SAT Question b74f2feb: Equivalent Expressions answer and explanation with Desmos solution

The rational exponent property is $\sqrt[n]{y^m}=y^{\frac{m}{n}}$, where $y>0$, $m$ and $n$ are integers, and $n>0$. This property can be applied to rewrite the given expression $6\sqrt[5]{3^5{x}^{45}}\cdot \sqrt[8]{2^{8}x}$ as $6\left(3^{\frac{5}{5}}\right)\left(x^{\frac{45}{5}}\right)\left(2^{\frac{8}{8}}\right)\left(x^{\frac{1}{8}}\right)$, or $6\left(3\right)\left(x^9\right)\left(2\right)\left(x^{\frac{1}{8}}\right)$. This expression can be rewritten by multiplying the constants, which gives $36\left(x^9\right)\left(x^{\frac{1}{8}}\right)$. The multiplication exponent property is $y^n\cdot y^m=y^{n+m}$, where $y>0$. This property can be applied to rewrite the expression $36\left(x^9\right)\left(x^{\frac{1}{8}}\right)$ as $36x^{9+\frac{1}{8}}$, or $36x^{\frac{73}{8}}$. Therefore, $6\sqrt[5]{3^5{x}^{45}}\cdot \sqrt[8]{2^{8}x}=36x^{\frac{73}{8}}$. It's given that $6\sqrt[5]{3^5{x}^{45}}\cdot \sqrt[8]{2^{8}x}$ is equivalent to $a{x}^b$; therefore, $a=36$ and $b=\frac{73}{8}$. It follows that $a+b=36+\frac{73}{8}$. Finding a common denominator on the right-hand side of this equation gives $a+b=\frac{288}{8}+\frac{73}{8}$, or $a+b=\frac{361}{8}$. Note that 361/8, 45.12, and 45.13 are examples of ways to enter a correct answer.