To solve the equation \( 2x^{3/2} = 16 \), follow these steps:
1. **Isolate the term with the variable:**
Divide both sides by 2:
\[
x^{3/2} = \frac{16}{2}
\]
\[
x^{3/2} = 8
\]
2. **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:
\[
8^{\frac{2}{3}} = (2)^2 = 4
\]
3. **Final Result:**
\[
x = 4
\]
Thus, the solution is \(\boxed{4}\).