To solve the equation \( x^{2/3} = 64 \), we can follow these steps:
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Raise both sides to the reciprocal of the exponent: Since the exponent on \( x \) is \( \frac{2}{3} \), we can raise both sides of the equation to \( \frac{3}{2} \):
\[ \left( x^{2/3} \right)^{3/2} = 64^{3/2} \]
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Simplify the left side: The left side simplifies to:
\[ x^{(2/3) \cdot (3/2)} = x^1 = x \]
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Calculate the right side: Now we need to calculate \( 64^{3/2} \). This can be done in two steps:
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First, find the square root of 64: \[ \sqrt{64} = 8 \]
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Then raise this result to the power of 3: \[ 8^3 = 512 \]
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Thus, we have:
\[ x = 512 \]
- Conclusion: The solution to the equation \( x^{2/3} = 64 \) is
\[ \boxed{512} \]
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