1. **State the problem:** Simplify the expression $$\sqrt{\frac{9 + 2\sqrt{20}}{18 - 2\sqrt{32}}}$$. 2. **Recall the formula and rules:** To simplify nested radicals, especially fractions with radicals, we often rationalize the denominator by multiplying numerator and denominator by the conjugate of the denominator. 3. **Simplify the radicals inside:** - $$\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}$$ - $$\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}$$ 4. **Rewrite the expression:** $$\sqrt{\frac{9 + 2 \times 2\sqrt{5}}{18 - 2 \times 4\sqrt{2}}} = \sqrt{\frac{9 + 4\sqrt{5}}{18 - 8\sqrt{2}}}$$ 5. **Rationalize the denominator:** Multiply numerator and denominator inside the square root by the conjugate of the denominator $$18 + 8\sqrt{2}$$: $$\sqrt{\frac{(9 + 4\sqrt{5})(18 + 8\sqrt{2})}{(18 - 8\sqrt{2})(18 + 8\sqrt{2})}}$$ 6. **Calculate the denominator:** $$(18)^2 - (8\sqrt{2})^2 = 324 - 64 \times 2 = 324 - 128 = 196$$ 7. **Calculate the numerator:** Expand: $$9 \times 18 + 9 \times 8\sqrt{2} + 4\sqrt{5} \times 18 + 4\sqrt{5} \times 8\sqrt{2}$$ Calculate each term: - $$9 \times 18 = 162$$ - $$9 \times 8\sqrt{2} = 72\sqrt{2}$$ - $$4\sqrt{5} \times 18 = 72\sqrt{5}$$ - $$4\sqrt{5} \times 8\sqrt{2} = 32\sqrt{10}$$ So numerator is: $$162 + 72\sqrt{2} + 72\sqrt{5} + 32\sqrt{10}$$ 8. **Rewrite the entire expression:** $$\sqrt{\frac{162 + 72\sqrt{2} + 72\sqrt{5} + 32\sqrt{10}}{196}} = \frac{1}{\sqrt{196}} \sqrt{162 + 72\sqrt{2} + 72\sqrt{5} + 32\sqrt{10}}$$ Since $$\sqrt{196} = 14$$, this is: $$\frac{1}{14} \sqrt{162 + 72\sqrt{2} + 72\sqrt{5} + 32\sqrt{10}}$$ 9. **Try to express the radicand as a perfect square:** Assume: $$\sqrt{162 + 72\sqrt{2} + 72\sqrt{5} + 32\sqrt{10}} = a + b\sqrt{2} + c\sqrt{5} + d\sqrt{10}$$ Square the right side: $$a^2 + 2b^2 + 5c^2 + 10d^2 + 2ab\sqrt{2} + 2ac\sqrt{5} + 2ad\sqrt{10} + 2bc\sqrt{10} + 2bd \times 2 + 2cd \times 5$$ Group terms: - Constant: $$a^2 + 2b^2 + 5c^2 + 10d^2$$ - $$\sqrt{2}$$ term: $$2ab + 4bd$$ - $$\sqrt{5}$$ term: $$2ac + 10cd$$ - $$\sqrt{10}$$ term: $$2ad + 2bc$$ Set equal to radicand coefficients: - Constant: 162 - $$\sqrt{2}$$: 72 - $$\sqrt{5}$$: 72 - $$\sqrt{10}$$: 32 Solve the system: $$\begin{cases} a^2 + 2b^2 + 5c^2 + 10d^2 = 162 \\ 2ab + 4bd = 72 \\ 2ac + 10cd = 72 \\ 2ad + 2bc = 32 \end{cases}$$ Try $a=9$, $b=4$, $c=2$, $d=2$: - Constant: $9^2 + 2(4^2) + 5(2^2) + 10(2^2) = 81 + 2(16) + 5(4) + 10(4) = 81 + 32 + 20 + 40 = 173$ (too high) Try $a=7$, $b=4$, $c=4$, $d=1$: - Constant: $49 + 2(16) + 5(16) + 10(1) = 49 + 32 + 80 + 10 = 171$ (still high) Try $a=9$, $b=2$, $c=4$, $d=1$: - Constant: $81 + 2(4) + 5(16) + 10(1) = 81 + 8 + 80 + 10 = 179$ (too high) Try $a=7$, $b=2$, $c=4$, $d=2$: - Constant: $49 + 2(4) + 5(16) + 10(4) = 49 + 8 + 80 + 40 = 177$ (too high) Try $a=6$, $b=3$, $c=3$, $d=2$: - Constant: $36 + 2(9) + 5(9) + 10(4) = 36 + 18 + 45 + 40 = 139$ (too low) Try $a=8$, $b=3$, $c=3$, $d=2$: - Constant: $64 + 18 + 45 + 40 = 167$ (too high) Try $a=7$, $b=3$, $c=3$, $d=2$: - Constant: $49 + 18 + 45 + 40 = 152$ (too low) Try $a=8$, $b=2$, $c=3$, $d=2$: - Constant: $64 + 8 + 45 + 40 = 157$ (too low) Try $a=9$, $b=3$, $c=2$, $d=1$: - Constant: $81 + 18 + 20 + 10 = 129$ (too low) Try $a=9$, $b=3$, $c=3$, $d=1$: - Constant: $81 + 18 + 45 + 10 = 154$ (too low) Try $a=9$, $b=4$, $c=3$, $d=1$: - Constant: $81 + 32 + 45 + 10 = 168$ (too high) Try $a=8$, $b=4$, $c=3$, $d=1$: - Constant: $64 + 32 + 45 + 10 = 151$ (too low) Try $a=7$, $b=4$, $c=2$, $d=1$: - Constant: $49 + 32 + 20 + 10 = 111$ (too low) Try $a=9$, $b=2$, $c=3$, $d=1$: - Constant: $81 + 8 + 45 + 10 = 144$ (too low) Try $a=9$, $b=4$, $c=2$, $d=2$: - Constant: $81 + 32 + 20 + 40 = 173$ (too high) Try $a=8$, $b=3$, $c=2$, $d=2$: - Constant: $64 + 18 + 20 + 40 = 142$ (too low) Try $a=7$, $b=3$, $c=2$, $d=2$: - Constant: $49 + 18 + 20 + 40 = 127$ (too low) Try $a=8$, $b=4$, $c=2$, $d=2$: - Constant: $64 + 32 + 20 + 40 = 156$ (too low) Try $a=9$, $b=3$, $c=2$, $d=2$: - Constant: $81 + 18 + 20 + 40 = 159$ (too low) Try $a=9$, $b=4$, $c=3$, $d=2$: - Constant: $81 + 32 + 45 + 40 = 198$ (too high) Try $a=7$, $b=4$, $c=3$, $d=2$: - Constant: $49 + 32 + 45 + 40 = 166$ (too high) Try $a=6$, $b=4$, $c=3$, $d=2$: - Constant: $36 + 32 + 45 + 40 = 153$ (too low) Try $a=5$, $b=4$, $c=3$, $d=2$: - Constant: $25 + 32 + 45 + 40 = 142$ (too low) Try $a=7$, $b=3$, $c=4$, $d=2$: - Constant: $49 + 18 + 80 + 40 = 187$ (too high) Try $a=6$, $b=3$, $c=4$, $d=2$: - Constant: $36 + 18 + 80 + 40 = 174$ (too high) Try $a=5$, $b=3$, $c=4$, $d=2$: - Constant: $25 + 18 + 80 + 40 = 163$ (too high) Try $a=4$, $b=3$, $c=4$, $d=2$: - Constant: $16 + 18 + 80 + 40 = 154$ (too low) Try $a=3$, $b=3$, $c=4$, $d=2$: - Constant: $9 + 18 + 80 + 40 = 147$ (too low) Try $a=2$, $b=3$, $c=4$, $d=2$: - Constant: $4 + 18 + 80 + 40 = 142$ (too low) Try $a=1$, $b=3$, $c=4$, $d=2$: - Constant: $1 + 18 + 80 + 40 = 139$ (too low) Try $a=0$, $b=3$, $c=4$, $d=2$: - Constant: $0 + 18 + 80 + 40 = 138$ (too low) Try $a=3$, $b=4$, $c=3$, $d=2$: - Constant: $9 + 32 + 45 + 40 = 126$ (too low) Try $a=4$, $b=4$, $c=3$, $d=2$: - Constant: $16 + 32 + 45 + 40 = 133$ (too low) Try $a=5$, $b=4$, $c=3$, $d=2$: - Constant: $25 + 32 + 45 + 40 = 142$ (too low) Try $a=6$, $b=4$, $c=3$, $d=2$: - Constant: $36 + 32 + 45 + 40 = 153$ (too low) Try $a=7$, $b=4$, $c=3$, $d=2$: - Constant: $49 + 32 + 45 + 40 = 166$ (too high) Try $a=8$, $b=4$, $c=3$, $d=2$: - Constant: $64 + 32 + 45 + 40 = 181$ (too high) Try $a=9$, $b=4$, $c=3$, $d=2$: - Constant: $81 + 32 + 45 + 40 = 198$ (too high) Since this is tedious, try a simpler approach: factor numerator and denominator inside the root separately. 10. **Rewrite numerator:** $$9 + 4\sqrt{5} = (\sqrt{5} + 2)^2$$ because: $$(\sqrt{5} + 2)^2 = 5 + 4\sqrt{5} + 4 = 9 + 4\sqrt{5}$$ 11. **Rewrite denominator:** $$18 - 8\sqrt{2} = (3 - 2\sqrt{2})^2$$ because: $$(3 - 2\sqrt{2})^2 = 9 - 12\sqrt{2} + 8 = 17 - 12\sqrt{2}$$ which is not equal to denominator. Try $(3 - \sqrt{2})^2 = 9 - 6\sqrt{2} + 2 = 11 - 6\sqrt{2}$ no. Try $(3 - 4\sqrt{2})^2 = 9 - 24\sqrt{2} + 32 = 41 - 24\sqrt{2}$ no. Try $(6 - 4\sqrt{2})^2 = 36 - 48\sqrt{2} + 32 = 68 - 48\sqrt{2}$ no. Try $(3 - 2\sqrt{2})^2$ again: $9 - 12\sqrt{2} + 8 = 17 - 12\sqrt{2}$ no. Try $(6 - 2\sqrt{2})^2 = 36 - 24\sqrt{2} + 8 = 44 - 24\sqrt{2}$ no. Try $(4 - 2\sqrt{2})^2 = 16 - 16\sqrt{2} + 8 = 24 - 16\sqrt{2}$ no. Try $(5 - 2\sqrt{2})^2 = 25 - 20\sqrt{2} + 8 = 33 - 20\sqrt{2}$ no. Try $(3 - \sqrt{2})^2 = 11 - 6\sqrt{2}$ no. Try $(3 - 2\sqrt{2})^2$ no. Try $(\sqrt{8} - 2)^2 = 8 - 4\sqrt{8} + 4 = 12 - 4\sqrt{8}$ no. Try $(\sqrt{18} - \sqrt{8})^2 = 18 - 2\sqrt{144} + 8 = 26 - 24$ no. Try $(3 - 2\sqrt{2})^2$ no. Try to factor denominator as $a - b\sqrt{2}$: $$(a - b\sqrt{2})^2 = a^2 + 2b^2 - 2ab\sqrt{2}$$ Set equal to $18 - 8\sqrt{2}$: $$a^2 + 2b^2 = 18$$ $$2ab = 8 \Rightarrow ab = 4$$ Try integer pairs for $a,b$: - $a=4$, $b=1$: $a^2 + 2b^2 = 16 + 2 = 18$, $ab=4$ correct. So denominator is: $$(4 - \sqrt{2})^2 = 18 - 8\sqrt{2}$$ 12. **Rewrite the original expression:** $$\sqrt{\frac{(\sqrt{5} + 2)^2}{(4 - \sqrt{2})^2}} = \sqrt{\left(\frac{\sqrt{5} + 2}{4 - \sqrt{2}}\right)^2} = \left|\frac{\sqrt{5} + 2}{4 - \sqrt{2}}\right|$$ Since all terms are positive, absolute value is not needed. 13. **Rationalize the denominator:** $$\frac{\sqrt{5} + 2}{4 - \sqrt{2}} \times \frac{4 + \sqrt{2}}{4 + \sqrt{2}} = \frac{(\sqrt{5} + 2)(4 + \sqrt{2})}{(4)^2 - (\sqrt{2})^2} = \frac{(\sqrt{5} + 2)(4 + \sqrt{2})}{16 - 2} = \frac{(\sqrt{5} + 2)(4 + \sqrt{2})}{14}$$ 14. **Expand numerator:** $$4\sqrt{5} + \sqrt{5}\sqrt{2} + 8 + 2\sqrt{2} = 4\sqrt{5} + \sqrt{10} + 8 + 2\sqrt{2}$$ 15. **Final simplified expression:** $$\frac{8 + 4\sqrt{5} + 2\sqrt{2} + \sqrt{10}}{14}$$ **Answer:** $$\boxed{\frac{8 + 4\sqrt{5} + 2\sqrt{2} + \sqrt{10}}{14}}$$