Solve the following equation with a rational exponent: x2/3=64 .(1 point)

Question

Responses

512
512

32
32

16
16

8

Answer 1

To solve the equation \( x^{2/3} = 64 \), we can start by isolating \( x \).

1. Raise both sides of the equation to the power of \( \frac{3}{2} \) to eliminate the exponent on \( x \):

\[ (x^{2/3})^{3/2} = 64^{3/2} \]

2. This simplifies the left-hand side:

\[ x^{(2/3) \cdot (3/2)} = x^1 = x \]

3. Now, we need to compute \( 64^{3/2} \):

   • First calculate \( 64^{1/2} \): \[ 64^{1/2} = 8 \]

   • Then raise it to the 3rd power: \[ 8^3 = 512 \]

4. Hence, we have: \[ x = 512 \]

Thus, the solution to the equation \( x^{2/3} = 64 \) is \( \boxed{512} \).