Solve the following equation with a rational exponent: 2x32=16

Question

2
x
3
2
=
16
.(1 point)
Responses

8
8

84
84

4
4

no solution

Answer 1

To solve the equation \( 2x^{\frac{3}{2}} = 16 \), follow these steps:

1. Isolate \( x^{\frac{3}{2}} \):

   \[ x^{\frac{3}{2}} = \frac{16}{2} \] \[ x^{\frac{3}{2}} = 8 \]

2. Raise both sides to the reciprocal of \( \frac{3}{2} \), which is \( \frac{2}{3} \):

   \[ x = 8^{\frac{2}{3}} \]

3. Calculate \( 8^{\frac{2}{3}} \):

   • First, find the cube root of 8, which is 2.
   • Then, square the result: \( 2^2 = 4 \).

Thus, the solution is:

\[ x = 4 \]

The answer is \( 4 \).