Question

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

84
84

no solution
no solution

8
8

4

Answer 1

To solve the equation \( 2x^{32} = 16 \), we can first express 16 in terms of a power of 2:

\[
16 = 2^4
\]

This allows us to rewrite the equation as:

\[
2x^{32} = 2^4
\]

Now, we can divide both sides by 2 (assuming \( x^{32} \neq 0 \)):

\[
x^{32} = 2^3
\]

This simplifies to:

\[
x^{32} = 8
\]

Now, to solve for \( x \), we take the 32nd root of both sides:

\[
x = 8^{\frac{1}{32}}
\]

Next, since \( 8 = 2^3 \), we can rewrite this as:

\[
x = (2^3)^{\frac{1}{32}} = 2^{\frac{3}{32}}
\]

Thus, the solution can be expressed as:

\[
x = 2^{\frac{3}{32}}
\]

This result does not match any of the given responses exactly, implying that there is no solution among the provided options.

Therefore, the correct response is:

**no solution**