Focus on $(x^2 y^5)^{3/4}$. Use $(ab)^m = a^m b^m$ and then $(a^m)^n = a^{mn}$ to find the new exponents on $x$ and $y$.

Once both numerator and denominator are written as powers of $x$ and $y$, divide by subtracting exponents for each base separately: $a^m / a^n = a^{m-n}$.

Be careful when subtracting fractions like $15/4 - 1/2$; rewrite $1/2$ with denominator 4 before subtracting.

In Desmos (calculator or graphing), type the original expression with specific positive values for $x$ and $y$, such as $x=16$ and $y=4$:

Record the numerical result that Desmos gives.

On new lines in Desmos, type each choice with the same $x$ and $y$ values, for example:

• $16^{1/2} \cdot 4^{7/2}$
• $16^{3/2} \cdot 4^{13/4}$
• $16 \cdot 4^{7/2}$
• $16 \cdot 4^{13/4}$ Compare each result with the value from the original expression; the choice that gives exactly the same number is the equivalent expression.

Start by rewriting the denominator using a fractional exponent.

• The square root $\sqrt{xy} = (xy)^{1/2} = x^{1/2} y^{1/2}$.

So the expression becomes

Use $(ab)^m = a^m b^m$ and $(a^m)^n = a^{mn}$ on the numerator.

Now multiply the exponents:

• For $x$: $2\cdot \tfrac{3}{4} = \tfrac{6}{4} = \tfrac{3}{2}$
• For $y$: $5\cdot \tfrac{3}{4} = \tfrac{15}{4}$

So the numerator simplifies to

Now the whole fraction is

When you divide like bases with exponents, subtract the exponents: $a^m / a^n = a^{m-n}$.

Handle $x$ and $y$ separately.

For $x$:

For $y$:

Putting these together, the simplified expression is

which corresponds to choice D.