1. 7^-2 * 7^3 2. 5^-3 : 5^-3 3. 10^10 * 10^-15 + 10^0 4. (2.25)^-3 : (1/3)^-4 5. 10^-6 : 100^-2 6. 2^8 : 2^9 : 2^-3 7. 3^25 / ((-3)^15 * (-3)^-4) 8. (a^2)^-5 * (a^3)^4 9. 24^-3 * 36^4 / (2^6 * 3^-1) 10. 3^8 * 54^-3 * 18^-4 * 72 / 24^-6 8) (a^-2)^-3 * (a^6)^-1 9) 16^-4 * 32^2 / (2^-5 * 2^8) 10) 20^-3 * (50^-1)^-3 / (5^-3)^-3 * 250^2

Решение:

1. \( 7^{-2} \cdot 7^{3} = 7^{-2+3} = 7^{1} = 7 \)

2. \( 5^{-3} : 5^{-3} = 5^{-3 - (-3)} = 5^{0} = 1 \)

3. \( 10^{10} \cdot 10^{-15} + 10^{0} = 10^{10-15} + 1 = 10^{-5} + 1 = \frac{1}{10^{5}} + 1 = \frac{1}{100000} + 1 = 1.00001 \)

4. \( (2.25)^{-3} : (\frac{1}{3})^{-4} = (\frac{9}{4})^{-3} : 3^{4} = (\frac{4}{9})^{3} \cdot \frac{1}{3^{4}} = \frac{4^{3}}{9^{3}} \cdot \frac{1}{3^{4}} = \frac{64}{(3^{2})^{3}} \cdot \frac{1}{3^{4}} = \frac{64}{3^{6}} \cdot \frac{1}{3^{4}} = \frac{64}{3^{10}} \)

5. \( 10^{-6} : 100^{-2} = 10^{-6} : (10^{2})^{-2} = 10^{-6} : 10^{-4} = 10^{-6 - (-4)} = 10^{-2} = \frac{1}{100} \)

6. \( 2^{8} : 2^{9} : 2^{-3} = 2^{8-9} : 2^{-3} = 2^{-1} : 2^{-3} = 2^{-1 - (-3)} = 2^{2} = 4 \)

7. \( \frac{3^{25}}{(-3)^{15} \cdot (-3)^{-4}} = \frac{3^{25}}{(-3)^{15-4}} = \frac{3^{25}}{(-3)^{11}} = \frac{3^{25}}{-(3^{11})} = -\frac{3^{25}}{3^{11}} = -3^{25-11} = -3^{14} \)

8. \( (a^{2})^{-5} \cdot (a^{3})^{4} = a^{-10} \cdot a^{12} = a^{-10+12} = a^{2} \)

9. \( \frac{24^{-3} \cdot 36^{4}}{2^{6} \cdot 3^{-1}} = \frac{(2^{3} \cdot 3)^{-3} \cdot (2^{2} \cdot 3^{2})^{4}}{2^{6} \cdot 3^{-1}} = \frac{2^{-9} \cdot 3^{-3} \cdot 2^{8} \cdot 3^{8}}{2^{6} \cdot 3^{-1}} = \frac{2^{-1} \cdot 3^{5}}{2^{6} \cdot 3^{-1}} = 2^{-1-6} \cdot 3^{5-(-1)} = 2^{-7} \cdot 3^{6} = \frac{3^{6}}{2^{7}} \)

10. \( \frac{3^{8} \cdot 54^{-3} \cdot 18^{-4} \cdot 72}{24^{-6}} = \frac{3^{8} \cdot (2 \cdot 3^{3})^{-3} \cdot (2 \cdot 3^{2})^{-4} \cdot (2^{3} \cdot 3)}{ (2^{3} \cdot 3)^{-6}} = \frac{3^{8} \cdot 2^{-3} \cdot 3^{-9} \cdot 2^{-4} \cdot 3^{-8} \cdot 2^{3} \cdot 3}{ 2^{-18} \cdot 3^{-6}} = \frac{2^{-3-4+3} \cdot 3^{8-9-8+1}}{2^{-18} \cdot 3^{-6}} = \frac{2^{-4} \cdot 3^{-8}}{2^{-18} \cdot 3^{-6}} = 2^{-4-(-18)} \cdot 3^{-8-(-6)} = 2^{14} \cdot 3^{-2} = \frac{2^{14}}{3^{2}} \)

8. \( (a^{-2})^{-3} \cdot (a^{6})^{-1} = a^{6} \cdot a^{-6} = a^{6-6} = a^{0} = 1 \)

9. \( \frac{16^{-4} \cdot 32^{2}}{2^{-5} \cdot 2^{8}} = \frac{(2^{4})^{-4} \cdot (2^{5})^{2}}{2^{-5+8}} = \frac{2^{-16} \cdot 2^{10}}{2^{3}} = \frac{2^{-6}}{2^{3}} = 2^{-6-3} = 2^{-9} = \frac{1}{2^{9}} = \frac{1}{512} \)

10. \( \frac{20^{-3} \cdot (50^{-1})^{-3}}{(5^{-3})^{-3} \cdot 250^{2}} = \frac{(2^{2} \cdot 5)^{-3} \cdot 50^{3}}{(5^{9}) \cdot (25 \cdot 10)^{2}} = \frac{2^{-6} \cdot 5^{-3} \cdot (2 \cdot 5^{2})^{3}}{5^{9} \cdot (5^{2} \cdot 2 \cdot 5)^{2}} = \frac{2^{-6} \cdot 5^{-3} \cdot 2^{3} \cdot 5^{6}}{5^{9} \cdot (2 \cdot 5^{3})^{2}} = \frac{2^{-3} \cdot 5^{3}}{5^{9} \cdot 2^{2} \cdot 5^{6}} = \frac{2^{-3} \cdot 5^{3}}{2^{2} \cdot 5^{15}} = 2^{-3-2} \cdot 5^{3-15} = 2^{-5} \cdot 5^{-12} = \frac{1}{2^{5} \cdot 5^{12}} \)

Ответ: 1. 7; 2. 1; 3. 1.00001; 4. 64/3^10; 5. 1/100; 6. 4; 7. -3^14; 8. a^2; 9. 3^6/2^7; 10. 2^14/3^2; 8. 1; 9. 1/512; 10. 1/(2^5 * 5^12).