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Section A.2 – Exponents and Radicals

Section Details.
  • Definition and properties of exponents
  • Simplifying expressions with exponents
  • Definition and properties of radicals
  • Evaluating and simplifying expressions with radicals
  • Rationalizing denominators
  • Definition and properties of fractional exponents
  • Evaluating and simplifying expressions with fractional exponents


Practice Problems


Directions. The following are review problems for the section. We recommend you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it. 

  1. Simplify the following expression. Write your answer so that each variable appears at most once, and all exponents are positive. \[-\dfrac{12(xy^{-1})^3(x^{-2}y^2)^2}{20(x^{-4})^{-2}(xy^{-3})^2}\]

    Answer: \(-\dfrac{3y^7}{5x^{11}}\)

    Solution:
    \[\begin{align}
    -\dfrac{12(xy^{-1})^3(x^{-2}y^2)^2}{20(x^{-4})^{-2}(xy^{-3})^2}
    &=-\dfrac{12(x^3y^{-3})(x^{-4}y^4)}{20(x^8)(x^2y^{-6})}\\
    &=-\dfrac{3\cdot4x^{-1}y}{5\cdot4x^{10}y^{-6}}\\
    &=-\dfrac{3y^7}{5x^{11}}
    \end{align}\]


    To see the full video page and find related videos, click the following link.
    WIR1 20B M150 V7

  2. Simplify each radical expression
    1. \(\dfrac{\sqrt[3]{-24x^4y^2z^6}}{\sqrt[3]{81xy}}\)

      Answer:  \(\dfrac{-2xz^2}{3}\cdot\sqrt[3]{y}\)

      Solution:
      \[\begin{align}
      \dfrac{\sqrt[3]{-24x^4y^2z^6}}{\sqrt[3]{81xy}}
      &=\dfrac{-2xz^2\sqrt[3]{3xy^2}}{3\sqrt[3]{3xy}}\\
      &=\dfrac{-2xz^2}{3}\cdot\sqrt[3]{\dfrac{3xy^2}{3xy}}\\
      &=\dfrac{-2xz^2}{3}\cdot\sqrt[3]{y}
      \end{align}\]


      To see the full video page and find related videos, click the following link.
      WIR1 20B M150 V8

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

      Answer: \((3|x|-1)\sqrt{x}\)

      Solution:
      \[\begin{align}
      \sqrt{x^3}+\sqrt{4x^3}-\sqrt{x}
      &=|x|\sqrt{x}+2|x|\sqrt{x}-\sqrt{x}\\
      &=(3|x|-1)\sqrt{x}
      \end{align}\]



      To see the full video page and find related videos, click the following link.
      WIR1 20B M150 V9

  3. Rationalize the denominator.
    1. \(\dfrac{x-y}{\sqrt{x}-\sqrt{y}}\)

      Answer: \(\sqrt{x}+\sqrt{y}\)

      Solution: 
      \[\begin{align}
      \dfrac{x-y}{\sqrt{x}-\sqrt{y}}
      &=\dfrac{x-y}{\sqrt{x}-\sqrt{y}}\cdot\dfrac{\sqrt{x}+\sqrt{y}}{\sqrt{x}+\sqrt{y}}\\
      &=\dfrac{(x-y)\cdot(\sqrt{x}+\sqrt{y})}{(\sqrt{x})^2-(\sqrt{y})^2}\\
      &=\dfrac{(x-y)\cdot(\sqrt{x}+\sqrt{y})}{x-y}\\
      &=\sqrt{x}+\sqrt{y}
      \end{align}\]



      To see the full video page and find related videos, click the following link.
      WIR1 20B M150 V11

    2. \(\dfrac{4\sqrt{6}+3\sqrt{3}}{3\sqrt{6}-4\sqrt{3}}\)

      Answer: \(18+\dfrac{25}{2}\sqrt{2}\)

      Solution:
      \[\begin{align}
      \frac{4\sqrt{6}+3\sqrt{3}}{3\sqrt{6}-4\sqrt{3}}
      &=\frac{4\sqrt{6}+3\sqrt{3}}{3\sqrt{6}-4\sqrt{4}}\cdot\dfrac{3\sqrt{6}+4\sqrt{3}}{3\sqrt{6}+4\sqrt{3}}\\
      &=\frac{72+16\sqrt{18}+9\sqrt{18}+36}{(3\sqrt{6})^2-(4\sqrt{3})^2}\\
      &=\frac{108+75\sqrt{2}}{54-48}\\
      &=\dfrac{108+75\sqrt{2}}{6}\\
      &=18+\dfrac{25}{2}\sqrt{2}
      \end{align} \]



      To see the full video page and find related videos, click the following link.
      WIR1 20B M150 V10

  4. ​Simplify the following expression. \[\left(\dfrac{a^{5/4}\cdot a^{-3/8}}{a^{-3/4}}\right)^{2/3}\]

    Answer: \(a^{13/12}\)

    Solution: 
    \[\begin{align}
    \left(\dfrac{a^{5/4}\cdot a^{-3/8}}{a^{-3/4}}\right)^{2/3}
    &=\left(\dfrac{a^{10/8}\cdot a^{-3/8}}{a^{-6/8}}\right)^{2/3}\\
    &=\left(\dfrac{a^{7/8}}{a^{-6/8}}\right)^{2/3}\\
    &=\left(a^{13/8}\right)^{2/3}\\
    &=a^{26/24}\\
    &=a^{13/12}
    \end{align}\]



    To see the full video page and find related videos, click the following link.
    WIR1 20B M150 V12

  5. Simplify the expression with positive exponents \[\displaystyle{\left(\frac{x^{-3}y^4}{5}\right)^{-3}}\]

    \(\dfrac{125x^9}{y^{12}}\)



    To see the full video page and find related videos, click the following link.
    WIR8 20B M150 V01