a^2+1/a^2=14 a is not equal to 0 4a^3+4/a^3 +2a+2/a​

Correct Question :-

If a² + 1/a² = 14, then find 4a³ + 4/a³ + 2a + 2/a.

Answer :-

4a³ + 4/a³ + 2a + 2/a = 216

Solution :-

4a³ + 4/a³ + 2a + 2/a

⇒ 4(a³ + 1/a³) + 2(a + 1/a)

First find the value of a + 1/a

a² + 1/a² = 14

Adding 2 on both sides

⇒ a² + 1/a² + 2 = 14 + 2

⇒ a² + 1/a² + 2 = 16

⇒ (a)² + (1/a)² + 2(x)(1/x) = 16

⇒ (a + 1/a)² = 16

[Since (a + b)² = a² + b² + 2ab]

⇒ a + 1/a = √16

⇒ a + 1/a = 4

Now find the value of a³ + 1/a³

Now cubing on both sides

(a + 1/a)³ = (4)³

⇒ (a + 1/a)³ = 64

⇒ (a)³ + (1/a)³ + 3(a)(1/a)(a + 1/a) = 64

[Since (a + b)³ = a³ + b³ + 3ab(a + b) ]

⇒ a³ + 1³/a³ + 3(a + 1/a) = 64

⇒ a³ + 1/a³ + 3(a + 1/a) = 64

⇒ a³ + 1/a³ + 3(4) = 64

[Since a + 1/a = 4]

⇒ a³ + 1/a³ + 12 = 64

⇒ a³ + 1/a³ = 64 - 12

⇒ a³ + 1/a³ = 52

Now Consider 4(a³ + 1/a³) + 2(a + 1/a)

Here we got to know

• a + 1/a = 4

• a³ + 1/a³ = 52

By substituting the values

= 4(52) + 2(4)

= 208 + 8

= 216

4(a³ + 1/a³) + 2(a + 1/a) = 216

⇒ 4a³ + 4/a³ + 2a + 2/a = 216