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 $4$.