Question 261·Hard·Advanced Math

A power of a power multiplies exponents. Give the outside $-3$ to both factors in the numerator:

Multiply those exponents:

The $b$ exponent is positive because two negatives multiply to a positive.

The outside $\frac32$ reaches both factors in the denominator, too:

Multiply those exponents:

For a quotient of powers with the same base, subtract the bottom exponent from the top one. Apply that rule separately to $a$ and $b$:

Keep the parentheses around $-3$: the denominator’s $a$ exponent is negative.

Subtracting negative $3$ means adding $3$:

Write $3$ in fourths:

Combine the fourths:

For $b$, subtract its denominator exponent $\frac34$. Write $6$ in fourths:

Combine the fourths:

The problem allows any positive $a$ and $b$, so choose $a=4$ and $b=2$. Type those values and the original expression into Desmos. It prints about $861.078$. This is a check for an arithmetic slip, not proof of equivalence: two different expressions can agree at one pair of values.

The exponent work gives $a^{9/4}b^{21/4}$. Type it beneath the original in Desmos; it also prints about $861.078$. The matching value checks your work, while the exponent rules show why the expressions are equivalent for all positive $a$ and $b$. Choice D.