Section A.2 – Exponents and Radicals
- 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
Exercises
Directions: You should try to solve each problem first, and then click "Reveal Answer" to check your answer. You can click "Watch Video" if you need help with a problem.
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:
\[\begin{align}
-\dfrac{12(xy^{-1})^3(x^{-2}y^2)^2}{20(x^{-4})^{-2}(xy^{-3})^2}
&=-\dfrac{12(x^3y^{-3})(x^{-4}y^4)}{20(x^8)(x^2y^{-6})}\\
&=-\dfrac{3\cdot4x^{-1}y}{5\cdot4x^{10}y^{-6}}\\
&=-\dfrac{3y^7}{5x^{11}}
\end{align}\]
Solution:
\[\begin{align}
-\dfrac{12(xy^{-1})^3(x^{-2}y^2)^2}{20(x^{-4})^{-2}(xy^{-3})^2}
&=-\dfrac{12(x^3y^{-3})(x^{-4}y^4)}{20(x^8)(x^2y^{-6})}\\
&=-\dfrac{3\cdot4x^{-1}y}{5\cdot4x^{10}y^{-6}}\\
&=-\dfrac{3y^7}{5x^{11}}
\end{align}\]
2. Simplify the radical expression: \(\dfrac{\sqrt[3]{-24x^4y^2z^6}}{\sqrt[3]{81xy}}\)
Answer: \(\dfrac{-2xz^2}{3}\cdot\sqrt[3]{y}\)
Solution:
\[\begin{align}
\dfrac{\sqrt[3]{-24x^4y^2z^6}}{\sqrt[3]{81xy}}
&=\dfrac{-2xz^2\sqrt[3]{3xy^2}}{3\sqrt[3]{3xy}}\\
&=\dfrac{-2xz^2}{3}\cdot\sqrt[3]{\dfrac{3xy^2}{3xy}}\\
&=\dfrac{-2xz^2}{3}\cdot\sqrt[3]{y}
\end{align}\]
Solution:
\[\begin{align}
\dfrac{\sqrt[3]{-24x^4y^2z^6}}{\sqrt[3]{81xy}}
&=\dfrac{-2xz^2\sqrt[3]{3xy^2}}{3\sqrt[3]{3xy}}\\
&=\dfrac{-2xz^2}{3}\cdot\sqrt[3]{\dfrac{3xy^2}{3xy}}\\
&=\dfrac{-2xz^2}{3}\cdot\sqrt[3]{y}
\end{align}\]
3. Simplify the radical expression: \(\sqrt{x^3}+\sqrt{4x^3}-\sqrt{x}\)
Answer: \((3|x|-1)\sqrt{x}\)
Solution:
\[\begin{align}
\sqrt{x^3}+\sqrt{4x^3}-\sqrt{x}
&=|x|\sqrt{x}+2|x|\sqrt{x}-\sqrt{x}\\
&=(3|x|-1)\sqrt{x}
\end{align}\]
Solution:
\[\begin{align}
\sqrt{x^3}+\sqrt{4x^3}-\sqrt{x}
&=|x|\sqrt{x}+2|x|\sqrt{x}-\sqrt{x}\\
&=(3|x|-1)\sqrt{x}
\end{align}\]
4. Rationalize the denominator: \(\dfrac{4\sqrt{6}+3\sqrt{3}}{3\sqrt{6}-4\sqrt{3}}\)
Answer: \(18+\dfrac{25}{2}\sqrt{2}\)
Solution:
\[\begin{align}
\frac{4\sqrt{6}+3\sqrt{3}}{3\sqrt{6}-4\sqrt{3}}
&=\frac{4\sqrt{6}+3\sqrt{3}}{3\sqrt{6}-4\sqrt{4}}\cdot\dfrac{3\sqrt{6}+4\sqrt{3}}{3\sqrt{6}+4\sqrt{3}}\\
&=\frac{72+16\sqrt{18}+9\sqrt{18}+36}{(3\sqrt{6})^2-(4\sqrt{3})^2}\\
&=\frac{108+75\sqrt{2}}{54-48}\\
&=\dfrac{108+75\sqrt{2}}{6}\\
&=18+\dfrac{25}{2}\sqrt{2}
\end{align} \]
5. Rationalize the denominator: \(\dfrac{x-y}{\sqrt{x}-\sqrt{y}}\)
Answer: \(\sqrt{x}+\sqrt{y}\)
Solution:
\[\begin{align}
\dfrac{x-y}{\sqrt{x}-\sqrt{y}}
&=\dfrac{x-y}{\sqrt{x}-\sqrt{y}}\cdot\dfrac{\sqrt{x}+\sqrt{y}}{\sqrt{x}+\sqrt{y}}\\
&=\dfrac{(x-y)\cdot(\sqrt{x}+\sqrt{y})}{(\sqrt{x})^2-(\sqrt{y})^2}\\
&=\dfrac{(x-y)\cdot(\sqrt{x}+\sqrt{y})}{x-y}\\
&=\sqrt{x}+\sqrt{y}
\end{align}\]
6. Simplify the following expression. \[\left(\dfrac{a^{5/4}\cdot a^{-3/8}}{a^{-3/4}}\right)^{2/3}\]
Answer: \(a^{13/12}\)
Solution:
\[\begin{align}
\left(\dfrac{a^{5/4}\cdot a^{-3/8}}{a^{-3/4}}\right)^{2/3}
&=\left(\dfrac{a^{10/8}\cdot a^{-3/8}}{a^{-6/8}}\right)^{2/3}\\
&=\left(\dfrac{a^{7/8}}{a^{-6/8}}\right)^{2/3}\\
&=\left(a^{13/8}\right)^{2/3}\\
&=a^{26/24}\\
&=a^{13/12}
\end{align}\]
Solution:
\[\begin{align}
\left(\dfrac{a^{5/4}\cdot a^{-3/8}}{a^{-3/4}}\right)^{2/3}
&=\left(\dfrac{a^{10/8}\cdot a^{-3/8}}{a^{-6/8}}\right)^{2/3}\\
&=\left(\dfrac{a^{7/8}}{a^{-6/8}}\right)^{2/3}\\
&=\left(a^{13/8}\right)^{2/3}\\
&=a^{26/24}\\
&=a^{13/12}
\end{align}\]
7. Simplify the expression with positive exponents \(\displaystyle{\left(\frac{x^{-3}y^4}{5}\right)^{-3}}\)
\(\dfrac{125x^9}{y^{12}}\)