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

Question

4
8
84
no solution

Answer 1

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.

Answer 2

Solve the following equation with a rational exponent: 5x43=80 .(1 point) Responses
8
12
16
40.32

Answer 3

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 \( x \approx 1 \).

Answer 4

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

Answer 5

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 \( x \).

Answer 6

Solve the following equation with a rational exponent: x23=64 .(1 point) Responses
512
8
16
32

Answer 7

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.