Simplifying Radicals

We will use the following conventions for simplifying expressions involving radicals:

1. Given the expression $a^{\frac {b}{c}}$, write this as ${\sqrt[{c}]{a^{b}}}$
2. No fractions under the radical sign
3. No radicals in the denominator
4. The radicand has no exponentiated factors with exponent greater than or equal to the index of the radical

Example: Simplify the expression $\left({\frac {1}{8}}\right)^{\frac {1}{2}}$ Using convention 1, we rewrite the given expression as

$\left({\frac {1}{8}}\right)^{\frac {1}{2}}={\sqrt[{2}]{\left({\frac {1}{8}}\right)^{1}}}={\sqrt {\frac {1}{8}}}$

The expression now violates convention 2. To get rid of the fraction in the radical, apply the rule $\left({\frac {a}{b}}\right)^{n}={\frac {a^{n}}{b^{n}}}$ and simplify the result:

${\sqrt {\frac {1}{8}}}={\frac {\sqrt {1}}{\sqrt {8}}}={\frac {1}{\sqrt {8}}}$

The resulting expression violates convention 3. To get rid of the radical in the denominator, multiply by ${\frac {\sqrt {8}}{\sqrt {8}}}$:

${\frac {1}{\sqrt {8}}}={\frac {1}{\sqrt {8}}}\cdot {\frac {\sqrt {8}}{\sqrt {8}}}={\frac {\sqrt {8}}{8}}$

Notice that $8=2^{3}$. Since the index of the radical is 2, our expression violates convention 4. We can reduce the exponent of the expression under the radical as follows:

${\frac {\sqrt {8}}{8}}={\frac {\sqrt {2^{3}}}{8}}={\frac {\sqrt {2^{2}\cdot 2}}{8}}={\frac {2\cdot {\sqrt {2}}}{8}}={\frac {\sqrt {2}}{4}}$

Resources

• Introduction to Radicals, Lumen Learning: Boundless Algebra
• Simplifying Square Roots, Khan Academy
• Useful Radical/Root Rules for Simplification, Mathwords.com
• Example Problems of Simplifying Radicals, LibreTexts