Solve the Nested Radical Equation: √√81 = ∛√x^6

To solve this problem, we'll follow these steps:

• Step 1: Simplify the left side of the equation, $\sqrt{\sqrt{81}}$.
• Step 2: Simplify the right side of the equation, $\sqrt[3]{\sqrt{x^6}}$.
• Step 3: Equate the simplified expressions and solve for $x$.

Now, let's simplify each side:

Step 1: Simplify $\sqrt{\sqrt{81}}$.

First, evaluate $\sqrt{81}$, which is $9$, since $9^2 = 81$.
Then, evaluate $\sqrt{9}$, which is $3$, since $3^2 = 9$.
So, $\sqrt{\sqrt{81}} = 3$.

Step 2: Simplify $\sqrt[3]{\sqrt{x^6}}$.

Express $\sqrt{x^6}$ as $(x^6)^{1/2} = x^{6/2} = x^3$.
Express $\sqrt[3]{x^3}$ as $(x^3)^{1/3} = x^{3/3} = x^1 = x$.
So, $\sqrt[3]{\sqrt{x^6}} = x$.

Step 3: Set the simplified expressions equal.

We have simplified both sides of the equation to get $3 = x$.
Therefore, the solution to the problem is $x = 3$.

Hence, the correct answer is $x = 3$.

Therefore, the correct choice is:

Choice 2: $x = 3$.