Solve Nested Roots: Simplifying ¹⁰√(⁵√100) Step by Step

Solve the following exercise:

$\sqrt[10]{\sqrt[5]{100}}=$

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

• Step 1: Use the formula for the nested radical: $\sqrt[10]{\sqrt[5]{100}} = \sqrt[10 \times 5]{100}$.
• Step 2: Calculate the product of the indices of the roots: $10 \times 5 = 50$.
• Step 3: Write the simplified expression: $\sqrt[50]{100}$.

Let's apply these steps to find the solution:

First, we apply the root of a root property:
$\sqrt[10]{\sqrt[5]{100}} = \sqrt[10 \times 5]{100}$

This simplifies to:
$\sqrt[50]{100}$

Therefore, the solution to the problem is $\sqrt[50]{100}$.