# To calculate: The single radical of the expression \sqrt{x\sqrt{x\sqrt{x}}}

To calculate: The single radical of the expression $\sqrt{x\sqrt{x\sqrt{x}}}$
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Step 1
Consider the expression $\sqrt{x\sqrt{x\sqrt{x}}}$
$\sqrt{x\sqrt{x\sqrt{x}}}={\left(x{\left(x\left({x}^{\frac{1}{2}}\right)\right)}^{\frac{1}{2}}\right)}^{\frac{1}{2}}$
Use the product property and the expression becomes,
${\left(x{\left(x\left({x}^{\frac{1}{2}}\right)\right)}^{\frac{1}{2}}\right)}^{\frac{1}{2}}={\left(x{\left({x}^{1+\frac{1}{2}}\right)}^{\frac{1}{2}}\right)}^{\frac{1}{2}}$
$={\left(x{\left({x}^{\frac{3}{2}}\right)}^{\frac{1}{2}}\right)}^{\frac{1}{2}}$

Add the powers of the same bases.

$={\left({x}^{\frac{7}{4}}\right)}^{\frac{1}{2}}$
$={x}^{\frac{7}{8}}$
The obtained expression with rational exponents can be rewritten into radicals as.
${x}^{\frac{7}{8}}={\left({x}^{7}\right)}^{\frac{1}{8}}$
$=\sqrt[8]{{x}^{7}}$
Therefore, the value of the expression is $=\sqrt[8]{{x}^{7}}$

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