Question

if a2 +1/a2=7 find a3+1/a3​

Answer 1

Answer:

Step-by-step explanation:

a² + 1/a² = 7, a³ + 1/a³ = ?

=> a² + 1/a² = (a + 1/a)² - 2

=> 7 = (a + 1/a)² - 2

=> (a + 1/a)² = 9

=> a + 1/a = ±3.

i.e. a + 1/a = +3 or a + 1/a = -3

We know that  (a + b)³ = a³ + b³ + 3ab(a+b) i.e. a³ + b³ = (a + b)³ - 3ab(a+b)

So, Let us first consider a + 1/a = 3

a³ + 1/a³ = (a + 1/a)³ - 3(a)(1/a)(a+1/a)

              = 3³ - 3 (3)

               = 27 - 9

               = 18.

Let us first consider a + 1/a = -3

a³ + 1/a³ = (a + 1/a)³ - 3(a)(1/a)(a+1/a)

              = (-3)³ - 3 (-3)

               = -27 + 9

               = -18.

Answer 2

given

a² + 1/a² =7

a² + 1/a² + 2 = 7+2 ( adding 2 both sides)

(a + 1/a)² = 9

a + 1/a = ± 3

so a + 1/a=3 --------(1)

& a + 1/a= -3 ---------(2)

Cubing both sides in equation (1) we get

( a+ 1/a)³ = 27

a³ + 1/a³ +3(a+1/a) = 27

a³ + 1/a³ +3×3 = 27 [ since a + 1/a =3]

a³ + 1/a³ =18

And cubing both sides equation (2)

(a+1/a)³ = -27

a³ + 1/a³ + 3(a+1/a) = -27

a³ + 1/a³ +3(-3) = -27 [since a+1/a= -3]

a³ + 1/a³ -9 = -27

a³ + 1/a³ = -27+9

a³ + 1/a³ = -18