Question

if a=3-2√2 find the value of a^4-1/a^4​

Answer 1

Step-by-step explanation:

a + 1/a = (3 - 2√2) + 1/(3 - 2√2)

= (3 - 2√2)² + 1 / (3 - 2√2)

= (9 + 8 - 12√2 + 1)/(3 - 2√2)

= (18 - 12√2) / (3 - 2√2)

= 6(3 - 2√2) / (3 - 2√2)

a + 1/a = 6

a - 1/a = √(a + 1/a)² - 4

= √36 - 4

= √32 = 4√2

(a + 1/a)(a - 1/a) = 6 × 4√2

a² - 1/a² = 24√2 -------(1)

a² + 1/a² + 2 = 36

a² + 1/a² = 34 --------(2)

Now , from equation (1) × (2)

(a² - 1/a²) × (a² + 1/a²) = 24√2 × 34

a⁴ - 1/a⁴ = 816√2

Answer 2

ANSWER :–

a⁴ - 1/a⁴ = - 816√2

EXPLANATION :–

GIVEN :–

• a = 3-2√2

TO FIND :–

Value of a⁴ - 1/a⁴.

SOLUTION :–

• According to the question –

⇨ a = 3-2√2

➨ Now let's find 1/a –

⇨ 1/a = [1/(3 - 2√2)][(3 + 2√2)(3 + 2√2)]

⇨ 1/a = (3 + 2√2)/(9 - 8)

⇨ 1/a = 3 + 2√2

➨ Now let's find (a² + 1/a²) –

⇨ (a + 1/a)² = a² + (1/a²) + 2(a)(1/a)

⇨ (6)² = a² + (1/a²) + 2

⇨ 36 = a² + (1/a²) + 2

⇨ a² + 1/a² = 34 —–———eq.(1)

➨ Now let's find (a² - 1/a²) –

⇨ (a² - 1/a²) = (a - 1/a)(a + 1/a)

⇨ a² - 1/a² = (-4√2)(6)

⇨ a² - 1/a² = -24√2 ——–—–eq.(2)

• Now multiply both equations —

➨ (a² + 1/a²)(a² - 1/a² ) = a⁴ - 1/a⁴

➨ a⁴ - 1/a⁴ = (34)(-24√2)

➨ a⁴ - 1/a⁴ = - 816√2