Задание 13. Найдите значение выражения: 1. (a^9)^3 * a^7 / a^29 при a=2; 2. (a^3)^8 * a^7 / a^29 при a=7; 3. (a^5)^5 * a^6 / a^27 при a=2; 4. (a^4)^4 * a^5 / a^18 при a=3; 5. (a^3)^5 * a^6 / a^19 при a=5; 6. (a^7)^3 * a^10 / a^28 при a=4; 7. (a^3)^6 * a^3 / a^17 при a=3; 8. (a^3)^4 * a^12 / a^21 при a=5; 9. a^23 * (b^5)^4 / (a*b)^20 при a=2, b=sqrt(2); 10. a^14 * (b^6)^2 / (a*b)^12 при a=6, b=sqrt(6); 11. a^17 * (b^5)^3 / (a*b)^15 при a=7, b=sqrt(7); 12. a^21 * (b^9)^2 / (a*b)^18 при a=5, b=sqrt(5); 13. a^21 * (b^6)^3 / (a*b)^18 при a=3, b=sqrt(3); 14. a^22 * (b^3)^6 / (a*b)^18 при a=2, b=sqrt(2); 15. a^18 * (b^8)^2 / (a*b)^16 при a=5, b=sqrt(5); 16. a^14 * (b^4)^3 / (a*b)^12 при a=3, b=sqrt(3).

Решение:

1. \( \frac{(a^9)^3 \cdot a^7}{a^{29}} = \frac{a^{27} \cdot a^7}{a^{29}} = \frac{a^{34}}{a^{29}} = a^5 \). При \( a = 2 \), \( a^5 = 2^5 = 32 \).
2. \( \frac{(a^3)^8 \cdot a^7}{a^{29}} = \frac{a^{24} \cdot a^7}{a^{29}} = \frac{a^{31}}{a^{29}} = a^2 \). При \( a = 7 \), \( a^2 = 7^2 = 49 \).
3. \( \frac{(a^5)^5 \cdot a^6}{a^{27}} = \frac{a^{25} \cdot a^6}{a^{27}} = \frac{a^{31}}{a^{27}} = a^4 \). При \( a = 2 \), \( a^4 = 2^4 = 16 \).
4. \( \frac{(a^4)^4 \cdot a^5}{a^{18}} = \frac{a^{16} \cdot a^5}{a^{18}} = \frac{a^{21}}{a^{18}} = a^3 \). При \( a = 3 \), \( a^3 = 3^3 = 27 \).
5. \( \frac{(a^3)^5 \cdot a^6}{a^{19}} = \frac{a^{15} \cdot a^6}{a^{19}} = \frac{a^{21}}{a^{19}} = a^2 \). При \( a = 5 \), \( a^2 = 5^2 = 25 \).
6. \( \frac{(a^7)^3 \cdot a^{10}}{a^{28}} = \frac{a^{21} \cdot a^{10}}{a^{28}} = \frac{a^{31}}{a^{28}} = a^3 \). При \( a = 4 \), \( a^3 = 4^3 = 64 \).
7. \( \frac{(a^3)^6 \cdot a^3}{a^{17}} = \frac{a^{18} \cdot a^3}{a^{17}} = \frac{a^{21}}{a^{17}} = a^4 \). При \( a = 3 \), \( a^4 = 3^4 = 81 \).
8. \( \frac{(a^3)^4 \cdot a^{12}}{a^{21}} = \frac{a^{12} \cdot a^{12}}{a^{21}} = \frac{a^{24}}{a^{21}} = a^3 \). При \( a = 5 \), \( a^3 = 5^3 = 125 \).
9. \( \frac{a^{23} \cdot (b^5)^4}{(a \cdot b)^{20}} = \frac{a^{23} \cdot b^{20}}{a^{20} \cdot b^{20}} = a^3 \). При \( a = 2 \), \( a^3 = 2^3 = 8 \).
10. \( \frac{a^{14} \cdot (b^6)^2}{(a \cdot b)^{12}} = \frac{a^{14} \cdot b^{12}}{a^{12} \cdot b^{12}} = a^2 \). При \( a = 6 \), \( a^2 = 6^2 = 36 \).
11. \( \frac{a^{17} \cdot (b^5)^3}{(a \cdot b)^{15}} = \frac{a^{17} \cdot b^{15}}{a^{15} \cdot b^{15}} = a^2 \). При \( a = 7 \), \( a^2 = 7^2 = 49 \).
12. \( \frac{a^{21} \cdot (b^9)^2}{(a \cdot b)^{18}} = \frac{a^{21} \cdot b^{18}}{a^{18} \cdot b^{18}} = a^3 \). При \( a = 5 \), \( a^3 = 5^3 = 125 \).
13. \( \frac{a^{21} \cdot (b^6)^3}{(a \cdot b)^{18}} = \frac{a^{21} \cdot b^{18}}{a^{18} \cdot b^{18}} = a^3 \). При \( a = 3 \), \( a^3 = 3^3 = 27 \).
14. \( \frac{a^{22} \cdot (b^3)^6}{(a \cdot b)^{18}} = \frac{a^{22} \cdot b^{18}}{a^{18} \cdot b^{18}} = a^4 \). При \( a = 2 \), \( a^4 = 2^4 = 16 \).
15. \( \frac{a^{18} \cdot (b^8)^2}{(a \cdot b)^{16}} = \frac{a^{18} \cdot b^{16}}{a^{16} \cdot b^{16}} = a^2 \). При \( a = 5 \), \( a^2 = 5^2 = 25 \).
16. \( \frac{a^{14} \cdot (b^4)^3}{(a \cdot b)^{12}} = \frac{a^{14} \cdot b^{12}}{a^{12} \cdot b^{12}} = a^2 \). При \( a = 3 \), \( a^2 = 3^2 = 9 \).

Ответ: 1. 32; 2. 49; 3. 16; 4. 27; 5. 25; 6. 64; 7. 81; 8. 125; 9. 8; 10. 36; 11. 49; 12. 125; 13. 27; 14. 16; 15. 25; 16. 9.