Question

Solve the following equation with a rational exponent:  x2/.3=64 .(1 point)Responses323216165125128

Answer 1

To solve the equation \( \frac{x^2}{0.3} = 64 \), we first isolate \( x^2 \):

1. Multiply both sides by 0.3:
\[
x^2 = 64 \times 0.3
\]

2. Calculate \( 64 \times 0.3 \):
\[
64 \times 0.3 = 19.2
\]
So we have:
\[
x^2 = 19.2
\]

3. Now take the square root of both sides:
\[
x = \sqrt{19.2}
\]

4. To simplify \( \sqrt{19.2} \), express 19.2 as a fraction:
\[
19.2 = \frac{192}{10} = \frac{96}{5}
\]

5. Now let's find:
\[
x = \sqrt{\frac{96}{5}} = \frac{\sqrt{96}}{\sqrt{5}} = \frac{4\sqrt{6}}{\sqrt{5}} = \frac{4\sqrt{30}}{5}
\]

Now, we can calculate the decimal value:
1. First, find the estimated value of \( \sqrt{30} \) which is approximately \( 5.477 \).
2. Thus:
\[
x \approx \frac{4 \times 5.477}{5} \approx \frac{21.908}{5} \approx 4.3816
\]

Since none of the exact values are integer values, let's compare \( \sqrt{19.2} \) with the options given.

We can check each of the provided answers to find the one that is closest to \( 4.3816 \):

- **32**
- **32**
- **16**
- **165**
- **125**
- **128**

None of the direct comparisons match, but it's likely there is an error in interpreting or calculating the options.

To estimate:
- **16** is the closest integer and is a value that can be considered within the rational interpretation of the value of \( x \).

If we pick amongst the integers, the answer to pick would be **16** as it fits as the nearest result when considering the roots.

Thus the best response to choose from the provided options is **16**.