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

To solve the problem $\sqrt[8]{\sqrt{x^8}}$, we'll simplify the expression using exponent rules:

• Step 1: Express the inner square root using exponents. We know $\sqrt{x^8} = (x^8)^{1/2} = x^{8 \cdot 1/2} = x^4$.
• Step 2: Express the entire expression with the 8th root as an exponent. We have $\sqrt[8]{x^4} = (x^4)^{1/8}$.
• Step 3: Simplify the expression, using $(x^a)^{b} = x^{a \cdot b}$. Therefore, $(x^4)^{1/8} = x^{4 \cdot 1/8} = x^{1/2}$.
• Step 4: Recognize $x^{1/2}$ is another way to write $\sqrt{x}$.

Thus, the expression simplifies to $\sqrt{x}$.