We need to simplify the expression $(16x^6)^{\frac{1}{2}}$.

1. Problem Description

We need to simplify the expression $(16x^6)^{\frac{1}{2}}$.

2. Solution Steps

We can apply the power of a product rule: $(ab)^n = a^n b^n$.

$(16x^6)^{\frac{1}{2}} = 16^{\frac{1}{2}} (x^6)^{\frac{1}{2}}$

$16^{\frac{1}{2}}$ is the square root of 16, which is

4. $16^{\frac{1}{2}} = \sqrt{16} = 4$

For the second term, we can apply the power of a power rule: $(a^m)^n = a^{m \cdot n}$.

$(x^6)^{\frac{1}{2}} = x^{6 \cdot \frac{1}{2}} = x^3$

Putting these results together:

$16^{\frac{1}{2}} (x^6)^{\frac{1}{2}} = 4x^3$

3. Final Answer

$4x^3$