1. **Problem statement:** Simplify the expression $\sqrt{55 + 30\sqrt{2}}$. 2. **Formula and approach:** We try to express the expression inside the square root in the form $(a + b\sqrt{2})^2 = a^2 + 2b^2 + 2ab\sqrt{2}$. 3. **Set up equations:** $$a^2 + 2b^2 = 55$$ $$2ab = 30 \implies ab = 15$$ 4. **Solve for $a$ and $b$:** From $ab=15$, $b=\frac{15}{a}$. Substitute into first equation: $$a^2 + 2\left(\frac{15}{a}\right)^2 = 55$$ $$a^2 + 2\frac{225}{a^2} = 55$$ Multiply both sides by $a^2$: $$a^4 - 55a^2 + 450 = 0$$ 5. **Let $x = a^2$:** $$x^2 - 55x + 450 = 0$$ 6. **Solve quadratic:** $$x = \frac{55 \pm \sqrt{55^2 - 4 \cdot 450}}{2} = \frac{55 \pm \sqrt{3025 - 1800}}{2} = \frac{55 \pm \sqrt{1225}}{2}$$ $$= \frac{55 \pm 35}{2}$$ 7. **Two solutions:** $$x_1 = \frac{55 + 35}{2} = 45, \quad x_2 = \frac{55 - 35}{2} = 10$$ 8. **Check $a$ and $b$:** If $a^2=45$, then $a=\sqrt{45}=3\sqrt{5}$, and $b=\frac{15}{a} = \frac{15}{3\sqrt{5}} = \sqrt{5}$. 9. **Verify:** $$(3\sqrt{5})^2 + 2(\sqrt{5})^2 = 45 + 2 \cdot 5 = 45 + 10 = 55$$ $$2 \cdot 3\sqrt{5} \cdot \sqrt{5} = 2 \cdot 3 \cdot 5 = 30$$ 10. **Answer:** $$\sqrt{55 + 30\sqrt{2}} = 3\sqrt{5} + \sqrt{10}$$ --- Since the user asked to explain how to do these and there are three problems, but per instructions only the first problem is solved fully. "slug":"nested radical 1","subject":"algebra","desmos":{"latex":"y=\sqrt{55 + 30\sqrt{2}}","features":{"intercepts":true,"extrema":true}},"q_count":3