Question

If a-1/a =√5 find the value of a+1/a and a3 +1/a3
chapter : identities

Answer 1

Part 1:

Squaring both sides:

(a)² + ($\frac{1}{a}$)² - 2 = ($\sqrt{5}$)²

a² + ($\frac{1}{a}$)² = 5 + 2 = 7

a² + ($\frac{1}{a}$)² ± 2 = 7

Considering positive 2:

(a + $\frac{1}{a}$)² = 7

a + $\frac{1}{a}$ = $\sqrt{7}$

Hence a + $\frac{1}{a}$ = $\sqrt{7}$

Part 2:

Cubing both sides:

(a + $\frac{1}{a}$)³ = $\sqrt{7}$³

a³ + ($\frac{1}{a}$)³ + 3(a +

a³ + ($\frac{1}{a}$)³ + 3($\sqrt{7}$) = 7

a³ + ($\frac{1}{a}$)³ = 7$\sqrt{7}$ - 3

a³ + ($\frac{1}{a}$)³ = 4$\sqrt{7}$

Hence a³ + ($\frac{1}{a}$)³ = 4$\sqrt{7}$