Question

Find the solution(s) of the following equation.

\[v^2 = \dfrac{25}{81}\]
Choose all answers that apply:
Choose all answers that apply:
(Choice A)

\[v = \dfrac{5}{81}\]
A

\[v = \dfrac{5}{81}\]
(Choice B)

\[v = -\dfrac{5}{81}\]
B

\[v = -\dfrac{5}{81}\]
(Choice C)

\[v = \sqrt{\dfrac{25}{81}}\]
C

\[v = \sqrt{\dfrac{25}{81}}\]
(Choice D)

\[v = -\sqrt{\dfrac{25}{81}}\]
D

\[v = -\sqrt{\dfrac{25}{81}}\]
(Choice E) None of the above
E
None of the above

Answer 1

To solve the equation \(v^2 = \dfrac{25}{81}\), we will take the square root of both sides.

1. Taking the square root of both sides, we have:
\[
v = \pm \sqrt{\dfrac{25}{81}}
\]

2. Now, we'll simplify \(\sqrt{\dfrac{25}{81}}\):
\[
\sqrt{\dfrac{25}{81}} = \dfrac{\sqrt{25}}{\sqrt{81}} = \dfrac{5}{9}
\]

3. Therefore, the solutions for \(v\) are:
\[
v = \dfrac{5}{9} \quad \text{and} \quad v = -\dfrac{5}{9}
\]

Now, we check the choices provided:

- **Choice A:** \(v = \dfrac{5}{81}\) (incorrect, as this is not the correct solution).
- **Choice B:** \(v = -\dfrac{5}{81}\) (incorrect, as this is not the correct solution).
- **Choice C:** \(v = \sqrt{\dfrac{25}{81}} = \dfrac{5}{9}\) (correct).
- **Choice D:** \(v = -\sqrt{\dfrac{25}{81}} = -\dfrac{5}{9}\) (correct).
- **Choice E:** None of the above (incorrect, since choices C and D are correct).

The correct choices are **C and D**.