Radical and Exponent Form
Simple overview of converting radical form to exponential form and vice versa.
Core Formula
To convert back and forth we use the following
\[\sqrt[n]{a^m}=a^{m/n}\]The index $n$ is the root, which becomes the denominator. The power within the radical $m$ becomes the numerator.
Watch this section: naming radical form at 0:00.
Simple Example $\sqrt{y}$
Consider
\[\sqrt{y}\]Since a square root has an unwritten index of $2$, and $y$ has an unwritten power of $1$, the rational exponent is
\[\sqrt{y} = \sqrt[2]{y} = y^{\frac{1}{2}}\]Watch this section: square root of $y$ at 0:30.
Cube Root With A Power
Consider
\[\left(\sqrt[3]{z}\right)^2\]Note:
\[\left(\sqrt[3]{z}\right)^2 = \left(\sqrt[3]{z^2}\right)\]and so
\[\left(\sqrt[3]{z}\right)^2=z^{2/3}\]Watch this section: cube root example at 1:33.
A Larger Index
What happens if the index is much larger?
\[\left(\sqrt[32]{y+1}\right)^5=(y+1)^{5/32}\]Regardless of the size of the root or internal power, the rule still remains the same.
Watch this section: thirty-second root example at 2:14.
Keep Track Of What Is Under The Radical
Consider the following expressions that use $z^3$ and a $+5$ in slightly different ways (mainly how they grouped under the radical).
If the radical covers only $z^3$, then only that part is rewritten:
\[\sqrt{z^3}+5=z^{3/2}+5\]The constant that is added outside the radical remains untouched.
Now, if the $+5$ was grouped with the $z^3$ under the radical, it stays grouped when converting to the exponential form:
\[\sqrt{z^3+5}=(z^3+5)^{1/2}\]Watch this section: square root expression at 2:45.
Timestamp Guide
Related Algebra Work
Use the Exponents topic page for more exponent lessons.
For rules that often show up after rational exponents, review the Exponent Power Rule and Negative Exponents.