Question

If a+1/a=2, find the value of
${a}^{4} + \frac{1}{a} {}^{4}$


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Answer 1

Solution :-

→ (a + 1/a) = 2

Squaring both sides, we get,

→ (a + 1/a)² = 2²

using (x + y)² = x² + y² + 2xy ,

→ a² + 1/a² + 2 * a * 1/a = 4

→ a² + 1/a² + 2 = 4

→ a² + 1/a² = 4 - 2

→ a² + 1/a² = 2

Now,

→ (a² + 1/a²) = 2

Squaring both sides, we get,

→ (a² + 1/a²)² = 2²

using (x + y)² = x² + y² + 2xy ,

→ a⁴ + 1/a⁴ + 2 * a² * 1/a² = 4

→ a⁴ + 1/a⁴ + 2 = 4

→ a⁴ + 1/a⁴ = 4 - 2

→ (a⁴ + 1/a⁴) = 2 (Ans.)

Answer 2

Question:-

If a + 1/a = 2, find a⁴ + 1/a⁴

Formula Used:-

(x + y)² = x² + y² + 2xy

Answer:-

1st method:-

2 = (a + 1/a)

Squaring both sides,

=> 2² = (a + 1/a)²

=> 4 = a² + 1/a² + (2 * a * 1/a)

=> a² + 1/a² = 4 - 2

=> a² + 1/a² = 2

2 = (a² + 1/a²)

Squaring both sides,

=> 2² = (a² + 1/a²)²

=> 4 = (a²)² + (1/a²)² + (2 * a² * 1/a²)

=> a⁴ + 1/a⁴ = 4 - 2

=> a⁴ + 1/a⁴ = 2

Ans. a⁴ + 1/a⁴ = 2

2nd Method:-

a + 1/a = 2

=> (a² + 1)/a = 2

=> a² - 2a + 1 = 0

=> a² - a - a + 1 = 0

=> a(a - 1) - 1(a - 1) = 0

=> (a - 1)(a - 1) = 0

So, a = 1

So, a⁴ + 1/a⁴

= (1)⁴ + 1/(1)⁴

= 2

Ans. a⁴ + 1/a⁴ = 2

From 2nd method, even if the question is asked, find the value of a^(9000) + 1/(a)^(9000), the answer will still be (1 + 1) = 2, as we know the value of a = 1

Similarly, if the question is asked a^(8000) - 1/(a)^(8000), the answer will be (1 - 1) = 0,

as we know the value of a = 1