Question:
If \[ x+\frac{1}{x}=4, \] then \[ x^4+\frac{1}{x^4}= \]
(a) 196
(b) 194
(c) 192
(d) 190
Solution:
First find \[ x^2+\frac{1}{x^2} \]
Using identity:
\[ \left(x+\frac{1}{x}\right)^2 = x^2+\frac{1}{x^2}+2 \]
Substituting the given value:
\[ (4)^2 = x^2+\frac{1}{x^2}+2 \]
\[ 16 = x^2+\frac{1}{x^2}+2 \]
\[ x^2+\frac{1}{x^2} = 14 \]
Now using identity:
\[ \left(x^2+\frac{1}{x^2}\right)^2 = x^4+\frac{1}{x^4}+2 \]
\[ (14)^2 = x^4+\frac{1}{x^4}+2 \]
\[ 196 = x^4+\frac{1}{x^4}+2 \]
\[ x^4+\frac{1}{x^4} = 196-2 \]
\[ =194 \]
Hence, the correct answer is:
\[ \boxed{194} \]