Question 36·Hard·Advanced Math

A negative exponent means a reciprocal, so flip the entire fraction before applying the power of $3$:

For a quotient of powers with the same base, subtract the bottom exponent from the top. Divide $8$ by $2$ as well: $\left(\frac{8a^{5/6}b^{-1/2}}{2a^{-1/4}b^{2/3}}\right)^3=\left(4a^{5/6-(-1/4)}b^{-1/2-2/3}\right)^3$. Subtract the bottom exponent for each variable, even when it is negative.

For $a$, subtracting a negative adds. Rewrite in twelfths, then add:

For $b$, rewrite in sixths, then subtract:

So the expression inside the outside power is now $4a^{13/12}b^{-7/6}$.

A power of a power multiplies exponents. The $3$ reaches the $4$, the $a$ power, and the $b$ power:

Multiply each exponent, keeping the minus sign on $b$:

For a check, not a proof, type $a=16$ and $b=4$, then type the original expression and $64a^{13/4}b^{-7/2}$. Desmos prints $4096$ for each. The pair is allowed because both values are positive; matching at one pair can catch a slip, but the exponent rules establish the equivalence.

Since $b^{-7/2}=1/b^{7/2}$, $b$ belongs below the fraction bar:

Split each exponent into a whole number and a fraction so you can read the roots:

$=\frac{64a^3a^{1/4}}{b^3b^{1/2}}$. The fractional parts are $\frac14$ for $a$ and $\frac12$ for $b$.

For positive values, $a^{1/4}=\sqrt[4]{a}$ and $b^{1/2}=\sqrt b$. Replace those powers with their radicals: $\frac{64a^3a^{1/4}}{b^3b^{1/2}}=\frac{64a^3\sqrt[4]{a}}{b^3\sqrt b}$. Read the denominator of each fractional exponent to identify its root. That gives the equivalent expression. Choice D.