If a^2+1/a^2=7 find the value of a+1/a, a-1/a and a^2-1/a^2 [S.Chand ICSE Class 9 Chapter 3 Exercise 3.1 Expansion and Factorisation Solution]
(i) $\left(a + \frac{1}{a}\right)$
$\left(a + \frac{1}{a}\right)^2$ = $a^2 + 2.a\ldotp \frac{1}{a} + \frac{1}{a^2}$
$\Rightarrow \left(a + \frac{1}{a}\right)^2 $=$ a^2 + \frac{1}{a^2} + 2$
$\Rightarrow \left(a + \frac{1}{a}\right)^2$ = 7 + 2
$\Rightarrow \left(a + \frac{1}{a}\right)^2$ = 9
$\Rightarrow \left(a + \frac{1}{a}\right)$ = $\pm 3$
(ii) $\left(a – \frac{1}{a}\right)$
$\left(a – \frac{1}{a}\right)^2$ = $a^2 – 2.a\ldotp \frac{1}{a} + \frac{1}{a^2}$
$\Rightarrow \left(a – \frac{1}{a}\right)^2$ =$ a^2 + \frac{1}{a^2} – 2$
$\Rightarrow \left(a – \frac{1}{a}\right)^2$ = 7 – 2
$\Rightarrow \left(a – \frac{1}{a}\right)^2$ = 5
$\Rightarrow \left(a – \frac{1}{a}\right)$ = $\pm \sqrt{5}$
(iii) $a^2 – \frac{1}{a^2}$
$a^2 – \frac{1}{a^2}$
= $\left(a + \frac{1}{a}\right)\left(a – \frac{1}{a}\right)$
= $\left( \pm 3\right)\left( \pm \sqrt{5}\right)$
= $\pm 3\sqrt{5}$