Solve the following equation with a rational exponent: x2/3=64 .(1 point) Responses 8 8 32 32 512 512 16

To solve the equation \( x^{\frac{2}{3}} = 64 \), we can first isolate \( x \) by raising both sides of the equation to the reciprocal of the exponent \( \frac{2}{3} \), which is \( \frac{3}{2} \).

\[
x = 64^{\frac{3}{2}}
\]

Now we can calculate \( 64^{\frac{3}{2}} \).

First, find the square root of \( 64 \):

\[
\sqrt{64} = 8
\]

Next, raise that result to the power of 3:

\[
8^3 = 512
\]

Therefore, the solution to the equation \( x^{\frac{2}{3}} = 64 \) is:

\[
x = 512
\]

Thus, among the response options, the answer is **512**.