Inside each set of parentheses, apply $(x^m)^n = x^{mn}$ separately to the $a$ part and the $b$ part.

After simplifying the numerator and denominator, you will have something like $a^{p} b^{q} / (a^{r} b^{s})$. Think about how exponents combine when you divide expressions with the same base.

In Desmos, on two separate lines, type something like a = 4 and b = 9 (any positive values work, because the expression is defined only for positive a and b).

On a new line, enter the original expression using a and b: (a^(1/4)*b^(-2))^(-3)/(a^(-2)*b^(1/2))^(3/2). Note the numerical value that Desmos shows for this line.

On four new lines, enter each choice written with a and b: (ab)^(1/4)/(a^2b^5), a^(3/4)*b^(21/4), a^(9/4)*b^(15/4), and a^(9/4)*b^(21/4). Compare their numerical values to the value of the original expression from step 2; the correct answer is the option whose value exactly matches the original expression.

Use these rules:

• Power to a power: $(x^m)^n = x^{mn}$ (multiply the exponents).
• Product of same base: $x^m x^n = x^{m+n}$.
• Quotient of same base: $x^m / x^n = x^{m-n}$. You will apply the power-to-a-power rule separately to $a$ and $b$ in the numerator and denominator, then use the quotient rule.

The numerator is $(a^{1/4} b^{-2})^{-3}$. Apply the power-to-a-power rule to each factor:

• For $a$: $(a^{1/4})^{-3} = a^{1/4 \,\cdot\, (-3)} = a^{-3/4}$.
• For $b$: $(b^{-2})^{-3} = b^{-2 \,\cdot\, (-3)} = b^{6}$. So the numerator simplifies to $a^{-3/4} b^{6}$.

The denominator is $(a^{-2} b^{1/2})^{3/2}$. Again apply the power-to-a-power rule to each factor:

• For $a$: $(a^{-2})^{3/2} = a^{-2 \,\cdot\, 3/2} = a^{-3}$.
• For $b$: $(b^{1/2})^{3/2} = b^{1/2 \,\cdot\, 3/2} = b^{3/4}$. So the denominator simplifies to $a^{-3} b^{3/4}$.

Now write the whole expression using your simplified numerator and denominator:

Apply the quotient rule for each base:

• For $a$: subtract exponents: $-3/4 - (-3) = -3/4 + 3 = -3/4 + 12/4 = 9/4$, so you get $a^{9/4}$.
• For $b$: subtract exponents: $6 - 3/4 = 24/4 - 3/4 = 21/4$, so you get $b^{21/4}$.

Therefore, the simplified expression is

which corresponds to choice D.