Solve Nested Radicals: Cube Root of Square Root of 16 × Square Root Problems

Complete the following exercise:

$\sqrt[3]{\sqrt{16}}\cdot\sqrt[]{\sqrt{16}}=$

To solve this problem, we will simplify the expression $\sqrt[3]{\sqrt{16}} \cdot \sqrt[]{\sqrt{16}}$ using the rules for exponents and roots.

First, consider the inner square root $\sqrt{16}$. We know that:
$\sqrt{16} = 16^{1/2}$

Next, we address the cube root term $\sqrt[3]{\sqrt{16}}$. Express $\sqrt{16}$ as $16^{1/2}$, then:

• $\sqrt[3]{\sqrt{16}} = \sqrt[3]{16^{1/2}}$ Convert to exponents: $\sqrt[3]{16^{1/2}} = (16^{1/2})^{1/3} = 16^{(1/2) \cdot (1/3)} = 16^{1/6}$
• The other term is $\sqrt{\sqrt{16}} = \sqrt{16^{1/2}}$ Convert to exponents: $\sqrt{16^{1/2}} = (16^{1/2})^{1/2} = 16^{(1/2) \cdot (1/2)} = 16^{1/4}$

Now, multiply these results:
$16^{1/6} \cdot 16^{1/4}$

Using the product rule for exponents $(a^m \cdot a^n = a^{m+n})$, combine the exponents:
$16^{1/6 + 1/4}$

Find the common denominator to add the fractions:

• $1/6 = 2/12$
• $1/4 = 3/12$ $1/6 + 1/4 = 2/12 + 3/12 = 5/12$

Thus, the expression becomes:
$16^{5/12}$

Therefore, the simplified expression is $16^{\frac{5}{12}}$.