To solve the problem, we need to find the value of \((2 + \sqrt{5})^4 - (2 - \sqrt{5})^4\).
- Let's apply the formula for the difference of powers:
a^4 - b^4 = (a^2 + b^2)(a^2 - b^2)
- Here, \(a = (2 + \sqrt{5})\) and \(b = (2 - \sqrt{5})\).
- First, find \(a^2\) and \(b^2\):
- \((2 + \sqrt{5})^2 = 2^2 + 2 \times 2 \times \sqrt{5} + (\sqrt{5})^2 = 4 + 4\sqrt{5} + 5 = 9 + 4\sqrt{5}\)
- \((2 - \sqrt{5})^2 = 2^2 - 2 \times 2 \times \sqrt{5} + (\sqrt{5})^2 = 4 - 4\sqrt{5} + 5 = 9 - 4\sqrt{5}\)
- Calculate \(a^2 + b^2\):
(9 + 4\sqrt{5}) + (9 - 4\sqrt{5}) = 9 + 9 = 18
- Calculate \(a^2 - b^2\):
(9 + 4\sqrt{5}) - (9 - 4\sqrt{5}) = 9 + 4\sqrt{5} - 9 + 4\sqrt{5} = 8\sqrt{5}
- Substitute these values back into the formula:
a^4 - b^4 = (a^2 + b^2)(a^2 - b^2) = 18 \times 8\sqrt{5} = 144\sqrt{5}
Thus, the value of \((2 + \sqrt{5})^4 - (2 - \sqrt{5})^4\) is \(\mathbf{144\sqrt{5}}\), which matches the correct option.
Similar Questions
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Important Questions from Identities
If \(2x + { {1} \over 3x}= 5, x ≠ 0\) , then what is the value of \(27x^3+{{1} \over 8x^3}\) ?
If x 2 + 4y 2 = 40, xy = 6 and x > 2y then the value of x - 2y is:
- \(\dfrac{(0.73)^3+(0.31)^3}{(0.73)^2-0.73\times0.31+(0.31)^2}\)
- \(\dfrac{(5.17-2.19)^2-(5.17+2.19)^2}{11.3223}\)
If a(a + b + c) 2= 1792; b(a + b + c) 2= 1536; c(a + b + c) 2= 768 then what will be the value of a?
