1.3.H: Homework

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    From LibreTexts, under CC BY 4.0, as the source states it; its equations are written in TeX, which the PDF shows typeset. This page at LibreTexts.

    Practice Makes Perfect

    Simplify Expressions with Exponents

    In the following exercises, simplify each expression with exponents.

    Exercise 1
    1. \(8^3\)
    2. \(8^1\)
    3. \((\frac{3}{4})^3\)
    4. \((\frac{2}{3})^-1\)
    Answer
    1. 512
    2. 8
    3. \(\frac{27}{64}\)
    4. \(\frac{3}{2}\)
    Exercise 2
    1. \((−4)^2\)
    2. \(−4^2\)
    Answer
    1. 16
    2. −16

    Simplify Expressions Using the Product Property for Exponents

    In the following exercises, simplify each expression using the Product Property for Exponents. Leave all answers in terms of positive exponents.

    Exercise 3
    1. \(3^{10}·3^6\)
    2. \(5·5^{4}\)
    Answer
    1. \(3^{16}\)
    2. \(5^5\)
    Exercise 4
    1. \(w^5·w\)
    2. \(u^{41}·u^{53}\)
    Answer
    1. \(w^6\)
    2. \(u^{94}\)
    Exercise 5

    \(y·y^3·y^5\)

    Answer

    \(y^9\)

    Simplify Expressions Using the Power Property for Exponents

    In the following exercises, simplify each expression using the Power Property for Exponents.

    Exercise 6
    1. \((b^2)^7\)
    2. \((3^8)^2\)
    Answer
    1. \(b^{14}\)
    2. \(3^{16}\)

    Simplify Expressions Using the Product to a Power Property

    In the following exercises, simplify each expression using the Product to a Power Property.

    Exercise 7
    1. \((5x)^2\)
    2. \((4ab)^2\)
    Answer
    1. \(25x^2\)
    2. \(16a^{2}b^{2}\)

    Simplify Expressions by Applying Several Properties

    In the following exercises, simplify each expression.

    Exercise 8
    1. \((w^4)^3·(w^5)^2\)
    2. \((2xy^4)^5\)
    Answer
    1. \(w^{22}\)
    2. \(32x^{5}y^{20}\)
    Exercise 9

    \((−2a^{3}b^2)^4\)

    Answer

    \(16a^{12}b^8\)

    Exercise 10

    \((\frac{2}{3}x^{2}y)^3\)

    Answer

    \(\frac{8}{27}x^{6}y^3\)

    Simplify Expressions Using the Quotient Property for Exponents

    In the following exercises, simplify.

    Exercise 11
    1. \(\dfrac{y^{20}}{y^{10}}\)
    2. \(\dfrac{7^{16}}{7^{2}}\)
    Answer
    1. \(y^{10}\)
    2. \(7^{14}\)

    Simplify Expressions with Zero Exponents

    In the following exercises, simplify.

    Exercise 12
    1. \(13^0\)
    2. \(k^{0}\)
    Answer
    1. 1
    2. 1

    Simplify Expressions Using the Quotient to a Power Property

    In the following exercises, simplify.

    Exercise 13
    1. \(\left(\dfrac{2}{5}\right)^{2}\)
    2. \(\left(\dfrac{x}{3}\right)^{4}\)
    3. \(\left(\dfrac{a}{b}\right)^{5}\)
    Answer
    1. \(\dfrac{4}{25}\)
    2. \(\dfrac{x^{4}}{81}\)
    3. \(\left(\dfrac{a}{b}\right)^{5}\)
    Exercise 14
    1. \(\left(\dfrac{a}{3 b}\right)^{4}\)
    2. \(\left(\dfrac{10}{3 q}\right)^{4}\)
    Answer
    1. \(\dfrac{x^{3}}{8 y^{3}}\)
    2. \(\dfrac{10,000}{81 q^{4}}\)

    Simplify Expressions by Applying Several Properties

    In the following exercises, simplify.

    Exercise 15

    \(\dfrac{\left(p^{3}\right)^{4}}{p^{5}}\)

    Answer

    \(p^{7}\)

    Exercise 16

    \(\left(\dfrac{q^{8}}{q^{2}}\right)^{3}\)

    Answer

    \(q^{18}\)

    Exercise 17

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

    Answer

    \(\dfrac{27 m^{15}}{125 n^{3}}\)

    Exercise 18

    \(\left(\dfrac{j^{2} j^{5}}{j^{4}}\right)^{3}\)

    Answer

    \(j^{9}\)

    Exercise 19

    \(\dfrac{54 x^{9} y^{3}}{-18 x^{6} y^{15}}\)

    Answer

    \(-\dfrac{3 x^{3}}{y^{12}}\)

    Exercise 20

    \(\dfrac{20 m^{8} n^{4}}{30 m^{5} n^{9}}\)

    Answer

    \(\dfrac{2 m^{3}}{3 n^{5}}\)

    Use the Definition of a Negative Exponent

    In the following exercises, simplify each expression.

    Exercise 21

    ⓐ \(a^{−2}\)

    ⓑ \(10^{−3}\)

    ⓒ \(\dfrac{1}{c^{−5}}\)

    ⓓ \(\dfrac{1}{3^{−2}}\)

    Answer

    ⓐ \(\dfrac{1}{a^{2}}\)

    ⓑ \(\dfrac{1}{1000}\)

    ⓒ \(c^{5}\)

    ⓓ \(9\)

    Exercise 22

    ⓐ \(\left(\dfrac{5}{8}\right)^{-2}\)

    ⓑ \(\left(−\dfrac{b}{a}\right)^{−2}\)

    Answer

    ⓐ \(\dfrac{64}{25}\)

    ⓑ \(\dfrac{a^{2}}{b^{2}}\)

    Exercise 23

    ⓐ \((−5)^{−2}\)

    ⓑ \(−5^{−2}\)

    ⓒ \(\left(−\dfrac{1}{5}\right)^{−2}\)

    ⓓ \(−\left(\dfrac{1}{5}\right)^{−2}\)

    Answer

    ⓐ \(\dfrac{1}{25}\)

    ⓑ \(−\dfrac{1}{25}\)

    ⓒ \(25\)

    ⓓ \(−25\)

    Exercise 24

    ⓐ \(3·5^{−1}\)

    ⓑ \((3·5)^{−1}\)

    Answer

    ⓐ \(\dfrac{3}{5}\)

    ⓑ \(\dfrac{1}{15}\)

    In the following exercises, simplify each expression by applying several properties.

    Exercise 25

    ⓐ \((pt^2)^{−1}\)

    ⓑ \(\dfrac{(m^2)(n^3)}{(m^5)(n^7)}\)

    ⓒ \(\left(\dfrac{2y^2}{x^5}\right)^{−3}\)

    Answer

    ⓐ \(\dfrac{1}{pt^2}\)

    ⓑ \(\dfrac{1}{m^{3}n^4}\)

    ⓒ \(\dfrac{x^{15}}{8y^6}\)

    Simplifying Expressions with Roots

    In the following exercises, simplify.

    Exercise 26

    a. \(\sqrt{64}\)

    b. \(-\sqrt{81}\)

    Answer

    a. \(8\)

    b. \(-9\)

    Exercise 27

    a. \(\sqrt{\frac{4}{9}}\)

    b. \(-\sqrt{0.01}\)

    Answer

    a. \(\frac{2}{3}\)

    b. \(-0.1\)

    Exercise 28

    a. \(\sqrt{-121}\)

    b. \(-\sqrt{289}\)

    Answer

    a. not a real number

    b. \(-17\)

    Exercise 29

    a. \(\sqrt[3]{216}\)

    b. \(\sqrt[4]{256}\)

    Answer

    a. \(6\)

    b. \(4\)

    Exercise 30

    15. a. \(\sqrt[3]{-8}\) b. \(\sqrt[4]{-81}\) c. \(\sqrt[5]{-32}\)

    Answer

    a. \(-2\) b. not a real number c. \(-2\)

    This page titled 1.3.H: Homework was last modified on Mon, 06 Jul 2026 03:20:47 GMT and is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Kiersten Purves.