Find the Following Values
\[ x^4+\frac{1}{x^4}=194 \]
Find:
\[ x^3+\frac{1}{x^3},\quad x^2+\frac{1}{x^2},\quad x+\frac{1}{x} \]
Solution:
Using identity:
\[ \left(x^2+\frac{1}{x^2}\right)^2 = x^4+\frac{1}{x^4}+2 \]
\[ \left(x^2+\frac{1}{x^2}\right)^2 = 194+2 \]
\[ \left(x^2+\frac{1}{x^2}\right)^2 = 196 \]
\[ x^2+\frac{1}{x^2} = 14 \]
Now,
\[ \left(x+\frac{1}{x}\right)^2 = x^2+\frac{1}{x^2}+2 \]
\[ \left(x+\frac{1}{x}\right)^2 = 14+2 \]
\[ \left(x+\frac{1}{x}\right)^2 = 16 \]
\[ x+\frac{1}{x} = 4 \]
Now using identity:
\[ a^3+b^3=(a+b)^3-3ab(a+b) \]
Here,
\[ a=x,\quad b=\frac{1}{x},\quad ab=1 \]
\[ x^3+\frac{1}{x^3} = \left(x+\frac{1}{x}\right)^3 -3\left(x+\frac{1}{x}\right) \]
\[ = (4)^3-3(4) \]
\[ = 64-12 \]
\[ =52 \]