Question:
If
\[ x^3+\frac{1}{x^3}=110, \] then \[ x+\frac{1}{x}= \]
(a) 5
(b) 10
(c) 15
(d) none of these
Solution:
Using identity:
\[ \left(x+\frac{1}{x}\right)^3 = x^3+\frac{1}{x^3} + 3\left(x+\frac{1}{x}\right) \]
Substituting the given value:
\[ \left(x+\frac{1}{x}\right)^3 = 110+3\left(x+\frac{1}{x}\right) \]
Let
\[ x+\frac{1}{x}=a \]
Then
\[ a^3=110+3a \]
\[ a^3-3a-110=0 \]
Checking the options:
\[ a=5 \]
\[ 5^3-3(5)-110 = 125-15-110 = 0 \]
Hence,
\[ x+\frac{1}{x}=5 \]
Therefore, the correct answer is:
\[ \boxed{5} \]