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

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

Since \( 16 = 2^4 \), we can rewrite the equation as:

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

Next, we can divide both sides of the equation by 2:

\[
x^{32} = 2^{4-1} = 2^3
\]

Now we have:

\[
x^{32} = 8
\]

To solve for \( x \), we raise both sides to the power of \( \frac{1}{32} \):

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

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

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

This gives us a real number value for \( x \) instead of one of the provided integer choices. However, let's approximate it.

Since \( 2^{\frac{3}{32}} \) is between 1 and 2, we can conclude that \( x \) does not match any of the integer options given (4, 8, 84, or no solution) precisely.

Thus, the answer is **no solution** within the context of the integer responses provided.

Solve the following equation with a rational exponent: 5x43=80 .(1 point) Responses

8
12
16
40.32

To solve the equation \( 5x^{43} = 80 \), we first isolate \( x^{43} \) by dividing both sides by 5:

\[
x^{43} = \frac{80}{5}
\]

Calculating the right side gives:

\[
x^{43} = 16
\]

Next, we need to solve for \( x \) by taking both sides to the power of \( \frac{1}{43} \):

\[
x = 16^{\frac{1}{43}}
\]

We can express 16 as a power of 2:

\[
16 = 2^4
\]

Thus, we have:

\[
x = (2^4)^{\frac{1}{43}} = 2^{\frac{4}{43}}
\]

Now, we approximate \( 2^{\frac{4}{43}} \). This exponent will yield a small value since \( \frac{4}{43} \) is less than 1.

To evaluate \( 2^{\frac{4}{43}} \) numerically, we can estimate:

- \( 2^{\frac{4}{43}} \) is a little more than 1, specifically much closer to 1 than to 2.

Now, looking at the choices provided (8, 12, 16, and 40.32), none of these options represent the approximate value of \( 2^{\frac{4}{43}} \), which is a small positive number slightly greater than 1.

Thus, it seems necessary to conclude that within the context of your options, none are correct. Therefore, if this is a multiple-choice examination, the answer would likely be **none of the above**. Alternatively, if you meant we should select a value close to what we computed, I would clarify that \( x \approx 1 \).

there is no none of the above just pick one that you think is right

Given the options provided (8, 12, 16, 40.32) and the fact that we calculated \( x \approx 2^{\frac{4}{43}} \), which is a value slightly greater than 1 but much less than 8, 12, or 16, the smallest among the options is 8.

If I must choose one of the provided options, I would select **8**, since it is the smallest and closest to our approximate value of \( x \).

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

512
8
16
32

To solve the equation \( x^{23} = 64 \), we first rewrite 64 as a power of 2:

\[
64 = 2^6
\]

Thus, we have:

\[
x^{23} = 2^6
\]

Next, we solve for \( x \) by raising both sides to the power of \( \frac{1}{23} \):

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

Now, we can approximate \( 2^{\frac{6}{23}} \). Since \( \frac{6}{23} \) is a small fraction and less than 1, we know \( x \) will be a number between 1 and 2.

### Evaluating Possible Answers
Let’s evaluate the choices given (512, 8, 16, 32):

1. **512** is \( 2^9 \) (too high).
2. **8** is \( 2^3 \) (too low).
3. **16** is \( 2^4 \) (too low).
4. **32** is \( 2^5 \) (too low).

Since all these options seem quite large compared to the approximate value of \( 2^{\frac{6}{23}} \), we note \( 2^{\frac{6}{23}} \) is definitely less than 2.

Among the provided choices, **8** is the smallest and seems the most reasonable to choose, since it is the least far above the value we calculated.

Thus, I would select **8** as the answer, since it is the only reasonable guess based on rational exponent estimates, although none are exactly correct.