Simplify the Nested Radical: Solving √⁶(√(25x⁶))

Complete the following exercise:

$\sqrt[6]{\sqrt{25x^6}}$

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

• Step 1: Convert the expression under the roots to fractional exponents.
• Step 2: Simplify the fractional exponents.
• Step 3: Return the expression to root form if needed.

Now, let's work through each step:

Step 1: We have the expression $\sqrt[6]{\sqrt{25x^6}}$. Begin by simplifying the inner square root:

$\sqrt{25x^6} = (25x^6)^{1/2} = 25^{1/2} \cdot (x^6)^{1/2}$.

We know $25^{1/2} = 5$ and $(x^6)^{1/2} = x^{6 \cdot 1/2} = x^3$.

So, $\sqrt{25x^6} = 5x^3$.

Step 2: Apply the outer sixth root:

$\sqrt[6]{5x^3} = (5x^3)^{1/6} = 5^{1/6} \cdot (x^3)^{1/6} = 5^{1/6} \cdot x^{1/2}$.

Convert $5^{1/6}$ as $\sqrt[12]{25}$:

Since $5^{1/6} = (5^{1/2})^{1/3} = \sqrt{5}^{1/3} = \sqrt[12]{25}$,

we have $5^{1/6} = \sqrt[12]{25}$.

Therefore, the solution becomes:

$\sqrt{x} \cdot \sqrt[12]{25}$.

Therefore, the solution to the problem is $\sqrt{x}\cdot\sqrt[12]{25}$.