Prealgebra 2e · free preview

Chapter 10 Review and Practice Test

Learning objectives: Review the key ideas of this chapter.; Apply the chapter's methods to review exercises.; Check your understanding with a practice test.

1. Learning objectives

Learning objectives

Learning objectives

  • Review the key ideas of this chapter.
  • Apply the chapter's methods to review exercises.
  • Check your understanding with a practice test.

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2. Worked example

Worked example

binomial

A binomial is a polynomial with exactly two terms.

degree of a constant

degree of a polynomial

The degree of a polynomial is the highest degree of all its terms.

degree of a term

The degree of a term of a polynomial is the exponent of its variable.

greatest common factor

The greatest common factor (GCF) of two or more expressions is the largest expression that is a factor of all the expressions.

monomial

negative exponent

polynomial

A polynomial is a monomial, or two or more monomials, combined by addition or subtraction.

scientific notation

scientific notation

trinomial

A trinomial is a polynomial with exactly three terms.

zero exponent

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Key Concepts

  • Exponential Notation This is read aa to the mth{m}^{\mathrm{th}} power.
  • Product Property of Exponents If aa is a real number and m,nm,n are counting numbers, then
am⋅an=am+n{a}^{m}\cdot{a}^{n}={a}^{m+n}

To multiply with like bases, add the exponents.

  • Power Property for Exponents If aa is a real number and m,nm,n are counting numbers, then
(am)n=am⋅n{({a}^{m})}^{n}={a}^{m\cdot{}n}
  • Product to a Power Property for Exponents If aa and bb are real numbers and mm is a whole number, then
(ab)m=ambm{(ab)}^{m}={a}^{m}{b}^{m}
  • Use the FOIL method for multiplying two binomials. Step 1. Multiply the First terms. Step 2. Multiply the Outer terms. Step 3. Multiply the Inner terms. Step 4. Multiply the Last terms. Step 5. Combine like terms, when possible.
  • Multiplying Two Binomials: To multiply binomials, use the: Distributive Property FOIL Method Vertical Method
  • Multiplying a Trinomial by a Binomial: To multiply a trinomial by a binomial, use the: Distributive Property Vertical Method
  • Equivalent Fractions Property If a, b, ca,\,b,\,c are whole numbers where b≠0, c≠0,b\ne{}0,\,c\ne{}0, then
ab=a ⋅ cb ⋅ c and a ⋅ cb ⋅ c=ab\frac{a}{b}=\frac{a\,\cdot\,c}{b\,\cdot\,c}\,\mathrm{and}\,\frac{a\,\cdot\,c}{b\,\cdot\,c}=\frac{a}{b}
  • Zero Exponent If aa is a non-zero number, then a0=1.{a}^{0}=1. Any nonzero number raised to the zero power is 1.1.
  • Quotient Property for Exponents If aa is a real number, a≠0,a\ne{}0, and m, nm,\,n are whole numbers, then
aman=am−n, m>n and aman=1an−m, n>m\frac{{a}^{m}}{{a}^{n}}={a}^{m-n},\,m>n\,\mathrm{and}\,\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}},\,n>m
  • Quotient to a Power Property for Exponents If aa and bb are real numbers, b≠0,b\ne{}0, and mm is a counting number, then
(ab)m= ambm{(\frac{a}{b})}^{m}=\,\frac{{a}^{m}}{{b}^{m}}

To raise a fraction to a power, raise the numerator and denominator to that power.

  • Summary of Exponent Properties If a,ba,b are real numbers and m,nm,n are integers, then
ProductPropertyam⋅an=am+nPowerProperty(am)n=am⋅nProducttoaPowerProperty(ab)m=ambmQuotientPropertyaman=am−n, a≠0ZeroExponentPropertya0=1, a≠0QuotienttoaPowerProperty(ab)m=ambm, b≠0DefinitionofNegativeExponenta−n=1an\begin{array}{lccl} \mathrm{Product Property} & & & {a}^{m}\cdot{a}^{n}={a}^{m+n} \\ \mathrm{Power Property} & & & {({a}^{m})}^{n}={a}^{m\cdot{}n} \\ \mathrm{Product to a Power Property} & & & {(ab)}^{m}={a}^{m}{b}^{m} \\ \mathrm{Quotient Property} & & & \frac{{a}^{m}}{{a}^{n}}={a}^{m-n},\,a\ne{}0 \\ \mathrm{Zero Exponent Property} & & & {a}^{0}=1,\,a\ne{}0 \\ \mathrm{Quotient to a Power Property} & & & {(\frac{a}{b})}^{m}=\frac{{a}^{m}}{{b}^{m}},\,b\ne{}0 \\ \mathrm{Definition of Negative Exponent} & & & {a}^{-n}=\frac{1}{{a}^{n}} \\ \end{array}
  • Convert from Decimal Notation to Scientific Notation: To convert a decimal to scientific notation: Step 1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10. Step 2. Count the number of decimal places, nn, that the decimal point was moved. Step 3. Write the number as a product with a power of 10.If the original number is greater than 1, the power of 10 will be 10n{10}^{n}. If the original number is between 0 and 1, the power of 10 will be 10−n{10}^{-n}. Step 4. Check.
  • Convert Scientific Notation to Decimal Form: To convert scientific notation to decimal form: Step 1. Determine the exponent, nn, on the factor 10. Step 2. Move the decimal nn places, adding zeros if needed. If the exponent is positive, move the decimal point nn places to the right. If the exponent is negative, move the decimal point ∣n∣|n| places to the left. Step 3. Check.
  • Find the greatest common factor. Step 1. Factor each coefficient into primes. Write all variables with exponents in expanded form. Step 2. List all factors—matching common factors in a column. In each column, circle the common factors. Step 3. Bring down the common factors that all expressions share. Step 4. Multiply the factors.
  • Distributive Property If aa, bb, cc are real numbers, then a(b+c)=ab+aca(b+c)=ab+ac and ab+ac=a(b+c)ab+ac=a(b+c)
  • Factor the greatest common factor from a polynomial. Step 1. Find the GCF of all the terms of the polynomial. Step 2. Rewrite each term as a product using the GCF. Step 3. Use the Distributive Property ‘in reverse’ to factor the expression. Step 4. Check by multiplying the factors.

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Review Exercises

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.

Determine the Degree of Polynomials

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

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

Evaluate a Polynomial for a Given Value of the Variable

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

Simplify Expressions with Exponents

In the following exercises, simplify.

Simplify Expressions Using the Product Property of Exponents

In the following exercises, simplify each expression.

Simplify Expressions Using the Power Property of Exponents

In the following exercises, simplify each expression.

Simplify Expressions Using the Product to a Power Property

In the following exercises, simplify each expression.

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

Multiply Monomials

In the following exercises, multiply the monomials.

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

Multiply a Binomial by a Binomial

In the following exercises, multiply the binomials using various methods.

Multiply a Trinomial by a Binomial

In the following exercises, multiply using any method.

Simplify Expressions Using the Quotient Property of Exponents

In the following exercises, simplify.

Simplify Expressions with Zero Exponents

In the following exercises, simplify.

Simplify Expressions Using the Quotient to a Power Property

In the following exercises, simplify.

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

Divide Monomials

In the following exercises, divide the monomials.

Use the Definition of a Negative Exponent

In the following exercises, simplify.

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

Convert from Decimal Notation to Scientific Notation

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

Convert Scientific Notation to Decimal Form

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

Multiply and Divide Using Scientific Notation

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

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

Factor the Greatest Common Factor from a Polynomial

In the following exercises, factor the greatest common factor from each polynomial.

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In the following exercises, simplify each expression.

In the following exercises, factor the greatest common factor from each polynomial.

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

3. Key Terms

Key Terms

binomial

A binomial is a polynomial with exactly two terms.

degree of a constant

degree of a polynomial

The degree of a polynomial is the highest degree of all its terms.

degree of a term

The degree of a term of a polynomial is the exponent of its variable.

greatest common factor

The greatest common factor (GCF) of two or more expressions is the largest expression that is a factor of all the expressions.

monomial

negative exponent

polynomial

A polynomial is a monomial, or two or more monomials, combined by addition or subtraction.

scientific notation

scientific notation

trinomial

A trinomial is a polynomial with exactly three terms.

zero exponent

##

Key Concepts

  • Exponential Notation This is read aa to the mth{m}^{\mathrm{th}} power.
  • Product Property of Exponents If aa is a real number and m,nm,n are counting numbers, then
am⋅an=am+n{a}^{m}\cdot{a}^{n}={a}^{m+n}

To multiply with like bases, add the exponents.

  • Power Property for Exponents If aa is a real number and m,nm,n are counting numbers, then
(am)n=am⋅n{({a}^{m})}^{n}={a}^{m\cdot{}n}
  • Product to a Power Property for Exponents If aa and bb are real numbers and mm is a whole number, then
(ab)m=ambm{(ab)}^{m}={a}^{m}{b}^{m}
  • Use the FOIL method for multiplying two binomials. Step 1. Multiply the First terms. Step 2. Multiply the Outer terms. Step 3. Multiply the Inner terms. Step 4. Multiply the Last terms. Step 5. Combine like terms, when possible.
  • Multiplying Two Binomials: To multiply binomials, use the: Distributive Property FOIL Method Vertical Method
  • Multiplying a Trinomial by a Binomial: To multiply a trinomial by a binomial, use the: Distributive Property Vertical Method
  • Equivalent Fractions Property If a, b, ca,\,b,\,c are whole numbers where b≠0, c≠0,b\ne{}0,\,c\ne{}0, then
ab=a ⋅ cb ⋅ c and a ⋅ cb ⋅ c=ab\frac{a}{b}=\frac{a\,\cdot\,c}{b\,\cdot\,c}\,\mathrm{and}\,\frac{a\,\cdot\,c}{b\,\cdot\,c}=\frac{a}{b}
  • Zero Exponent If aa is a non-zero number, then a0=1.{a}^{0}=1. Any nonzero number raised to the zero power is 1.1.
  • Quotient Property for Exponents If aa is a real number, a≠0,a\ne{}0, and m, nm,\,n are whole numbers, then
aman=am−n, m>n and aman=1an−m, n>m\frac{{a}^{m}}{{a}^{n}}={a}^{m-n},\,m>n\,\mathrm{and}\,\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}},\,n>m
  • Quotient to a Power Property for Exponents If aa and bb are real numbers, b≠0,b\ne{}0, and mm is a counting number, then
(ab)m= ambm{(\frac{a}{b})}^{m}=\,\frac{{a}^{m}}{{b}^{m}}

To raise a fraction to a power, raise the numerator and denominator to that power.

  • Summary of Exponent Properties If a,ba,b are real numbers and m,nm,n are integers, then
ProductPropertyam⋅an=am+nPowerProperty(am)n=am⋅nProducttoaPowerProperty(ab)m=ambmQuotientPropertyaman=am−n, a≠0ZeroExponentPropertya0=1, a≠0QuotienttoaPowerProperty(ab)m=ambm, b≠0DefinitionofNegativeExponenta−n=1an\begin{array}{lccl} \mathrm{Product Property} & & & {a}^{m}\cdot{a}^{n}={a}^{m+n} \\ \mathrm{Power Property} & & & {({a}^{m})}^{n}={a}^{m\cdot{}n} \\ \mathrm{Product to a Power Property} & & & {(ab)}^{m}={a}^{m}{b}^{m} \\ \mathrm{Quotient Property} & & & \frac{{a}^{m}}{{a}^{n}}={a}^{m-n},\,a\ne{}0 \\ \mathrm{Zero Exponent Property} & & & {a}^{0}=1,\,a\ne{}0 \\ \mathrm{Quotient to a Power Property} & & & {(\frac{a}{b})}^{m}=\frac{{a}^{m}}{{b}^{m}},\,b\ne{}0 \\ \mathrm{Definition of Negative Exponent} & & & {a}^{-n}=\frac{1}{{a}^{n}} \\ \end{array}
  • Convert from Decimal Notation to Scientific Notation: To convert a decimal to scientific notation: Step 1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10. Step 2. Count the number of decimal places, nn, that the decimal point was moved. Step 3. Write the number as a product with a power of 10.If the original number is greater than 1, the power of 10 will be 10n{10}^{n}. If the original number is between 0 and 1, the power of 10 will be 10−n{10}^{-n}. Step 4. Check.
  • Convert Scientific Notation to Decimal Form: To convert scientific notation to decimal form: Step 1. Determine the exponent, nn, on the factor 10. Step 2. Move the decimal nn places, adding zeros if needed. If the exponent is positive, move the decimal point nn places to the right. If the exponent is negative, move the decimal point ∣n∣|n| places to the left. Step 3. Check.
  • Find the greatest common factor. Step 1. Factor each coefficient into primes. Write all variables with exponents in expanded form. Step 2. List all factors—matching common factors in a column. In each column, circle the common factors. Step 3. Bring down the common factors that all expressions share. Step 4. Multiply the factors.
  • Distributive Property If aa, bb, cc are real numbers, then a(b+c)=ab+aca(b+c)=ab+ac and ab+ac=a(b+c)ab+ac=a(b+c)
  • Factor the greatest common factor from a polynomial. Step 1. Find the GCF of all the terms of the polynomial. Step 2. Rewrite each term as a product using the GCF. Step 3. Use the Distributive Property ‘in reverse’ to factor the expression. Step 4. Check by multiplying the factors.

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Review Exercises

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.

Determine the Degree of Polynomials

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

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

Evaluate a Polynomial for a Given Value of the Variable

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

Simplify Expressions with Exponents

In the following exercises, simplify.

Simplify Expressions Using the Product Property of Exponents

In the following exercises, simplify each expression.

Simplify Expressions Using the Power Property of Exponents

In the following exercises, simplify each expression.

Simplify Expressions Using the Product to a Power Property

In the following exercises, simplify each expression.

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

Multiply Monomials

In the following exercises, multiply the monomials.

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

Multiply a Binomial by a Binomial

In the following exercises, multiply the binomials using various methods.

Multiply a Trinomial by a Binomial

In the following exercises, multiply using any method.

Simplify Expressions Using the Quotient Property of Exponents

In the following exercises, simplify.

Simplify Expressions with Zero Exponents

In the following exercises, simplify.

Simplify Expressions Using the Quotient to a Power Property

In the following exercises, simplify.

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

Divide Monomials

In the following exercises, divide the monomials.

Use the Definition of a Negative Exponent

In the following exercises, simplify.

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

Convert from Decimal Notation to Scientific Notation

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

Convert Scientific Notation to Decimal Form

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

Multiply and Divide Using Scientific Notation

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

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

Factor the Greatest Common Factor from a Polynomial

In the following exercises, factor the greatest common factor from each polynomial.

##

In the following exercises, simplify each expression.

In the following exercises, factor the greatest common factor from each polynomial.

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

4. Practice 1

  1. y2+8y−20{y}^{2}+8y-20
  1. 9x3−19{x}^{3}-1
  1. 4p+11p4p+11p
  1. Add 4n5, −n5, −6n54{n}^{5},\,-{n}^{5},\,-6{n}^{5}
  1. (4a2+9a−11)+(6a2−5a+10)(4{a}^{2}+9a-11)+(6{a}^{2}-5a+10)

5. Practice 2

  1. (y2−3y+12)+(5y2−9)({y}^{2}-3y+12)+(5{y}^{2}-9)
  1. Find the sum of 8q3−278{q}^{3}-27 and q2+6q−2{q}^{2}+6q-2
  1. 200x−15 x2200x-\frac{1}{5}\,{x}^{2} when x=15x=15
  1. 5+40x−12 x25+40x-\frac{1}{2}\,{x}^{2} when x=−4x=-4
  1. A pair of glasses is dropped off a bridge 640640 feet above a river. The polynomial −16t2+640-16{t}^{2}+640 gives the height of the glasses tt seconds after they were dropped. Find the height of the glasses when t=6.t=6.

6. Practice 3

  1. p3⋅p10{p}^{3}\cdot{p}^{10}
  1. a⋅a2⋅a3a\cdot{a}^{2}\cdot{a}^{3}
  1. (y4)3{({y}^{4})}^{3}
  1. (32)5{({3}^{2})}^{5}
  1. (8n)2{(8n)}^{2}

7. Practice 4

  1. (2ab)8{(2ab)}^{8}
  1. (3a5)3{(3{a}^{5})}^{3}
  1. (x3)5(x2)3{({x}^{3})}^{5}{({x}^{2})}^{3}
  1. (−6p4)(9p)(-6{p}^{4})(9p)
  1. (8x2y5)(7xy6)(8{x}^{2}{y}^{5})(7x{y}^{6})

8. Practice 5

  1. 7(10−x)7(10-x)
  1. −5y(125y3−1)-5y(125{y}^{3}-1)
  1. (a+5)(a+2)(a+5)(a+2)
  1. (3x+1)(2x−7)(3x+1)(2x-7)
  1. (n+8)(n+1)(n+8)(n+1)

9. Practice 6

  1. (5u−3)(u+8)(5u-3)(u+8)
  1. (p+4)(p+7)(p+4)(p+7)
  1. (3c+1)(9c−4)(3c+1)(9c-4)
  1. (x+1)(x2−3x−21)(x+1)({x}^{2}-3x-21)
  1. (m+6)(m2−7m−30)(m+6)({m}^{2}-7m-30)

10. Practice 7

  1. 2822\frac{{2}^{8}}{{2}^{2}}
  1. n3n12\frac{{n}^{3}}{{n}^{12}}
  1. (35)2{(\frac{3}{5})}^{2}
  1. (5mn)3{(\frac{5m}{n})}^{3}
  1. (a3)2a4\frac{{({a}^{3})}^{2}}{{a}^{4}}

11. Practice 8

  1. (xx9)5{(\frac{x}{{x}^{9}})}^{5}
  1. (n5)3(n2)8\frac{{({n}^{5})}^{3}}{{({n}^{2})}^{8}}
  1. 72p12÷8p372{p}^{12}\mathrm{÷}8{p}^{3}
  1.  45y6−15y10\frac{\,45{y}^{6}}{-15{y}^{10}}
  1. 28a9b7a4b3\frac{28{a}^{9}b}{7{a}^{4}{b}^{3}}

12. Practice 9

  1. (5m9n3)(8m3n2)(10mn4)(m2n5)\frac{(5{m}^{9}{n}^{3})(8{m}^{3}{n}^{2})}{(10m{n}^{4})({m}^{2}{n}^{5})}
  1. 6−2{6}^{-2}
  1. 5⋅2−45\cdot{2}^{-4}
  1. x−3⋅x9{x}^{-3}\cdot{x}^{9}
  1. (uv−3)(u−4v−2)(u{v}^{-3})({u}^{-4}{v}^{-2})

13. Practice 10

  1. (k−2)−3{({k}^{-2})}^{-3}
  1. b8 b−2\frac{{b}^{8}}{\,{b}^{-2}}
  1. 5,300,0005,300,000
  1. The thickness of a piece of paper is about 0.0970.097 millimeter.
  1. 2.9 × 1042.9\,\times\,{10}^{4}

14. Practice 11

  1. (3 × 107)(2 × 10−4)(3\,\times\,{10}^{7})(2\,\times\,{10}^{-4})
  1. 6 × 1092 × 10−1\frac{6\,\times\,{10}^{9}}{2\,\times\,{10}^{-1}}
  1. 12x2, 20x3, 36x412{x}^{2},\,20{x}^{3},\,36{x}^{4}
  1. 16u−2416u-24
  1. 6p2+6p6{p}^{2}+6p

15. Practice 12

  1. −9a5−9a3-9{a}^{5}-9{a}^{3}
  1. 5y2−55y+455{y}^{2}-55y+45
  1. For the polynomial 8y4−3y2+18{y}^{4}-3{y}^{2}+1
  1. ⓐ Is it a monomial, binomial, or trinomial?
  1. ⓑ What is its degree?
  1. (10x2−3x+5)−(4x2−6)(10{x}^{2}-3x+5)-(4{x}^{2}-6)
  1. n⋅n4n\cdot{n}^{4}

16. Practice 13

  1. (8xy3)(−6x4y6)(8x{y}^{3})(-6{x}^{4}{y}^{6})
  1. (s+8)(s+9)(s+8)(s+9)
  1. (11a−6)(5a−1)(11a-6)(5a-1)
  1. (4a+9b)(6a−5b)(4a+9b)(6a-5b)
  1. (x3⋅x9x5)2{(\frac{{x}^{3}\cdot{x}^{9}}{{x}^{5}})}^{2}

17. Practice 14

  1. 24r3s 6r2s7\frac{24{r}^{3}s}{\,6{r}^{2}{s}^{7}}
  1. (15xy3−35x2y)÷5xy(15x{y}^{3}-35{x}^{2}y)\mathrm{÷}5xy
  1. (2y)−3{(2y)}^{-3}
  1. x4 x−5\frac{{x}^{4}}{\,{x}^{-5}}
  1. −6x2−30x-6{x}^{2}-30x

18. Practice 15

  1. 9 × 1043 × 10−1\frac{9\,\times\,{10}^{4}}{3\,\times\,{10}^{-1}}

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