Find the Value
\[ x^2+\frac{1}{x^2}=51 \]
Find:
\[ x^3-\frac{1}{x^3} \]
Solution:
Using identity:
\[ \left(x-\frac{1}{x}\right)^2 = x^2+\frac{1}{x^2}-2 \]
\[ \left(x-\frac{1}{x}\right)^2 = 51-2 \]
\[ \left(x-\frac{1}{x}\right)^2 = 49 \]
\[ x-\frac{1}{x} = 7 \]
Now using identity:
\[ a^3-b^3=(a-b)^3+3ab(a-b) \]
Here,
\[ a=x,\quad b=\frac{1}{x},\quad ab=1 \]
\[ x^3-\frac{1}{x^3} = \left(x-\frac{1}{x}\right)^3 +3\left(x-\frac{1}{x}\right) \]
\[ = (7)^3+3(7) \]
\[ = 343+21 \]
\[ =364 \]