Simplifying Radicals

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

1. Given the expression $a^{\frac {b}{c}}$, write this as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\frac{1}{8}\right)^\frac{1}{2}} Using convention 1, we rewrite the given expression as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}} and simplify the result:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\sqrt{8}}{\sqrt{8}}} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{\sqrt{8}} = \frac{1}{\sqrt{8}}\cdot\frac{\sqrt{8}}{\sqrt{8}} = \frac{\sqrt{8}}{8}}

Notice that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\sqrt{8}}{8} = \frac{\sqrt{2^3}}{8} = \frac{\sqrt{2^2\cdot2}}{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