To solve the equation \( x^{2/3} = 64 \), follow these steps:
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Isolate \( x \): Start by raising both sides of the equation to the power of \( \frac{3}{2} \) to eliminate the rational exponent.
\[ \left( x^{2/3} \right)^{3/2} = 64^{3/2} \]
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Simplify the left side: The left side simplifies as follows: \[ x^{(2/3) \cdot (3/2)} = x^1 = x \]
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Calculate the right side: Now we need to calculate \( 64^{3/2} \):
- First, find \( \sqrt{64} \): \[ \sqrt{64} = 8 \]
- Next, calculate \( 8^3 \): \[ 8^3 = 512 \]
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Set the equation: So, we have: \[ x = 512 \]
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Conclusion: The solution to the equation \( x^{2/3} = 64 \) is: \[ x = 512 \]