 # To calculate:To convert the radical expression \sqrt{\sqrt{64x^{6}}} jernplate8 2021-10-21 Answered

To calculate:
To convert the radical expression $\sqrt{\sqrt{64{x}^{6}}}$ to rational exponent form and simplify.

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Calculation:
According to exponential rules, $\sqrt[n]{a}={a}^{\frac{1}{n}}$
Applying the above rule:
$\sqrt{\sqrt{64{x}^{6}}}=\left(\sqrt{64{x}^{6}}{\right)}^{\frac{1}{2}}$
Applying the rule again:
$\sqrt{\sqrt{64{x}^{6}}}=\left(\left(64{x}^{6}{\right)}^{\frac{1}{3}}{\right)}^{\frac{1}{2}}$
According to the power rule, ${\left({a}^{m}\right)}^{n}={a}^{mn}$
Applying the above rule:
$\sqrt{\sqrt{64{x}^{6}}}=\left(64{x}^{6}{\right)}^{\left(\frac{1}{3}\right)\left(\frac{1}{2}\right)}$
$\sqrt{\sqrt{64{x}^{6}}}=\left(64{x}^{6}{\right)}^{\left(\frac{1}{6}\right)}$
This can be written as
$\sqrt{\sqrt{64{x}^{6}}}=\left({2}^{6}{x}^{6}{\right)}^{\left(\frac{1}{6}\right)}$
According to product rule, ${\left(ab\right)}^{n}={a}^{n}\cdot {b}^{n}$
Applying the above rule:
$\sqrt{\sqrt{64{x}^{6}}}=\left({2}^{6}{\right)}^{\left(\frac{1}{6}\right)}\left({x}^{6}{\right)}^{\left(\frac{1}{6}\right)}$
Applying the power rule:
$\sqrt{\sqrt{64{x}^{6}}}={2}^{\left(6\right)\left(\frac{1}{6}\right)}{x}^{\left(6\right)\left(\frac{1}{6}\right)}$
$\sqrt{\sqrt{64{x}^{6}}}={2}^{1}{x}^{1}$
$\sqrt{\sqrt{64{x}^{6}}}=2x$