To solve the equation \(2x^{3/2}=16\), we start by isolating \(x^{3/2}\).
- Divide both sides by 2:
\[ x^{3/2} = \frac{16}{2} = 8 \]
- To eliminate the rational exponent, raise both sides 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 to \(x\):
\[ x = 8^{\frac{2}{3}} \]
- Now calculate \(8^{\frac{2}{3}}\):
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First, calculate \(8^{\frac{1}{3}}\): since \(2^3 = 8\), we know \(8^{\frac{1}{3}} = 2\).
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Now raise this result to the power of 2:
\[ 8^{\frac{2}{3}} = (8^{\frac{1}{3}})^2 = 2^2 = 4 \]
So the solution to the equation is
\[ \boxed{4} \]
Thus, the correct response is \(4\).
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