Solve Nested Roots: Simplifying ⁷√(⁴√14) Step by Step

Solve the following exercise:

$\sqrt[7]{\sqrt[4]{14}}=$

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

• Step 1: Identify the inner and outer roots.
• Step 2: Apply the root of a root formula.
• Step 3: Calculate the combined root.

Now, let's work through each step:

Step 1: The expression given is $\sqrt[7]{\sqrt[4]{14}}$. The inner root is $\sqrt[4]{14}$, and the outer root is $\sqrt[7]{(\cdot)}$.

Step 2: Use the formula for the root of a root: $\sqrt[n]{\sqrt[m]{x}} = \sqrt[n \cdot m]{x}$.

Step 3: Plug in our values: $n = 7$ and $m = 4$. Thus, we have:

$\sqrt[7]{\sqrt[4]{14}} = \sqrt[7 \times 4]{14} = \sqrt[28]{14}$

Therefore, the simplified form of the given expression is $\sqrt[28]{14}$.