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Задание 1:

1. \(200\); 2) \(\frac12\); 3) \(20\); 4) \(32\); 5) \(3\); 6) \(7\); 7) \(7\); 8) \(42\); 9) \(5\); 10) \(312\); 11) \(49\); 12) \(243\); 13) \(\frac{49}{10}\); 14) \(2\); 15) \(7\); 16) \(8\); 17) \(23\).

Задание 2:

1. \(25\); 2) \(7\); 3) \(64\); 4) \(81\); 5) \(32\); 6) \(8\); 7) \(36\); 8) \(64\); 9) \(128\); 10) \(\frac1{16384}\); 11) \(24\); 12) \(81\); 13) \(125\); 14) \(49\); 15) \(1000\); 16) \(27\); 17) \(216\).

Ключевые правила

\[ \sqrt{u^2}=|u|,\qquad a^m\cdot a^n=a^{m+n},\qquad a^m:a^n=a^{m-n}, \] \[ (a^m)^n=a^{mn},\qquad (p-q)(p+q)=p^2-q^2. \]

Задание 1

1. \[ \sqrt{\frac{64a^{10}}{a^6}} =\sqrt{64a^4}=8a^2=8\cdot25=200. \]

2. \[ \sqrt{\frac{81x^2}{y^4}} =\frac{9|x|}{y^2} =\frac{9\cdot2}{6^2} =\frac12. \]

3. \[ \sqrt{\frac1{100}x^4y^6} =\frac{|x^2y^3|}{10} =\frac{25\cdot8}{10}=20. \]

4. \[ \sqrt{(-a)^6a^4} =\sqrt{a^{10}}=|a^5|=2^5=32. \]

5. Используем формулу квадрата суммы: \[ a^2+18ab+81b^2=(a+9b)^2. \] Так как \(a=2\frac4{13}=\frac{30}{13}\), \(b=\frac1{13}\), то \[ \sqrt{(a+9b)^2} =\left|\frac{30}{13}+\frac9{13}\right| =3. \]

6. \[ a^2+16ab+64b^2=(a+8b)^2, \] \[ \sqrt{(a+8b)^2}=|9+8(-2)|=|-7|=7. \]

7. \[ (\sqrt{28}-\sqrt7)\sqrt7 =(2\sqrt7-\sqrt7)\sqrt7=7. \]

8. \[ \sqrt{7\cdot12}\cdot\sqrt{21} =\sqrt{84\cdot21} =\sqrt{1764}=42. \]

9. \[ \frac{\sqrt{30}\cdot\sqrt{15}}{\sqrt{18}} =\sqrt{\frac{30\cdot15}{18}} =\sqrt{25}=5. \]

10. \[ 4\sqrt{13}\cdot2\sqrt3\cdot\sqrt{39} =8\sqrt{13\cdot3\cdot39} =8\sqrt{1521}=312. \]

11. \[ \sqrt{7^4}=7^2=49. \]

12. \[ \sqrt{9^5}=\sqrt{9^4\cdot9}=9^2\cdot3=243. \]

13. \[ \frac{(7\sqrt{11})^2}{110} =\frac{49\cdot11}{110} =\frac{49}{10}. \]

14. \[ \frac{48}{(2\sqrt6)^2} =\frac{48}{4\cdot6}=2. \]

15. \[ (\sqrt{23}-4)(\sqrt{23}+4)=23-16=7. \]

16. \[ (\sqrt{15}-\sqrt7)(\sqrt{15}+\sqrt7)=15-7=8. \]

17. \[ (\sqrt{14}-3)^2+6\sqrt{14} =14-6\sqrt{14}+9+6\sqrt{14}=23. \]

Задание 2

Знак «\(:\)» означает деление.

1. \[ \frac{a^7a^6}{a^{11}}=a^{7+6-11}=a^2=5^2=25. \]

2. \[ \frac{a^{12}a^{-6}}{a^5}=a^{12-6-5}=a=7. \]

3. \[ \frac{(a^5)^4}{a^{17}}=a^{20-17}=a^3=4^3=64. \]

4. \[ a^{10}a^5:a^{11}=a^{10+5-11}=a^4=3^4=81. \]

5. \[ a^{27}a^{-12}:a^{10}=a^{27-12-10}=a^5=2^5=32. \]

6. \[ a^{-23}(a^4)^6=a^{-23+24}=a=8. \]

7. \[ (a^2)^{-7}:a^{-16}=a^{-14-(-16)}=a^2=6^2=36. \]

8. \[ \frac{(a^3)^9a^{11}}{a^{36}} =a^{27+11-36}=a^2=8^2=64. \]

9. \[ \frac{a^{19}(b^4)^3}{(ab)^{12}} =\frac{a^{19}b^{12}}{a^{12}b^{12}} =a^7=2^7=128. \]

10. Так как \(64=4^3\), то \[ \frac{4^5}{64^4} =\frac{4^5}{4^{12}} =4^{-7} =\frac1{4^7} =\frac1{16384}. \]

11. Так как \(12=2\cdot6\), то \[ \frac{12^7}{2^5\cdot6^6} =\frac{2^7\cdot6^7}{2^5\cdot6^6} =2^2\cdot6=24. \]

12. Так как \(72=8\cdot9\), то \[ \frac{8^{11}\cdot9^{13}}{72^{11}} =\frac{8^{11}\cdot9^{13}}{8^{11}\cdot9^{11}} =9^2=81. \]

13. \[ \frac{(5\cdot8)^9}{5^6\cdot8^9} =\frac{5^9\cdot8^9}{5^6\cdot8^9} =5^3=125. \]

14. \[ 7^{-13}(7^5)^3=7^{-13+15}=7^2=49. \]

15. \[ \frac{(10^4)^{-6}}{10^{-27}} =\frac{10^{-24}}{10^{-27}} =10^3=1000. \]

16. \[ \frac1{3^{-18}}\cdot\frac1{3^{15}} =3^{18}\cdot3^{-15} =3^3=27. \]

17. \[ \frac{6^{-8}\cdot6^{21}}{6^{10}} =6^{-8+21-10} =6^3=216. \]