Question

Solve the following equation with a rational exponent: x23=64
.(1 point)
Responses

512
512

32
32

16
16

8

Answer 1

To solve the equation \( x^{23} = 64 \), we need to isolate \( x \). We can do this by applying both sides of the equation with the exponent \( \frac{1}{23} \).

Starting with the equation:

\[
x^{23} = 64
\]

We take the \( \frac{1}{23} \)th power of both sides:

\[
x = 64^{\frac{1}{23}}
\]

Next, we express \( 64 \) in terms of a power of \( 2 \):

\[
64 = 2^6
\]

Thus, substituting this into the equation gives:

\[
x = (2^6)^{\frac{1}{23}} = 2^{\frac{6}{23}}
\]

We can convert this back to a numerical answer, but to find an approximate decimal value:

Calculating \( 2^{\frac{6}{23}} \) yields a number between \( 1 \) and \( 2 \).

To match your answer choices, we need to approximate:

- \( 8 = 2^3 \)
- \( 16 = 2^4 \)
- \( 32 = 2^5 \)
- \( 512 = 2^9 \)

Since \( \frac{6}{23} \) is less than \( 1 \), the value is closer to \( 2^0 = 1 \).

However, we cannot select one of the options accurately as no value matches directly; it appears none of the provided responses (512, 32, 16, 8) equals \( 2^{\frac{6}{23}} \).

Hence, since we are approximating a rational expression and the specific answer is not directly present in the options, if I had to choose one for a reasonable estimate, I would assess it but also on the basis that \( 2^{\frac{6}{23}} < 2 \).

If we must choose one from the original choices without matching exactly, we might opt for the response that represents numbers around \( 1 \) but choose not to ignore that the solution should rather lead us to look for \( 2^{\frac{6}{23}} \) being not equated fairly to any option therein.

In conclusion, the direct output of \( x \) from the equation is \( x = 2^{\frac{6}{23}} \), which does not match neatly with the selection given but leans towards less than \( 8 \).