To solve the equation \( 2x^{3/2} = 16 \), we will isolate \( x^{3/2} \) and then solve for \( x \).
- Start by dividing both sides of the equation by 2:
\[ x^{3/2} = \frac{16}{2} \] \[ x^{3/2} = 8 \]
- Now, we need to eliminate the rational exponent by raising both sides to the reciprocal of \( \frac{3}{2} \), which is \( \frac{2}{3} \):
\[ (x^{3/2})^{2/3} = 8^{2/3} \]
- On the left side, the exponents multiply:
\[ x^{(3/2) \cdot (2/3)} = x^1 = x \]
- Now we simplify the right side \( 8^{2/3} \). First, express 8 as \( 2^3 \):
\[ 8^{2/3} = (2^3)^{2/3} \]
- Now applying the power of a power rule gives:
\[ (2^3)^{2/3} = 2^{3 \cdot \frac{2}{3}} = 2^2 = 4 \]
- We find that:
\[ x = 4 \]
Thus, the solution to the equation \( 2x^{3/2} = 16 \) is \( \boxed{4} \).