Simplify the Nested Radical: Finding the Value of ∜(√(x⁸))

The 8 in $x^8$ and the 8th root don't directly cancel because there's a square root in between. You must simplify $\sqrt{x^8}$ first to get $x^4$, then apply the 8th root.

Convert to fractional exponents when dealing with nested radicals or complex root expressions. It makes the algebra much cleaner: $\sqrt[n]{a} = a^{1/n}$.

They're actually the same! $\sqrt[8]{\sqrt{x^8}} = (x^8)^{1/2 \cdot 1/8} = x^{8/16} = x^{1/2}$, and $\sqrt[16]{x^8} = x^{8/16} = x^{1/2}$. Both equal $\sqrt{x}$.

Because $\sqrt{x^8} = (x^8)^{1/2} = x^{8 \cdot 1/2} = x^4$. The square root means raise to the power of 1/2, so you multiply the exponents: $8 \times \frac{1}{2} = 4$.

It's much harder! You'd have to think about what number, when raised to the 8th power, gives you $\sqrt{x^8}$. Using exponent rules makes nested radicals much more manageable.