1. **Stating the problem:** We are given the system: $$a + b = \sqrt{3}, \quad a - b = \sqrt{2}$$ and the equations: $$8(a^2 + b^2)(a^3 - b^3) = 55\sqrt{2}$$ $$16a^2b^2(a^4 + b^4) = 49$$ We want to understand how to work with these expressions and possibly find values or simplify. 2. **Step 1: Find $a$ and $b$ from the system** Add the two equations: $$a + b = \sqrt{3}$$ $$a - b = \sqrt{2}$$ Adding gives: $$2a = \sqrt{3} + \sqrt{2} \implies a = \frac{\sqrt{3} + \sqrt{2}}{2}$$ Subtracting gives: $$2b = \sqrt{3} - \sqrt{2} \implies b = \frac{\sqrt{3} - \sqrt{2}}{2}$$ 3. **Step 2: Calculate $a^2$ and $b^2$** $$a^2 = \left(\frac{\sqrt{3} + \sqrt{2}}{2}\right)^2 = \frac{(\sqrt{3} + \sqrt{2})^2}{4} = \frac{3 + 2\sqrt{6} + 2}{4} = \frac{5 + 2\sqrt{6}}{4}$$ $$b^2 = \left(\frac{\sqrt{3} - \sqrt{2}}{2}\right)^2 = \frac{(\sqrt{3} - \sqrt{2})^2}{4} = \frac{3 - 2\sqrt{6} + 2}{4} = \frac{5 - 2\sqrt{6}}{4}$$ 4. **Step 3: Calculate $a^2 + b^2$** $$a^2 + b^2 = \frac{5 + 2\sqrt{6}}{4} + \frac{5 - 2\sqrt{6}}{4} = \frac{5 + 5}{4} = \frac{10}{4} = \frac{5}{2}$$ 5. **Step 4: Calculate $a^3 - b^3$** Use the identity: $$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$$ We know $a - b = \sqrt{2}$. Calculate $ab$: $$ab = \left(\frac{\sqrt{3} + \sqrt{2}}{2}\right)\left(\frac{\sqrt{3} - \sqrt{2}}{2}\right) = \frac{(\sqrt{3})^2 - (\sqrt{2})^2}{4} = \frac{3 - 2}{4} = \frac{1}{4}$$ Calculate $a^2 + ab + b^2$: $$a^2 + ab + b^2 = \frac{5}{2} + \frac{1}{4} = \frac{10}{4} + \frac{1}{4} = \frac{11}{4}$$ Therefore: $$a^3 - b^3 = \sqrt{2} \times \frac{11}{4} = \frac{11\sqrt{2}}{4}$$ 6. **Step 5: Verify the first big equation** $$8(a^2 + b^2)(a^3 - b^3) = 8 \times \frac{5}{2} \times \frac{11\sqrt{2}}{4} = 8 \times \frac{55\sqrt{2}}{8} = 55\sqrt{2}$$ This matches the given equation. 7. **Step 6: Calculate $a^4 + b^4$** Use the identity: $$a^4 + b^4 = (a^2 + b^2)^2 - 2a^2b^2$$ Calculate $a^2b^2 = (ab)^2 = \left(\frac{1}{4}\right)^2 = \frac{1}{16}$ Calculate: $$a^4 + b^4 = \left(\frac{5}{2}\right)^2 - 2 \times \frac{1}{16} = \frac{25}{4} - \frac{2}{16} = \frac{25}{4} - \frac{1}{8} = \frac{50}{8} - \frac{1}{8} = \frac{49}{8}$$ 8. **Step 7: Verify the second big equation** $$16a^2b^2(a^4 + b^4) = 16 \times \frac{1}{16} \times \frac{49}{8} = 1 \times \frac{49}{8} = \frac{49}{8}$$ The given equation states it equals 49, but the calculation yields $\frac{49}{8}$. This suggests a possible typo or misinterpretation in the problem statement. **Summary:** - We solved for $a$ and $b$. - Verified the first equation exactly. - Calculated $a^4 + b^4$ and found a discrepancy in the second equation. This is how you approach such problems: solve the system, use algebraic identities, substitute, and verify step-by-step.