \(\PageIndex{6}\)

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

    Answer

    16\(a^{6} b^{10}\)

    Exercise \(\PageIndex{7}\)

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

    Exercise \(\PageIndex{8}\)

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

    Answer

    \(3 k^{3}-21 k^{2}+39 k\)

    Exercise \(\PageIndex{9}\)

    \((m+6)(m+12)\)

    Exercise \(\PageIndex{10}\)

    (v−9)(9v−5)

    Answer

    \(9 v^{2}-86 v+45\)

    Exercise \(\PageIndex{11}\)

    (4c−11)(3c−8)

    Exercise \(\PageIndex{12}\)

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

    Answer

    \(n^{3}-11 n^{2}+34 n-24\)

    Exercise \(\PageIndex{13}\)

    \((2 x-15 y)(5 x+7 y)\)

    Exercise \(\PageIndex{14}\)

    \((7 p-5)(7 p+5)\)

    Answer

    \(49 p^{2}-25\)

    Exercise \(\PageIndex{15}\)

    \((9 v-2)^{2}\)

    Exercise \(\PageIndex{16}\)

    \(\frac{3^{8}}{3^{10}}\)

    Answer

    \(\frac{1}{9}\)

    Exercise \(\PageIndex{17}\)

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

    Exercise \(\PageIndex{18}\)

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

    Answer

    1

    Exercise \(\PageIndex{19}\)

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

    Exercise \(\PageIndex{20}\)

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

    Answer

    \(6 x+21-\frac{3}{x}\)

    Exercise \(\PageIndex{21}\)

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

    Exercise \(\PageIndex{22}\)

    \(\frac{64 x^{3}-1}{4 x-1}\)

    Answer

    \(16 x^{2}+4 x+1\)

    Exercise \(\PageIndex{23}\)

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

    Exercise \(\PageIndex{24}\)

    \(5^{-2}\)

    Answer

    \(\frac{1}{25}\)

    Exercise \(\PageIndex{25}\)

    \((4 m)^{-3}\)

    Exercise \(\PageIndex{26}\)

    \(q^{-4} \cdot q^{-5}\)

    Answer

    \(\frac{1}{q^{9}}\)

    Exercise \(\PageIndex{27}\)

    \(\frac{n^{-2}}{n^{-10}}\)

    Exercise \(\PageIndex{28}\)

    Convert 83,000,000 to scientific notation.

    Answer

    \(8.3 \times 10^{7}\)

    Exercise \(\PageIndex{29}\)

    Convert \(6.91 \times 10^{-5}\) to decimal form.

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

    Exercise \(\PageIndex{30}\)

    \(\left(3.4 \times 10^{9}\right)\left(2.2 \times 10^{-5}\right)\)

    Answer

    74,800

    Exercise \(\PageIndex{31}\)

    \(\frac{8.4 \times 10^{-3}}{4 \times 10^{3}}\)

    Exercise \(\PageIndex{32}\)

    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.

    Answer

    424 feet

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