# rational exponents simplify

In this section we are going to be looking at rational exponents. This is the currently selected item. RATIONAL EXPONENTS. The Power Property for Exponents says that $$\left(a^{m}\right)^{n}=a^{m \cdot n}$$ when $$m$$ and $$n$$ are whole numbers. Watch the recordings here on Youtube! We will rewrite each expression first using $$a^{-n}=\frac{1}{a^{n}}$$ and then change to radical form. To simplify with exponents, don't feel like you have to work only with, or straight from, the rules for exponents. Simplify Rational Exponents. The general form for converting between a radical expression with a radical symbol and one with a rational exponent is How To: Given an expression with a rational exponent, write the expression as a radical. Now that we have looked at integer exponents we need to start looking at more complicated exponents. The cube root of −8 is −2 because (−2) 3 = −8. Examples: x1 = x 71 = 7 531 = 53 01 = 0 Nine Exponent Rules Evaluations. Come to Algebra-equation.com and read and learn about operations, mathematics and … Using Rational Exponents. We can look at $$a^{\frac{m}{n}}$$ in two ways. Radicals - Rational Exponents Objective: Convert between radical notation and exponential notation and simplify expressions with rational exponents using the properties of exponents. $$\left(27 u^{\frac{1}{2}}\right)^{\frac{2}{3}}$$. Except where otherwise noted, textbooks on this site They may be hard to get used to, but rational exponents can actually help simplify some problems. We do not show the index when it is $$2$$. OpenStax is part of Rice University, which is a 501(c)(3) nonprofit. If $$a$$ and $$b$$ are real numbers and $$m$$ and $$n$$ are rational numbers, then, $$\frac{a^{m}}{a^{n}}=a^{m-n}, a \neq 0$$, $$\left(\frac{a}{b}\right)^{m}=\frac{a^{m}}{b^{m}}, b \neq 0$$. When we simplify radicals with exponents, we divide the exponent by the index. ${x}^{\frac{2}{3}}$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$, [ "article:topic", "Rational Exponents", "license:ccby", "showtoc:no", "transcluded:yes", "authorname:openstaxmarecek", "source[1]-math-5169" ], $$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$, 5.4: Add, Subtract, and Multiply Radical Expressions, Simplify Expressions with $$a^{\frac{1}{n}}$$, Simplify Expressions with $$a^{\frac{m}{n}}$$, Use the Properties of Exponents to Simplify Expressions with Rational Exponents, Simplify expressions with $$a^{\frac{1}{n}}$$, Simplify expressions with $$a^{\frac{m}{n}}$$, Use the properties of exponents to simplify expressions with rational exponents, $$\sqrt{\left(\frac{3 a}{4 b}\right)^{3}}$$, $$\sqrt{\left(\frac{2 m}{3 n}\right)^{5}}$$, $$\left(\frac{2 m}{3 n}\right)^{\frac{5}{2}}$$, $$\sqrt{\left(\frac{7 x y}{z}\right)^{3}}$$, $$\left(\frac{7 x y}{z}\right)^{\frac{3}{2}}$$, $$x^{\frac{1}{6}} \cdot x^{\frac{4}{3}}$$, $$\frac{x^{\frac{2}{3}}}{x^{\frac{5}{3}}}$$, $$y^{\frac{3}{4}} \cdot y^{\frac{5}{8}}$$, $$\frac{d^{\frac{1}{5}}}{d^{\frac{6}{5}}}$$, $$\left(32 x^{\frac{1}{3}}\right)^{\frac{3}{5}}$$, $$\left(x^{\frac{3}{4}} y^{\frac{1}{2}}\right)^{\frac{2}{3}}$$, $$\left(81 n^{\frac{2}{5}}\right)^{\frac{3}{2}}$$, $$\left(a^{\frac{3}{2}} b^{\frac{1}{2}}\right)^{\frac{4}{3}}$$, $$\frac{m^{\frac{2}{3}} \cdot m^{-\frac{1}{3}}}{m^{-\frac{5}{3}}}$$, $$\left(\frac{25 m^{\frac{1}{6}} n^{\frac{11}{6}}}{m^{\frac{2}{3}} n^{-\frac{1}{6}}}\right)^{\frac{1}{2}}$$, $$\frac{u^{\frac{4}{5}} \cdot u^{-\frac{2}{5}}}{u^{-\frac{13}{5}}}$$, $$\left(\frac{27 x^{\frac{4}{5}} y^{\frac{1}{6}}}{x^{\frac{1}{5}} y^{-\frac{5}{6}}}\right)^{\frac{1}{3}}$$. 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