To solve the equation \( 2x^{3/2} = 16 \), we can follow these steps:
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Isolate the term with the exponent: \[ x^{3/2} = \frac{16}{2} \] \[ x^{3/2} = 8 \]
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Raise both sides to the power of \( \frac{2}{3} \) to eliminate the exponent: \[ \left( x^{3/2} \right)^{\frac{2}{3}} = 8^{\frac{2}{3}} \]
This simplifies to: \[ x = 8^{\frac{2}{3}} \]
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Calculate \( 8^{\frac{2}{3}} \):
- First, find the cube root of 8: \( 8^{1/3} = 2 \).
- Then, square the result: \( 2^2 = 4 \).
Therefore, \[ x = 4 \]
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Check the solution: Substitute \( x = 4 \) back into the original equation: \[ 2(4)^{3/2} = 16 \] Calculate \( (4^{3/2}) \): \[ 4^{3/2} = (4^{1/2})^3 = 2^3 = 8 \] Then, \[ 2 \times 8 = 16, \] which matches the original equation.
Thus, the solution is \[ \boxed{4}. \]