Simplify the Nested Radical: Sixth Root of Square Root of x^12

Complete the following exercise:

$\sqrt[6]{\sqrt{x^{12}}}=$

To solve $\sqrt[6]{\sqrt{x^{12}}}$, we will follow these steps:

• Step 1: Simplify the inner radical expression $\sqrt{x^{12}}$.
• Step 2: Use the property of roots, expressing $\sqrt{x^{12}}$ as a power of $x$.
• Step 3: Use the result from step 1 in the outer root expression.
• Step 4: Simplify the entire expression using exponent rules.

Now, let's perform each of these steps:

Step 1: Simplify $\sqrt{x^{12}}$.
$\sqrt{x^{12}} = x^{12/2} = x^6$.

Step 2: Simplify the outer expression $\sqrt[6]{x^6}$.
$\sqrt[6]{x^6} = (x^6)^{1/6}$.

Step 3: Apply the exponent rule $(a^m)^{n} = a^{m \times n}$.
$(x^6)^{1/6} = x^{6 \times 1/6} = x^1 = x$.

Therefore, the simplified expression is $\boxed{x}$.

Thus, the solution to $\sqrt[6]{\sqrt{x^{12}}}$ is $x$.