Questions: Solve the equation and check for extraneous solutions. x^(2/3) = 16

Solution Steps

To solve the equation \( x^{\frac{2}{3}} = 16 \), we need to isolate \( x \). We can do this by raising both sides of the equation to the power of \(\frac{3}{2}\). After finding the solution, we should check if it satisfies the original equation to ensure there are no extraneous solutions.

To solve the equation \( x^{\frac{2}{3}} = 16 \), we raise both sides to the power of \(\frac{3}{2}\): \[ x = 16^{\frac{3}{2}} \]

Calculate \( 16^{\frac{3}{2}} \): \[ 16^{\frac{3}{2}} = (16^{\frac{1}{2}})^3 = 4^3 = 64 \] Thus, \( x = 64 \).

Verify if \( x = 64 \) satisfies the original equation: \[ (64)^{\frac{2}{3}} = 16 \] Calculate \( 64^{\frac{2}{3}} \): \[ 64^{\frac{2}{3}} = (64^{\frac{1}{3}})^2 = 4^2 = 16 \] Since \( 64^{\frac{2}{3}} = 16 \), the solution \( x = 64 \) is valid.

Final Answer