Problem
Solve the following equation by making an appropriate substitution. \[ 9 x^{\frac{2}{3}}-44 x^{\frac{1}{3}}-5=0 \] Make an appropriate substitution and rewrite the equation in quadratic form. Let $u=\square$, then the quadratic equation in $u$ is $\square$. Solve the equation. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is \{\} . (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. The solution set is the empty set.
Solution
Step 1 ：Given the equation \(9 x^{\frac{2}{3}}-44 x^{\frac{1}{3}}-5=0\)
Step 2 ：Make an appropriate substitution and rewrite the equation in quadratic form. Let \(u=x^{\frac{1}{3}}\), then the quadratic equation in \(u\) is \(9u^2 - 44u - 5 = 0\)
Step 3 ：Solve the quadratic equation for \(u\). The solutions are \(u = -\frac{1}{9}\) and \(u = 5\)
Step 4 ：Substitute back to find the value of \(x\). The solutions for \(x\) are \(x = (-\frac{1}{9})^3 = -\frac{1}{729}\) and \(x = 5^3 = 125\)
Step 5 ：Final Answer: The solution set is \(\boxed{\{-\frac{1}{729}, 125\}}\)
