1. **State the problem:** We are given two expressions: $$a = (\sqrt{5} + \sqrt{2})^{-2}$$ and $$b = \frac{\sqrt{5} - \sqrt{2}}{3}$$ We want to simplify and understand these expressions. 2. **Simplify expression for $a$:** Recall that $x^{-2} = \frac{1}{x^2}$, so $$a = \frac{1}{(\sqrt{5} + \sqrt{2})^2}$$ 3. **Expand the denominator:** $$(\sqrt{5} + \sqrt{2})^2 = (\sqrt{5})^2 + 2 \cdot \sqrt{5} \cdot \sqrt{2} + (\sqrt{2})^2 = 5 + 2\sqrt{10} + 2 = 7 + 2\sqrt{10}$$ 4. **Rewrite $a$:** $$a = \frac{1}{7 + 2\sqrt{10}}$$ 5. **Rationalize the denominator:** Multiply numerator and denominator by the conjugate $7 - 2\sqrt{10}$: $$a = \frac{1}{7 + 2\sqrt{10}} \times \frac{7 - 2\sqrt{10}}{7 - 2\sqrt{10}} = \frac{7 - 2\sqrt{10}}{(7)^2 - (2\sqrt{10})^2}$$ 6. **Calculate denominator:** $$(7)^2 - (2\sqrt{10})^2 = 49 - 4 \times 10 = 49 - 40 = 9$$ 7. **Simplify $a$:** $$a = \frac{7 - 2\sqrt{10}}{9}$$ 8. **Expression for $b$ is already simplified:** $$b = \frac{\sqrt{5} - \sqrt{2}}{3}$$ 9. **Summary:** $$a = \frac{7 - 2\sqrt{10}}{9}, \quad b = \frac{\sqrt{5} - \sqrt{2}}{3}$$ These are the simplified forms of $a$ and $b$.