Final Answer:
1. Rational: $\frac{3}{2}$, Irrational: $\sqrt{2}$
2. Rational: $2.5$, Irrational: $\sqrt{6} + 0.1$ (any number like $\sqrt{6} + \frac{1}{10}$ works)
3. $5\sqrt{3} + 4 + 2\sqrt{6} + \sqrt{3} + 5\sqrt{6} = 6\sqrt{3} + 4 + 7\sqrt{6}$
4. $(3\sqrt{2} - 5\sqrt{5}) - (6\sqrt{2} + 3\sqrt{5}) = -3\sqrt{2} - 8\sqrt{5}$
5. $(\sqrt{3} + \sqrt{2})(2\sqrt{5} + 7\sqrt{3}) = 2\sqrt{15} + 7\cdot3 + 2\sqrt{10} + 7\sqrt{6} = 21 + 2\sqrt{15} + 2\sqrt{10} + 7\sqrt{6}$
6. $\frac{6\sqrt{2}}{3\sqrt{2}} = 2$
7. $10.03\overline{2} = \frac{9029}{900}$
8. $0.06\overline{5} = \frac{59}{900}$
9. $125^{-\frac{1}{3}} \times (125^{\frac{1}{3}} - 125^{\frac{2}{3}}) = \frac{1}{5} \times (5 - 25) = \frac{1}{5} \times (-20) = -4$
10. $\frac{\sqrt{3} + \sqrt{2}}{\sqrt{6} - \sqrt{2}} = \frac{(\sqrt{3} + \sqrt{2})(\sqrt{6} + \sqrt{2})}{(\sqrt{6} - \sqrt{2})(\sqrt{6} + \sqrt{2})} = \frac{3\sqrt{2} + \sqrt{6} + 2\sqrt{3} + 2}{4} = \frac{2 + 3\sqrt{2} + \sqrt{6} + 2\sqrt{3}}{4}$
11. $\frac{a^{\frac{1}{3}} \cdot 27^{-\frac{1}{2}}}{a^{\frac{2}{3}} \cdot a^{-\frac{1}{3}}} = \frac{a^{\frac{1}{3}}}{a^{\frac{1}{3}}} \cdot \frac{1}{\sqrt{27}} = \frac{1}{3\sqrt{3}} = \frac{\sqrt{3}}{9}$
12. $x = 4$
13. $(2.2\overline{3} - 0.3\overline{2}1) = \frac{201}{90} - \frac{29}{90} = \frac{172}{90} = \frac{86}{45}$
14. $\sqrt{17}$ lies between 4 and 5, closer to 4.1 (since $4.1^2 = 16.81$, $4.2^2 = 17.64$)
15. $\sqrt{13} \approx 3.6$ (since $3.6^2 = 12.96$, $3.61^2 = 13.0321$)
16. $a = 11$, $b = 1$
17. $\frac{1}{\sqrt{x}} = \frac{1}{3}$
18. $\frac{2\sqrt{3}}{3}$
19. $0$
20. $2$
Parent Tip: Review the logic above to help your child master the concept of number system worksheet.
