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
- If $a + b + c = 10$ and $ab + bc + ca = 31$, find the value of $a^2 + b^2 + c^2$
If $a + b + c = 0$, then find the value of $\frac{(a^2+b^2+c^2)^2}{a^2b^2+b^2c^2+c^2a^2}$
- If $x^4 + \frac{1}{x^4} = 322$, then $x^3 - \frac{1}{x^3} = ?$
- If $(x + \frac{1}{x}) = 7$, then $(x - \frac{1}{x})$ is equal to:
- If $x + y + z = 0$, then the value of $\frac{x^2}{yz} + \frac{y^2}{zx} + \frac{z^2}{xy}$ is:
- What is the value of the following expression:
$(1 + x)(1 + x^2)(1 + x^4)(1 + x^8)(1 - x)$ - If $a + b + c = 17$, $abc = 168$, and $ab + bc + ca = 94$, then $a^3 + b^3 + c^3 = ?$
- If $a = \frac{b^2}{b - a}$, then the value of $a^3 + b^3$ is:
- If $ab = 10$ and $a^{2} + b^{2} = 29$, then $(a - b)^{2} = ?$
- Find the value of $\frac{(74 + 47)^2 + (74 - 47)^2}{74^2 + 47^2}$
Important Questions from Identities
If x 2 + 4y 2 = 40, xy = 6 and x > 2y then the value of x - 2y is:
- \(\dfrac{(5.17-2.19)^2-(5.17+2.19)^2}{11.3223}\)
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?
If a + b + c = 5, a 2+ b 2+ c 2= 27, a 3 + b 3 + c 3 = 125, then the value of \(\frac{abc}{5}\) is:
(999) 2- 2 2=
