To find the square root of \(-8\) and the cube root of \(-8\), we can express these values in terms of complex numbers.
- Square root of \(-8\):
The square root of a negative number can be expressed using the imaginary unit \(i\), where \(i = \sqrt{-1}\).
\[ \sqrt{-8} = \sqrt{8} \cdot i = \sqrt{4 \cdot 2} \cdot i = 2\sqrt{2} \cdot i \]
Therefore,
\[ \sqrt{-8} = 2\sqrt{2} i \]
- Cube root of \(-8\):
The cube root of a negative number can be calculated directly. The cube root of \(-8\) is:
\[ \sqrt[3]{-8} = -2 \]
Putting it all together, we have:
- The square root of \(-8 = 2\sqrt{2} i\)
- The cube root of \(-8 = -2\)
So:
- Square root of \(-8\): \(2\sqrt{2} i\)
- Cube root of \(-8\): \(-2\)