1. **State the problem:** Given $a = \frac{1+\sqrt{5}}{2}$, find the value of $a^6 - 8a^2$. 2. **Recall the formula and properties:** The number $a = \frac{1+\sqrt{5}}{2}$ is the golden ratio, often denoted by $\phi$. It satisfies the quadratic equation: $$a^2 = a + 1$$ This relation will help simplify higher powers of $a$. 3. **Find $a^2$:** From the definition, $$a^2 = a + 1$$ 4. **Find $a^3$:** Multiply both sides of $a^2 = a + 1$ by $a$: $$a^3 = a \cdot a^2 = a(a + 1) = a^2 + a$$ Substitute $a^2 = a + 1$: $$a^3 = (a + 1) + a = 2a + 1$$ 5. **Find $a^4$:** Multiply $a^3$ by $a$: $$a^4 = a \cdot a^3 = a(2a + 1) = 2a^2 + a$$ Substitute $a^2 = a + 1$: $$a^4 = 2(a + 1) + a = 2a + 2 + a = 3a + 2$$ 6. **Find $a^5$:** Multiply $a^4$ by $a$: $$a^5 = a \cdot a^4 = a(3a + 2) = 3a^2 + 2a$$ Substitute $a^2 = a + 1$: $$a^5 = 3(a + 1) + 2a = 3a + 3 + 2a = 5a + 3$$ 7. **Find $a^6$:** Multiply $a^5$ by $a$: $$a^6 = a \cdot a^5 = a(5a + 3) = 5a^2 + 3a$$ Substitute $a^2 = a + 1$: $$a^6 = 5(a + 1) + 3a = 5a + 5 + 3a = 8a + 5$$ 8. **Calculate $a^6 - 8a^2$:** Substitute the expressions found: $$a^6 - 8a^2 = (8a + 5) - 8(a + 1) = 8a + 5 - 8a - 8 = 5 - 8 = -3$$ **Final answer:** $$a^6 - 8a^2 = -3$$ This corresponds to option A.