Solve the following equation with a rational exponent: x23=64

Question

.(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 $8$.