Big Fact: Radicals are really just exponents! Whoahhhhh!
Proof: Consider the equation $(\sqrt{x})^2=x$.
Using rules of exponents (gulp!), we now simplify $(x^{\frac{1}{2}})^2$
The Facts
1. $\sqrt{x}$ is a positive quantity that when squared gives $x.$
2. $x^{\frac{1}{2}}$ is a positive quantity that when squared gives $x.$
Conclusion?
Conclusion:
$\displaystyle \Huge{x^{\frac{1}{2}}=\sqrt{x}}$
Duhn! Duhn! Duuuuhhhun!
Definition: Rational Exponents.
$$x^{\frac{m}{n}}=\sqrt[n]{x^m}$$
Equivalently,
$$x^{\frac{m}{n}}=\left(\sqrt[n]{x}\right)^m$$
Examples: Rewrite in radical form and simplify.
$16^{\frac{3}{4}}$
$$
\begin{array}{lll}
\displaystyle 16^{\frac{3}{4}}&\displaystyle=\left(\sqrt[4]{16}\right)^3 &\mbox{}\\
\displaystyle &\displaystyle=\left(2\right)^3 &\mbox{}\\
\displaystyle &\displaystyle=8 &\mbox{}\\
\end{array}
$$
Note that $8=2\cdot 2 \cdot 2$ is $\displaystyle \frac{3}{4}$ of the factors of $16.$
$$
16 = 2 \cdot \underbrace{\color{magenta}{2 \cdot 2 \cdot 2}}_{=8}
$$
$\displaystyle \left(\frac{125}{216}\right)^{\frac{2}{3}}$
$$
\begin{array}{lll}
\displaystyle \left(\frac{125}{216}\right)^{\frac{2}{3}}&\displaystyle=\left(\sqrt[3]{\frac{125}{216}}\right)^2 &\mbox{}\\
\displaystyle &\displaystyle=\left(\frac{\sqrt[3]{125}}{\sqrt[3]{216}}\right)^2 &\mbox{optional step}\\
\displaystyle &\displaystyle=\left(\frac{5}{6}\right)^2 &\mbox{}\\
\displaystyle &\displaystyle=\frac{25}{36} &\mbox{}\\
\end{array}
$$
Good News! All the familiar (I hope) rules of exponents apply to rational exponents!
The Product Rule: $x^m \cdot x^n=x^{m+n}$
The Quotient Rule: $\frac{x^m}{x^n}=x^{m-n}$
The Power Rule: $(x^m)^n=x^{m \cdot n}$
Power of a Product: $(xy)^n=x^n y^n$
Power of a Quotient: $\left(\frac{x}{y}\right)^n=\frac{x^n}{y^n}$
Upstairs/Downstairs 1: $\frac{1}{x^{-n}}=x^{n}$
Upstairs/Downstairs 2: $x^{-n}=\frac{1}{x^n}$
Fraction Flip: $\left(\frac{x}{y}\right)^{-n}=\left(\frac{y}{x}\right)^{n}$
Examples: Use exponent rules to rewrite in radical form and simplify.
$36^{-\frac{3}{2}}$
$$
\begin{array}{lll}
\displaystyle 36^{-\frac{3}{2}}&\displaystyle=\frac{1}{36^{\frac{3}{2}}} &\mbox{}\\
\displaystyle &\displaystyle=\frac{1}{\left(\sqrt{36}\right)^3} &\mbox{}\\
\displaystyle &\displaystyle=\frac{1}{\left(6\right)^3} &\mbox{}\\
\displaystyle &\displaystyle=\frac{1}{216} &\mbox{}\\
\end{array}
$$
$(0.49)^{-\frac{3}{2}}$
$$
\begin{array}{lll}
\displaystyle (0.49)^{-\frac{3}{2}}&\displaystyle=\left(\frac{49}{100}\right)^{-\frac{3}{2}} &\mbox{}\\
\displaystyle &\displaystyle=\left(\frac{100}{49}\right)^{\frac{3}{2}} &\mbox{}\\
\displaystyle &\displaystyle=\left(\sqrt{\frac{100}{49}}\right)^3 &\mbox{}\\
\displaystyle &\displaystyle=\left(\frac{10}{7}\right)^3 &\mbox{}\\
\displaystyle &\displaystyle=\frac{1000}{343} &\mbox{}\\
\end{array}
$$
Examples: Rewrite the radical in exponential form.
$\displaystyle \sqrt[5]{a^2}$
$$\sqrt[5]{a^2}=a^{\frac{2}{5}}$$
$\displaystyle \left(\sqrt[3]{5ab}\right)^5$
$$\left(\sqrt[3]{5ab}\right)^5=(5ab)^{\frac{5}{3}}$$
$\displaystyle\sqrt{\left(\frac{7xy}{z}\right)^3}$
$$
\sqrt{\left(\frac{7xy}{z}\right)^3}=\left(\frac{7xy}{z}\right)^{\frac{3}{2}}
$$
$\displaystyle \sqrt[b]{(xy)^2}$
$$
\sqrt[b]{(xy)^2}=(xy)^{\frac{2}{b}}
$$
DON'T Write this Down in Your Notes!
Notice that our Big Facts about radicals are just the rules of exponents at work:
$$\sqrt[n]{ab}=(ab)^{1/n}=a^{1/n}b^{1/n}=\sqrt[n]{a}\sqrt[n]{b}$$ $$\sqrt[n]{\frac{a}{b}}=\left(\frac{a}{b}\right)^{1/n}=\frac{a^{1/n}}{b^{1/n}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}$$
A Veritable Variety of Examples
$\displaystyle a^{\frac{5}{2}}a^{\frac{3}{4}}$
$$
\begin{array}{lll}
\displaystyle a^{\frac{5}{2}}a^{\frac{3}{4}}&\displaystyle= a^{\frac{5}{2}+\frac{3}{4}}&\mbox{}\\
\displaystyle &\displaystyle=a^{\frac{13}{4}} &\mbox{}\\
\end{array}
$$
$\displaystyle \left(27s^{\frac{3}{4}}t^{\frac{1}{2}}\right)^{\frac{2}{3}}$
$$
\begin{array}{lll}
\displaystyle \left(27s^{\frac{3}{4}}t^{\frac{1}{2}}\right)^{\frac{2}{3}}&\displaystyle=\left(27\right)^{\frac{2}{3}}\left(s^{\frac{3}{4}}\right)^{\frac{2}{3}}\left(t^{\frac{1}{2}}\right)^{\frac{2}{3}} &\mbox{a power of a product is the product of the powers!}\\
\displaystyle &\displaystyle=9s^{\frac{3}{4}\cdot \frac{2}{3}}t^{\frac{1}{2}\cdot \frac{2}{3}} &\mbox{power of a power means multiply}\\
\displaystyle &\displaystyle=9s^{\frac{1}{2}}t^{\frac{1}{3}} &\mbox{simplify exponents}\\
\end{array}
$$
$\displaystyle \left(\frac{16 z^{\frac{5}{2}}}{625 z^{\frac{3}{5}}}\right)^{\frac{3}{4}}$
$$
\begin{array}{lll}
\displaystyle \left(\frac{16 z^{\frac{5}{2}}}{625 z^{\frac{3}{5}}}\right)^{\frac{3}{4}}&\displaystyle=\frac{\left(16 z^{\frac{5}{2}}\right)^{\frac{3}{4}}}{\left(625 z^{\frac{3}{5}}\right)^{\frac{3}{4}}} &\mbox{a power of a quotient is the quotient of the powers!}\\
\displaystyle &\displaystyle=\frac{\left(16\right)^{\frac{3}{4}} \left(z^{\frac{5}{2}}\right)^{\frac{3}{4}}}{\left(625\right)^{\frac{3}{4}} \left(z^{\frac{3}{5}}\right)^{\frac{3}{4}}} &\mbox{a power of a product is the product of the powers!}\\
\displaystyle &\displaystyle=\frac{8 z^{\frac{5}{2}\cdot \frac{3}{4}}}{125 z^{\frac{3}{5}\cdot \frac{3}{4}}} &\mbox{a power of a power means multiply}\\
\displaystyle &\displaystyle=\frac{8 z^{\frac{15}{8}}}{125 z^{\frac{9}{20}}} &\mbox{}\\
\displaystyle &\displaystyle=\frac{8 z^{\frac{15}{8}-\frac{9}{20}}}{125} &\mbox{quotient rule}\\
\displaystyle &\displaystyle=\frac{8 z^{\frac{57}{40}}}{125} &\mbox{}\\
\end{array}
$$
Scenic Alternative
$$
\begin{array}{lll}
\displaystyle \left(\frac{16 z^{\frac{5}{2}}}{625 z^{\frac{3}{5}}}\right)^{\frac{3}{4}}&\displaystyle= \left(\frac{16 z^{\frac{5}{2}-\frac{3}{5}}}{625}\right)^{\frac{3}{4}} &\mbox{do quotient rule first!}\\
\displaystyle &\displaystyle= \left(\frac{16 z^{\frac{19}{10}}}{625}\right)^{\frac{3}{4}} &\mbox{}\\
\displaystyle &\displaystyle= \frac{\left(16 z^{\frac{19}{10}}\right)^{\frac{3}{4}}}{625^{\frac{3}{4}}} &\mbox{a power of a quotient is the quotient of the powers!}\\
\displaystyle &\displaystyle= \frac{\left(16\right)^{\frac{3}{4}}\left(z^{\frac{19}{10}}\right)^{\frac{3}{4}}}{125} &\mbox{a power of a product is the product of the powers!}\\
\displaystyle &\displaystyle= \frac{8z^{\frac{19}{10}\cdot \frac{3}{4}}}{125} &\mbox{a power of a power means multiply}\\
\displaystyle &\displaystyle= \frac{8z^{\frac{57}{40}}}{125} &\mbox{same answer!}\\
\end{array}
$$
$\displaystyle \frac{\left(125 x^{\frac{3}{2}} y^{\frac{1}{4}}\right)^{\frac{1}{2}}\left(5 x^{\frac{3}{5}} y^{\frac{1}{5}}\right)^{\frac{1}{2}}}{\left(x^{-\frac{3}{5}}y^{-\frac{1}{4}}\right)^{-\frac{5}{3}}}$
$$
\begin{array}{lll}
\displaystyle \frac{\left(125 x^{\frac{3}{2}} y^{\frac{1}{4}}\right)^{\frac{1}{2}}\left(5 x^{\frac{3}{5}} y^{\frac{1}{5}}\right)^{\frac{1}{2}}}{\left(x^{-\frac{3}{5}}y^{-\frac{1}{4}}\right)^{-\frac{5}{3}}}&\displaystyle=\frac{\left(125\right)^{\frac{1}{2}} \left(x^{\frac{3}{2}}\right)^{\frac{1}{2}} \left(y^{\frac{1}{4}}\right)^{\frac{1}{2}}\left(5\right)^{\frac{1}{2}} \left(x^{\frac{3}{5}}\right)^{\frac{1}{2}} \left(y^{\frac{1}{5}}\right)^{\frac{1}{2}}}{\left(x^{-\frac{3}{5}}\right)^{-\frac{5}{3}}\left(y^{-\frac{1}{4}}\right)^{-\frac{5}{3}}} &\mbox{a power of a product is the product of the powers!}\\
\displaystyle &\displaystyle=\frac{\left(125\right)^{\frac{1}{2}} x^{\frac{3}{2}\cdot \frac{1}{2}} y^{\frac{1}{4}\cdot \frac{1}{2}}\left(5\right)^{\frac{1}{2}} x^{\frac{3}{5}\cdot \frac{1}{2}} y^{\frac{1}{5}\cdot \frac{1}{2}}}{x^{-\frac{3}{5}\cdot\left(-\frac{5}{3}\right)}y^{-\frac{1}{4}\cdot\left(-\frac{5}{3}\right)}} &\mbox{a power of a power means multiply}\\
\displaystyle &\displaystyle=\frac{\left(5\right)^{\frac{3}{2}} x^{\frac{3}{4}} y^{\frac{1}{8}}\left(5\right)^{\frac{1}{2}} x^{\frac{3}{10}} y^{\frac{1}{10}}}{x^{1}y^{\frac{5}{12}}} &\mbox{recognize like bases; simplify}\\
\displaystyle &\displaystyle=\frac{\left(5\right)^{\frac{3}{2}} \left(5\right)^{\frac{1}{2}} x^{\frac{3}{4}}x^{\frac{3}{10}} y^{\frac{1}{8}} y^{\frac{1}{10}} }{xy^{\frac{5}{12}}} &\mbox{get like bases together}\\
\displaystyle &\displaystyle=\frac{\left(5\right)^{\frac{3}{2}+\frac{1}{2}} x^{\frac{3}{4}+\frac{3}{10}} y^{\frac{1}{8}+\frac{1}{10}} }{xy^{\frac{5}{12}}} &\mbox{like bases in product add exponents}\\
\displaystyle &\displaystyle=\frac{\left(5\right)^{2} x^{\frac{21}{20}} y^{\frac{9}{40}}}{xy^{\frac{5}{12}}} &\mbox{}\\
\displaystyle &\displaystyle=25 x^{\frac{21}{20}-1} y^{\frac{9}{40}-\frac{5}{12}} &\mbox{like bases in quotient subtract exponents}\\
\displaystyle &\displaystyle=25 x^{\frac{1}{20}} y^{-\frac{23}{120}} &\mbox{}\\
\displaystyle &\displaystyle=\frac{25 x^{\frac{1}{20}}}{y^{\frac{23}{120}}} &\mbox{negative means put factors downstairs}\\
\end{array}
$$