SAT Question 42f19012: Equivalent Expressions answer and explanation with Desmos solution

Choice B is correct. Since $\frac{12}{12}=1$, multiplying the exponent of the given expression by $\frac{12}{12}$ yields an equivalent expression: $a^{\left(\frac{11}{12}\right)\left(\frac{12}{12}\right)}=a^{\left(\frac{132}{144}\right)}$. Since $\frac{132}{144}=132\left(\frac{1}{144}\right)$, the expression $a^{\frac{132}{144}}$ can be rewritten as $a^{\left(132\right)\left(\frac{1}{144}\right)}$. Applying properties of exponents, this expression can be rewritten as ${\left(a^{132}\right)}^{\frac{1}{144}}$. An expression of the form ${\left(m\right)}^{\frac{1}{k}}$, where $m>0$ and $k>0$, is equivalent to $\sqrt[k]{m}$. Therefore, ${\left(a^{132}\right)}^{\frac{1}{144}}$ is equivalent to $\sqrt[144]{a^{132}}$.