Question

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

4
4

84
84

no solution
no solution

8

Answer 1

To solve the equation \(2x^{32} = 16\), we'll start by isolating \(x^{32}\).

1. Divide both sides of the equation by 2:
\[
x^{32} = \frac{16}{2}
\]
\[
x^{32} = 8
\]

2. Now we can express \(8\) as a power of \(2\):
\[
8 = 2^3
\]
So, we have:
\[
x^{32} = 2^3
\]

3. To solve for \(x\), we can take both sides to the power of \(\frac{1}{32}\):
\[
x = (2^3)^{\frac{1}{32}} = 2^{\frac{3}{32}}
\]

Now we can evaluate \(2^{\frac{3}{32}}\). However, this expression doesn't simplify to a nice rational number.

To check our options:
- \(4\) is equivalent to \(2^2\), which is not equal to \(2^{\frac{3}{32}}\).
- \(84\) is not relevant here.
- \(8\) is \(2^3\), which also does not simplify correctly to \(x\).

Since \(2^{\frac{3}{32}}\) is not a whole number, the closest valid response given the options would be:
- **no solution** because we don't have any simple rational numbers represented in the options that are equal to \(2^{\frac{3}{32}}\).

Thus, the answer is:

**no solution**