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Mathematics LibreTexts

1.4.1: Exercises 1.4

  • Page ID
    62181
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    Terms and Concepts

    Exercise \(\PageIndex{1}\)

    Do exponent rules apply to root functions? Explain.

    Answer

    Yes, a root function is just a power function with a fractional exponent.

    Exercise \(\PageIndex{2}\)

    Explain why a negative exponents moves the term to the denominator and gives it a positive exponent.

    Answer

    A positive exponent means we are multiplying that term repeatedly, a negative exponent means we are dividing by that term repeatedly.

    Exercise \(\PageIndex{3}\)

    Is \(3x(2x+3)^{-5/3}\) in radical or exponential form?

    Answer

    Exponential form

    Exercise \(\PageIndex{4}\)

    Is \(\displaystyle \frac{3x}{\sqrt[3]{(2x+3)^5}}\) in radical or exponential form?

    Answer

    Radical form

    Problems

    In exercises \(\PageIndex{5}\) - \(\PageIndex{7}\), write the given term without using exponents.

    Exercise \(\PageIndex{5}\)

    \(\displaystyle (8x_1-5x_2+11)^{-1/3}\)

    Answer

    \(\displaystyle \frac{1}{\sqrt[3]{8x_1-5x_2+11}}\)

    Exercise \(\PageIndex{6}\)

    \(\displaystyle (-2x+y)^{-1/5}\)

    Answer

    \(\displaystyle \frac{1}{\sqrt[5]{-2x+y}}\)

    Exercise \(\PageIndex{7}\)

    \(\displaystyle (5x-2)^{1/4}\)

    Answer

    \(\displaystyle \sqrt[4]{5x-2}\)

    In exercises \(\PageIndex{8}\) - \(\PageIndex{10}\), simplify and write the given term without using radicals.

    Exercise \(\PageIndex{8}\)

    \(\displaystyle \Bigg( \sqrt{x} + \frac{1}{\sqrt{x}} \Bigg)^2\)

    Answer

    \(\displaystyle x + 2 + \frac{1}{x}\)

    Exercise \(\PageIndex{9}\)

    \(\displaystyle (\sqrt{x})^2 + \Bigg(\frac{1}{\sqrt{x}} \Bigg)^2\)

    Answer

    \(\displaystyle x + \frac{1}{x}\)

    Exercise \(\PageIndex{10}\)

    \(\displaystyle \Bigg(\sqrt[3]{x} +1 \Bigg)^3\)

    Answer

    \(\displaystyle x + 3x^{2/3} + 3x^{1/3} + 1\)

    In exercises \(\PageIndex{11}\) - \(\PageIndex{17}\), simplify the given term and write your answer without negative exponents.

    Exercise \(\PageIndex{11}\)

    \(\displaystyle \Bigg( \frac{-5x^{-1/4}y^3}{x^{1/4}y^{1/2}}\Bigg)^2\)

    Answer

    \(\displaystyle \frac{25y^5}{x}, y \neq 0\)

    Exercise \(\PageIndex{12}\)

    \(\displaystyle \Bigg( \frac{-2x^{2/3}y^2}{x^{-2}y^{1/2}}\Bigg)^6\)

    Answer

    \(\displaystyle 64x^{16}y^9; x,y \neq 0\)

    Exercise \(\PageIndex{13}\)

    \(\displaystyle \Bigg( \frac{-3s^{2/3}t^2}{4s^3t^{5/3}}\Bigg)^3\)

    Answer

    \(\displaystyle \frac{-27t}{64s^7}, t\neq 0\)

    Exercise \(\PageIndex{14}\)

    \(\displaystyle -3(x^2+4x+4)^{-4}(2x+4)\)

    Answer

    \(\displaystyle \frac{-6}{(x+2)^7}\)

    Exercise \(\PageIndex{15}\)

    \(\displaystyle \frac{1}{3}(x^4)^{-2/3}(4x^3)\)

    Answer

    \(\displaystyle \frac{4x^{1/3}}{3}\)

    Exercise \(\PageIndex{16}\)

    \(\displaystyle \frac{(e^{x+3})^2}{e^{-x}}\)

    Answer

    \(\displaystyle e^{3x+6}\)

    Exercise \(\PageIndex{17}\)

    \(\displaystyle \frac{e^{x^2-1}}{e^{x+1}}\)

    Answer

    \(\displaystyle e^{x^2-x-2}\)

    In exercises \(\PageIndex{18}\) - \(\PageIndex{20}\), simplify and write the given term in exponential form.

    Exercise \(\PageIndex{18}\)

    \(\displaystyle \frac{4x-1}{\sqrt[3]{(3x+2)^2}}\)

    Answer

    \(\displaystyle (4x-1)(3x+2)^{-2/3}\)

    Exercise \(\PageIndex{19}\)

    \(\displaystyle \sqrt[3]{\Bigg(\frac{e^{4\theta-6}y^2}{e^{\theta}y^{-4}}\Bigg)}\)

    Answer

    \(\displaystyle e^{\theta-2}y^2, y \neq 0\)

    Exercise \(\PageIndex{20}\)

    \(\displaystyle \sqrt[4]{\Bigg(\frac{x^2y^5}{y^{-3}}\Bigg)^2}\)

    Answer

    \(\displaystyle xy^4, y\neq 0\)


    This page titled 1.4.1: Exercises 1.4 is shared under a CC BY-NC license and was authored, remixed, and/or curated by Amy Givler Chapman, Meagan Herald, Jessica Libertini.

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