Question 132·Medium·Advanced Math

The given restrictions allow $a=b=1$. Type $a_1=1$ and $b_1=1$, then type the original expression with $a_1,b_1$ in place of $a,b$. Desmos prints $3$. Since every variable power is $1$ here, that result is the coefficient, the number multiplying the variable powers. One input cannot establish their exponents.

The $\frac{2}{3}$ is outside the parentheses, so give it to each factor: $27$, $a^{-3}$, and $b^9$. A power raised to a power multiplies its exponents:

Multiply the exponents on $a$ and $b$. Keep the minus sign on $a$: a negative exponent means a reciprocal, not an instruction to make the exponent positive:

At $a=b=1$, the original expression reduces to $\frac{27^{2/3}}{3}$. Desmos showed that this equals $3$, so replace only the numerical part:

To divide powers of the same base, subtract the bottom exponent. The bottom exponent is $-1$, so subtracting it adds $1$:

Simplify the exponent: $3a^{-2}b^{6-(-1)}=3a^{-2}b^7$. This matches for every nonzero $a$ and $b$. Choice D.