To solve the equation \( x^{2/3} = 64 \), we can follow these steps:
- Isolate \( x \): To eliminate the exponent \( \frac{2}{3} \), we can raise both sides of the equation to the reciprocal of \( \frac{2}{3} \), which is \( \frac{3}{2} \).
\[ (x^{2/3})^{3/2} = 64^{3/2} \]
- Simplify the left side: Using the property of exponents, we have:
\[ x^{(2/3) \cdot (3/2)} = x^1 = x \]
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Calculate the right side: Now we need to calculate \( 64^{3/2} \).
- First, we can find the square root of \( 64 \): \[ \sqrt{64} = 8 \]
- Next, we raise \( 8 \) to the power of \( 3 \): \[ 8^3 = 512 \]
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Combine the results: Now we have:
\[ x = 512 \]
Thus, the solution to the equation \( x^{2/3} = 64 \) is \( x = 512 \).
The correct answer from the options provided is \( 512 \).