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High School
Algebra
Algebra I
Exponents and radicals

Kye

Answered question

2021-09-07

Simplify the radical expression by changing the expression to exponential form. Write the answer in radical form when appropriate. Assume that the variable represents positive real numbers.
$\sqrt{\sqrt[4]{a}}$

Answer & Explanation

Caren

Skilled2021-09-08Added 96 answers

Solution:
Think about the following expression: $x = \sqrt{\sqrt[4]{a}}$ where a is the positive real number
Apply radical rule $\sqrt[n]{a} = a^{\frac{1}{n}}$ we have $\sqrt[4]{a} = a^{\frac{1}{4}}$
So $x = \sqrt{\sqrt[4]{a}} = {(a^{\frac{1}{4}})}^{\frac{1}{2}}$
Apply exponent rule ${(a^{b})}^{c} = a^{b c}$ we have
$x = {(a^{\frac{1}{4}})}^{\frac{1}{2}} = a^{\frac{1}{4} \cdot \frac{1}{2}}$
We also have $\frac{1}{4} \cdot \frac{1}{2} = \frac{1}{8}$ so
$x = a^{\frac{1}{4} \cdot \frac{1}{2}} = a^{\frac{1}{8}}$
Since $\sqrt[n]{a} = a^{\frac{1}{n}}$ so
$x = a^{\frac{1}{8}} = \sqrt[8]{a}$

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