Question:
If \[ a^2+\frac{1}{a^2}=102 \] find:
\[ a-\frac{1}{a} \]
Solution:
Using identity:
\[ \left(a-\frac{1}{a}\right)^2 = a^2+\frac{1}{a^2}-2 \]
Substituting the given value:
\[ \left(a-\frac{1}{a}\right)^2 = 102-2 \]
\[ \left(a-\frac{1}{a}\right)^2 = 100 \]
\[ a-\frac{1}{a} = \sqrt{100} \]
\[ =10 \]