# 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}

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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}

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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}

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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}

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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}

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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}

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

$$\dfrac{125x^9}{y^{12}}$$

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