Solve Nested Radicals: Sixth Root of √64 × Fourth Root of ∛16

Complete the following exercise:

$\sqrt[6]{\sqrt{64}}\cdot\sqrt[4]{\sqrt[3]{16}}=$

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

• Step 1: Simplify $\sqrt{64}$.
• Step 2: Simplify with $\sqrt[6]{}$.
• Step 3: Simplify $\sqrt[3]{16}$.
• Step 4: Simplify with $\sqrt[4]{}$.
• Step 5: Combine the simplified results.

Now, let's work through each step:

Step 1: Simplify $\sqrt{64}$.
Since $64 = 8^2$, we have $\sqrt{64} = 8$.

Step 2: Simplify $\sqrt[6]{8}$ using exponent rules:
$\sqrt[6]{8} = 8^{1/6}$.

Step 3: Simplify $\sqrt[3]{16}$.
Since $16 = 2^4$, then $\sqrt[3]{16} = (2^4)^{1/3} = 2^{4/3}$.

Step 4: Simplify $\sqrt[4]{2^{4/3}}$ using exponent rules:
We have $\sqrt[4]{2^{4/3}} = (2^{4/3})^{1/4} = 2^{4/12} = 2^{1/3}$.

Step 5: Combine simplified results:
We have $8^{1/6} \cdot 2^{1/3} = (2^3)^{1/6} \cdot 2^{1/3} = 2^{3/6} \cdot 2^{1/3} = 2^{1/2} \cdot 2^{1/3} = 2^{(1/2 + 1/3)} = 2^{(3/6 + 2/6)} = 2^{5/6}.$

Convert to a common root form:
The result is equivalent to $2^{5/6} = \sqrt[12]{2^{10}} = \sqrt[12]{1024}$.

Therefore, the solution to the problem is $\sqrt[12]{1024}$.