Solve the following equation with a rational exponent: 2x3/2=16 .(1 point)

Question

Responses

84
84

8
8

no solution
no solution

4

Answer 1

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

1. Divide both sides by 2: \[ x^{3/2} = \frac{16}{2} = 8 \]

2. To eliminate the rational exponent, we raise both sides to the power of \( \frac{2}{3} \): \[ x = 8^{\frac{2}{3}} \]

3. Now, we need to calculate \( 8^{\frac{2}{3}} \):

   • First, find \( 8^{\frac{1}{3}} \), which is the cube root of 8. Since \( 2^3 = 8 \), we have: \[ 8^{\frac{1}{3}} = 2 \]
   • Next, we square this result: \[ (8^{\frac{1}{3}})^2 = 2^2 = 4 \]

Thus, the solution to the equation \( 2x^{3/2} = 16 \) is \( x = 4 \).

The correct response is: 4