Problem 135 - Algebra and Trigonometry

Question

Solve for \(x: \sqrt[3]{x \sqrt{x}}=9\)

Step-by-Step Solution

Answer

After calculating \(9^{6/5}\), you will find that \(x \approx 13.39\)

Step 1: Rewrite the expression

The first step is to rewrite the expression in a easier to manage form. The expression can be rewritten as \(x^{1/3} * x^{1/2} = 9\). Simplifying the exponents gives \(x^{5/6} = 9\).

Step 2: Eliminate the root

Raise both sides of the equation to the power that will cancel out the fraction in the exponent. Since the exponent is \(5/6\), raising to the power of \(\frac{6}{5}\) will eliminate the fraction. \((x^{5/6})^{6/5} = 9^{6/5}\). This simplifies to \(x=9^{6/5}\)

Step 3: Evaluate

At this point evaluate \(9^{6/5}\). First, compute the fifth root of 9, then raise the result to the sixth power. Finally, you will get the value of x. It's simpler to use a calculator for this step.