If $a = \frac{\sqrt{5} + 1}{\sqrt{5} - 1}$ and $b = \frac{\sqrt{5} - 1}{\sqrt{5} + 1}$, the value of $\left(\frac{a^2 + ab + b^2}{a^2 - ab + b^2}\right)$ is

### Concept & Logic This requires simplifying complex algebraic expressions by evaluating the sum and product of reciprocal surd fractions. By rearranging the numerator and denominator, we can express the entire fraction in terms of $(a^2 + b^2)$ and $ab$. ### Step-by-Step Solution * **Given:** $a = \frac{\sqrt{5} + 1}{\sqrt{5} - 1}$ and $b = \frac{\sqrt{5} - 1}{\sqrt{5} + 1}$ * Notice that $b$ is the reciprocal of $a$. Therefore, the product $ab = 1$. * Now, rationalize $a$ by multiplying the numerator and denominator by $(\sqrt{5} + 1)$: * $a = \frac{(\sqrt{5} + 1)^2}{(\sqrt{5} - 1)(\sqrt{5} + 1)} = \frac{5 + 1 + 2\sqrt{5}}{5 - 1} = \frac{6 + 2\sqrt{5}}{4} = \frac{3 + \sqrt{5}}{2}$ * Similarly, rationalizing $b$ gives the conjugate: * $b = \frac{3 - \sqrt{5}}{2}$ * Find the sum $(a + b)$: * $a + b = \frac{3 + \sqrt{5} + 3 - \sqrt{5}}{2} = \frac{6}{2} = 3$ * Find the sum of their squares using the identity $a^2 + b^2 = (a+b)^2 - 2ab$: * $a^2 + b^2 = (3)^2 - 2(1) = 9 - 2 = 7$ * Now, substitute these values into the target expression: * Target Expression: $\frac{a^2 + ab + b^2}{a^2 - ab + b^2}$ * Rearrange to group terms: $\frac{(a^2 + b^2) + ab}{(a^2 + b^2) - ab}$ * Substitute $(a^2 + b^2) = 7$ and $ab = 1$: * $\frac{7 + 1}{7 - 1} = \frac{8}{6}$ * Simplify the fraction: $\frac{4}{3}$ ### Exam Strategy & Shortcut Use the direct sum formula for reciprocal surds of the form $a = \frac{\sqrt{x} + \sqrt{y}}{\sqrt{x} - \sqrt{y}}$. Note that $1$ can be written as $\sqrt{1}$. The sum $a + b = 2 \cdot \frac{x + y}{x - y}$. Here, $x = 5$ and $y = 1$. $a + b = 2 \cdot \frac{5 + 1}{5 - 1} = 2 \cdot \frac{6}{4} = 3$. Since $ab = 1$, we instantly know $a^2 + b^2 = 3^2 - 2 = 7$. Plug directly into $\frac{7 + 1}{7 - 1} = \frac{8}{6} = \frac{4}{3}$. This solves the problem in under 30 seconds. ### Common Pitfall A common mistake is incorrectly expanding the terms in the target expression or failing to recognize that $a$ and $b$ are reciprocals. If you miss that $ab = 1$ immediately, you will waste valuable time multiplying complex fractions. ### Final Answer **Therefore, the correct answer is 4/3.**