Question

Which of the expressions represent solutions to the equation y3 = 64
?
​Select each correct answer.
A
−643-\sqrt[3]{64}−
3

64
​

B
643\sqrt[3]{64}
3

64
​

C
−8-8−8
D
888
E
−4-4−4
F
444

Answer 1

To solve the equation \( y^3 = 64 \), we need to find the cube roots of 64.

Calculating the cube root of 64, we have:

\[
y = \sqrt[3]{64} = 4
\]

Since raising to the third power is unique for real numbers, the solution \( y = 4 \) gives us the primary real solution.

However, since \( y^3 = 64 \) can also derive complex solutions, the other roots can be found using the fact that any non-zero number \( r \) has \( n \) roots. Specifically, we can find the two additional cube roots in the complex plane.

The complex cube roots can be given by:

1. \( y = 4 \)
2. \( y = 4 \cdot \text{cis}\left(\frac{2\pi}{3}\right) \) (which gives us a second root), and
3. \( y = 4 \cdot \text{cis}\left(\frac{4\pi}{3}\right) \) (the third root).

For our case of \( y^3 = 64 \):
- The real solution is **4**
- The negative of the real root is **-4** which is not actually a solution in this case.

Now let's evaluate the given choices:
- **A**: \( -\sqrt[3]{64} = -4 \) (not a solution)
- **B**: \( \sqrt[3]{64} = 4 \) (a solution)
- **C**: \( -8 \) (not a solution)
- **D**: \( 8 \) (not a solution)
- **E**: \( -4 \) (not a solution)
- **F**: \( 4 \) (a solution)

The correct answers to the expression representing solutions to the equation \( y^3 = 64 \) are:
- **B** \( \sqrt[3]{64} \)
- **F** \( 4 \)