Question

Find the real solution(s) of the following equation.
\[n^3 = 64\]
Choose all answers that apply:
Choose all answers that apply:
(Choice A)
\[n = \sqrt[3]{64}\]
A
\[n = \sqrt[3]{64}\]
(Choice B)
\[n = -\sqrt[3]{64}\]
B
\[n = -\sqrt[3]{64}\]
(Choice C)
\[n = 4\]
C
\[n = 4\]
(Choice D)
\[n = -4\]
D
\[n = -4\]
(Choice E) None of the above
E
None of the above

Answer 1

To solve the equation \( n^3 = 64 \), we first need to find the cube root of 64.

Calculating the cube root:
\[
n = \sqrt[3]{64}
\]

We also know that:
\[
64 = 4^3
\]

Thus, we find that:
\[
n = \sqrt[3]{64} = 4
\]

Now, we check if there are any other possible values for \( n \). The equation \( n^3 = 64 \) also implies that the only real cube root of a positive number is positive. Therefore:
\[
n = -\sqrt[3]{64}
\]
is not a valid solution since it results in \( (-4)^3 = -64 \).

Next, evaluating the choices:

- **Choice A**: \( n = \sqrt[3]{64} \) is true, since \( \sqrt[3]{64} = 4 \).
- **Choice B**: \( n = -\sqrt[3]{64} \) is false, since the cube of \( -4 \) does not equal \( 64 \).
- **Choice C**: \( n = 4 \) is true, as we found this to be the solution.
- **Choice D**: \( n = -4 \) is false, since \( (-4)^3 = -64 \), which does not satisfy the equation.
- **Choice E**: This choice is also false because we have valid solutions with choices A and C.

Thus, the correct choices are:
- **Choice A**: \( n = \sqrt[3]{64} \)
- **Choice C**: \( n = 4 \)

So the answers are **A** and **C**.