To solve the equation \( 2x^{3/2} = 16 \), follow these steps:
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Isolate the term with the variable:
Divide both sides by 2: \[ x^{3/2} = \frac{16}{2} \] \[ x^{3/2} = 8 \]
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Remove the rational exponent:
To get rid of the exponent \(\frac{3}{2}\), we can raise both sides of the equation to the reciprocal of \(\frac{3}{2}\), which is \(\frac{2}{3}\): \[ (x^{3/2})^{\frac{2}{3}} = 8^{\frac{2}{3}} \]
The left side simplifies: \[ x^{(3/2) \cdot (2/3)} = x^{1} = x \]
Now we need to calculate \(8^{\frac{2}{3}}\). This can be done as follows: \[ 8^{\frac{2}{3}} = (8^{\frac{1}{3}})^2 \]
Since \(8^{\frac{1}{3}} = 2\) (because \(2^3 = 8\)), we have: \[ 8^{\frac{2}{3}} = (2)^2 = 4 \]
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Final Result: \[ x = 4 \]
Thus, the solution is \(\boxed{4}\).
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