Find the real solution(s) of the following equation.

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

GPT-4o mini · Answer

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 : $ 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 : $ n = \sqrt[3]{64} $ - **Choice C**: $ n = 4 $ So the answers are ** and **C**.