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

Question

.(1 point)
Responses

4
4

84
84

8
8

no solution
no solution

Answer 1

To solve the equation \( 2x^{32} = 16 \), we can follow these steps:

1. First, express \( 16 \) as a power of \( 2 \): \[ 16 = 2^4 \]

2. Rewrite the equation: \[ 2x^{32} = 2^4 \]

3. Divide both sides by \( 2 \): \[ x^{32} = \frac{2^4}{2} = 2^{4-1} = 2^3 \]

4. Now we have: \[ x^{32} = 2^3 \]

5. To isolate \( x \), take the \( 32^{nd} \) root of both sides: \[ x = (2^3)^{\frac{1}{32}} = 2^{\frac{3}{32}} \]

6. Since \( 2^{\frac{3}{32}} \) is not any of the options provided (4, 84, 8, or no solution), the closest conclusion we can draw is that there is no rational solution among the options given.

Therefore, the answer is no solution.