To solve the equation \( n^5 + 10 = -20 \), we first isolate \( n^5 \):
\[ n^5 + 10 = -20 \]
Subtract 10 from both sides:
\[ n^5 = -20 - 10 \]
This simplifies to:
\[ n^5 = -30 \]
Now, we take the fifth root of both sides to solve for \( n \):
\[ n = \sqrt[5]{-30} \]
The value of \( n \) is approximately \( -2.245 \) (but we don't have this in the provided responses as an option).
Reviewing the options provided:
- -150
- -50
- -6
- -2
We check which value would obey \( n^5 = -30 \):
-
\( n = -2 \): \[ (-2)^5 = -32 \] This is not equal to -30.
-
\( n = -6 \): \[ (-6)^5 = -7776
\] This is not equal to -30. -
\( n = -50 \): \[ (-50)^5 = -31250000
\] This is not equal to -30. -
\( n = -150 \): \[ (-150)^5 = -75937500000
\] This is not equal to -30.
None of the provided options exactly solve the equation \( n^5 + 10 = -20 \) since \( n = \sqrt[5]{-30} \) is neither of the values provided.
However, among the options, the closest approximation to \( \sqrt[5]{-30} \) is \( -2 \).
So, the best choice based on the options given is:
\[ \text{The solution is: } -2 \]