SAT Question 565428e1: Equivalent Expressions answer and explanation with Desmos solution

An expression of the form $\sqrt[n]{a^m}$, where $m$ and $n$ are integers greater than $1$ and $a\ge 0$, is equivalent to $a^{\frac{m}{n}}$. Therefore, the expression on the left-hand side of the given equation is equivalent to $p^{\frac{2}{3}}$; thus, $p^{\frac{2}{3}}=t^{\frac{9}{7}}$. If $t=p^{3n-1}$, where $n$ is a constant, then substituting $p^{3n-1}$ for $t$ in the equation yields $p^{\frac{2}{3}}=p^{\frac{9}{7}(3n-1)}$. It's given that $p>1$; therefore, $\frac{2}{3}=\frac{9}{7}(3n-1)$. Multiplying each side of this equation by $21$ yields $14=27(3n-1)$. Distributing multiplication over subtraction yields $14=81n-27$. Adding $27$ to each side of this equation yields $41=81n$. Dividing each side of this equation by $81$ yields $\frac{41}{81}=n$. Therefore, if $t=p^{3n-1}$, where $n$ is a constant, the value of $n$ is $\frac{41}{81}$. Note that .5061, .5062, and 41/81 are examples of ways to enter a correct answer.