Find the Value
\[ x^4+\frac{1}{x^4}=119 \]
Find:
\[ x^3-\frac{1}{x^3} \]
Solution:
Using identity:
\[ \left(x^2+\frac{1}{x^2}\right)^2 = x^4+\frac{1}{x^4}+2 \]
\[ \left(x^2+\frac{1}{x^2}\right)^2 = 119+2 \]
\[ \left(x^2+\frac{1}{x^2}\right)^2 = 121 \]
\[ x^2+\frac{1}{x^2} = 11 \]
Now,
\[ \left(x-\frac{1}{x}\right)^2 = x^2+\frac{1}{x^2}-2 \]
\[ \left(x-\frac{1}{x}\right)^2 = 11-2 \]
\[ \left(x-\frac{1}{x}\right)^2 = 9 \]
\[ x-\frac{1}{x} = 3 \]
Now using identity:
\[ a^3-b^3=(a-b)^3+3ab(a-b) \]
Here,
\[ a=x,\quad b=\frac{1}{x},\quad ab=1 \]
\[ x^3-\frac{1}{x^3} = \left(x-\frac{1}{x}\right)^3 +3\left(x-\frac{1}{x}\right) \]
\[ = (3)^3+3(3) \]
\[ = 27+9 \]
\[ =36 \]