For $(16p^6q^2)^{1/4}$, apply the $1/4$ power separately to $16$, $p^6$, and $q^2$. What is $16^{1/4}$? How do you multiply exponents on $p$ and $q$?

Rewrite $\sqrt{pq}$ as $(pq)^{1/2}$ and then as $p^{1/2}q^{1/2}$. After that, you can subtract exponents for $p$ and $q$ when dividing.

Remember: when dividing like bases, use $a^m / a^n = a^{m-n}$. Apply this separately to $p$ and to $q$, and see what is left.

In Desmos, type the original expression using exponents: (16*p^6*q^2)^(1/4) / sqrt(p*q).

On new lines, enter p = 9 and q = 4 (or any other positive values). Desmos will now show a single numerical value for the original expression; note this value.

On separate lines, enter each option using the same $p$ and $q$: p, 2*p, 2*p*q, and 2*sqrt(p). Compare the numerical values to the value from the original expression and see which option matches. You can change $p$ and $q$ to other positive values to confirm your result stays consistent.

Write each radical using exponents.

• The numerator is $(16p^6q^2)^{1/4}$.
• The denominator is $\sqrt{pq} = (pq)^{1/2}$.

So the whole expression becomes

Now you can use exponent rules instead of radical rules.

Apply the power to each factor inside $(16p^6q^2)^{1/4}$:

• $16^{1/4}$ is the fourth root of $16$, which is $2$.
• $p^{6 \cdot \frac{1}{4}} = p^{6/4} = p^{3/2}$.
• $q^{2 \cdot \frac{1}{4}} = q^{2/4} = q^{1/2}$.

So the numerator becomes

Now the expression is

Rewrite $(pq)^{1/2}$ by applying the exponent to each factor:

Now the whole expression is

Both numerator and denominator are now products of powers of $p$ and $q$.

When you divide powers with the same base, subtract the exponents:

For $p$:

For $q$:

So

Thus the original expression simplifies to $2p$, which matches choice B.