Question:
If
\[ x^4+\frac{1}{x^4}=194, \] then \[ x^3+\frac{1}{x^3}= \]
(a) 76
(b) 52
(c) 64
(d) none of these
Solution:
Using identity:
\[ x^4+\frac{1}{x^4} = \left(x^2+\frac{1}{x^2}\right)^2-2 \]
Substituting the given value:
\[ 194 = \left(x^2+\frac{1}{x^2}\right)^2-2 \]
\[ \left(x^2+\frac{1}{x^2}\right)^2 = 196 \]
\[ x^2+\frac{1}{x^2} = 14 \]
Now using:
\[ x^2+\frac{1}{x^2} = \left(x+\frac{1}{x}\right)^2-2 \]
\[ 14 = \left(x+\frac{1}{x}\right)^2-2 \]
\[ \left(x+\frac{1}{x}\right)^2 = 16 \]
\[ x+\frac{1}{x} = 4 \]
Now using identity:
\[ x^3+\frac{1}{x^3} = \left(x+\frac{1}{x}\right)^3 – 3\left(x+\frac{1}{x}\right) \]
\[ =4^3-3(4) \]
\[ =64-12 \]
\[ =52 \]
Hence, the correct answer is:
\[ \boxed{52} \]