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

Step 1
Consider the expression ${\displaystyle \sqrt{x\sqrt{x\sqrt{x}}}}$
${\displaystyle \sqrt{x\sqrt{x\sqrt{x}}}={\left(x{\left(x\left(x^{\frac{1}{2}}\right)\right)}^{\frac{1}{2}}\right)}^{\frac{1}{2}}}$
Use the product property and the expression becomes,
${\displaystyle {\left(x{\left(x\left(x^{\frac{1}{2}}\right)\right)}^{\frac{1}{2}}\right)}^{\frac{1}{2}}={\left(x{\left(x^{1+\frac{1}{2}}\right)}^{\frac{1}{2}}\right)}^{\frac{1}{2}}}$
${\displaystyle ={\left(x{\left(x^{\frac{3}{2}}\right)}^{\frac{1}{2}}\right)}^{\frac{1}{2}}}$
${\displaystyle {(x\cdot x^{\frac{3}{4}})}^{\frac{1}{2}}}$
Add the powers of the same bases.
${\displaystyle {(x\cdot x^{\frac{3}{4}})}^{\frac{1}{2}}={\left(x^{1+\frac{3}{4}}\right)}^{\frac{1}{2}}}$
${\displaystyle ={\left(x^{\frac{7}{4}}\right)}^{\frac{1}{2}}}$
${\displaystyle =x^{\frac{7}{8}}}$
The obtained expression with rational exponents can be rewritten into radicals as.
${\displaystyle x^{\frac{7}{8}}={\left(x^7\right)}^{\frac{1}{8}}}$
$=\sqrt[8]{x^7}$
Therefore, the value of the expression is $=\sqrt[8]{x^7}$