Question

Solve the following equation with a rational exponent: 2x32=16.(1 point)
Responses

8
8

no solution
no solution

84
84

4

Answer 1

To solve the equation \( 2x^{32} = 16 \), we can start by rewriting 16 as a power of 2.

We know that:
\[
16 = 2^4
\]

Thus, we can rewrite the equation:
\[
2x^{32} = 2^4
\]

Now, we can divide both sides by 2 (assuming \( x^{32} \) is not affecting the simplification):
\[
x^{32} = \frac{2^4}{2} = 2^{4-1} = 2^3
\]

This simplifies to:
\[
x^{32} = 2^3
\]

Now we can write it in terms of rational exponents:
\[
x = (2^3)^{\frac{1}{32}}
\]

This simplifies to:
\[
x = 2^{\frac{3}{32}}
\]

Now, we can compute this value:
\[
2^{\frac{3}{32}} \approx 1.0905
\]

Since the question seems to indicate selecting a specific value, none of the provided options (8, no solution, 84, 4) equals \( 2^{\frac{3}{32}} \).

However, since this is an exponential equation, it could have complex solutions as well, but strictly looking at the real solutions provided, we conclude that none of the options match our calculation.

So the answer is:
- **no solution** in the context of the options given.