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if a = √(7 + 2√12)  and b = √(7 – 2√12) then find the value of (a3 + b3)


1. 50
2. 48
3. 56
4. 58
5. 52

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Best answer
Correct Answer - Option 5 : 52

Given:

a = √(7 + 2√12)  and b = √(7 – 2√12)

Formula used:

(a + b)2 = a2 + b2 + 2ab

(a – b)2 = a2 + b2 – 2ab

(a + b)3 = a3 + b3 + 3ab(a + b)

Calculation:

a = √(7 + 2√12)

⇒ √(4 + 3 + 2 × √4 × √3)

⇒ √[(2)2 + (√3)3 + 2 × 2 × √3]

⇒ √(2 + √3)2

⇒ (2 + √3)

Similarly b = (2 – √3)           

⇒ a + b = 2 + √3 + 2 – √3 = 4

⇒ ab = (2 + √3)(2 – √3) = (2)2 – (√3)2 = 4 – 3 = 1

(a + b)3 = a3 + b3 + 3ab(a + b)

a3 + b3 = (a + b)3 – 3ab(a + b)

⇒ (4)3 – 3 × 1 × 4

⇒ 64 – 12

⇒ 52

The value of a3 + b3 is 52.

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