Question

if a+1/a=2 , find : a4 +1/a4

Answer 1

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Answer 2

Correct Question :-

If a + 1/a = 2, find $\sf a^4 + \dfrac{1}{a^4}$

Answer :-

$\tt a^4 + \dfrac{1}{a^4} =2$

Solution :-

a + 1/a = 2

Squaring on both sides

(a + 1/a)² = (2)²

⇒ (a + 1/a)² = 4

⇒ (a)² + (1/a)² + 2(a)(1/a) = 4

[Since (a + b)² = a² + b² + 2ab]

⇒ a² + 1²/a² + 2 = 4

⇒ a² + 1/a² + 2 = 4

⇒ a² + 1/a² = 4 - 2

⇒ a² + 1/a² = 2

Squaring on both sides

(a² + 1/a²)² = (2)²

(a² + 1/a²)² = 4

⇒ (a²)² + (1/a²)² + 2(a²)(1/a²) = 4

[Since (a + b)² = a² + b² + 2ab]

⇒ (a²)² + (1/a²)² + 2 = 4

$\sf \implies a^{2(2)} + \dfrac{1^{2} }{ ({a}^{2})^2 } + 2 = 4$

$\sf \implies a^4 + \dfrac{1}{a^{2(2)}} + 2 = 4$

$\sf \implies a^4 + \dfrac{1}{a^4} + 2 = 4$

$\sf \implies a^4 + \dfrac{1}{a^4} = 4 - 2$

$\sf \implies a^4 + \dfrac{1}{a^4} =2$

$\bf \therefore a^4 + \dfrac{1}{a^4} =2$

Identity used :-

• (a + b)² = a² + b² + 2ab