Chapter 6 Review Exercises

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