Для решения задач на упрощение выражений с корнями и степенями воспользуемся свойствами:
1) $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$, $\sqrt{a} / \sqrt{b} = \sqrt{a / b}$, $\sqrt{a^2} = |a| = a$ (при $a \ge 0$).
2) $a^n \cdot a^m = a^{n+m}$, $a^n / a^m = a^{n-m}$, $(a^n)^m = a^{n \cdot m}$.
**Задание 4**
Все выражения имеют вид $\sqrt{\frac{A \cdot B}{C}}$.
1) $\sqrt{\frac{25a^9 \cdot 16b^8}{a^5b^8}} = \sqrt{25 \cdot 16 \cdot a^4} = 5 \cdot 4 \cdot a^2 = 20a^2$. При $a=4$: $20 \cdot 16 = 320$.
2) $\sqrt{\frac{16a^9 \cdot 4b^3}{a^5b^3}} = \sqrt{64a^4} = 8a^2$. При $a=9$: $8 \cdot 81 = 648$.
3) $\sqrt{\frac{4a^{11} \cdot 9b^4}{a^7b^4}} = \sqrt{36a^4} = 6a^2$. При $a=7$: $6 \cdot 49 = 294$.
4) $\sqrt{\frac{25a^8 \cdot 36b^6}{a^5b^4}} = \sqrt{900 \cdot a^3 \cdot b^2} = 30a \cdot |b| \cdot \sqrt{a}$. При $a=4, b=9$: $30 \cdot 4 \cdot 9 \cdot \sqrt{4} = 1080 \cdot 2 = 2160$.
5) $\sqrt{\frac{16a^5 \cdot 36b}{ab}} = \sqrt{16 \cdot 36 \cdot a^4} = 4 \cdot 6 \cdot a^2 = 24a^2$. При $a=7$: $24 \cdot 49 = 1176$.
6) $\sqrt{\frac{4a^5 \cdot 25b^7}{a^2b^7}} = \sqrt{100a^3} = 10a \sqrt{a}$. При $a=9$: $10 \cdot 9 \cdot 3 = 270$.
7) $\sqrt{\frac{36a \cdot 9b^5}{ab}} = \sqrt{36 \cdot 9 \cdot b^4} = 6 \cdot 3 \cdot b^2 = 18b^2$. При $b=4$: $18 \cdot 16 = 288$.
8) $\sqrt{\frac{25a^8 \cdot 9b^5}{a^4b^5}} = \sqrt{225a^4} = 15a^2$. При $a=7$: $15 \cdot 49 = 735$.
**Задание 5**
1) $(5+\sqrt{2})^2 + (5-\sqrt{2})^2 = 25+10\sqrt{2}+2 + 25-10\sqrt{2}+2 = 54$.
2) $(4+\sqrt{7})^2 + (4-\sqrt{7})^2 = 16+8\sqrt{7}+7 + 16-8\sqrt{7}+7 = 46$.
3) $(3+\sqrt{2})^2 + (3-\sqrt{2})^2 = 9+6\sqrt{2}+2 + 9-6\sqrt{2}+2 = 22$.
4) $(4+\sqrt{5})^2 + (4-\sqrt{5})^2 = 16+8\sqrt{5}+5 + 16-8\sqrt{5}+5 = 42$.
5) $(5+\sqrt{7})^2 + (5-\sqrt{7})^2 = 25+10\sqrt{7}+7 + 25-10\sqrt{7}+7 = 64$.
6) $(3+\sqrt{5})^2 + (3-\sqrt{5})^2 = 9+6\sqrt{5}+5 + 9-6\sqrt{5}+5 = 28$.
7) $\sqrt{(-17)^2} = |-17| = 17$. 8) 11. 9) 19. 10) 23. 11) 5. 12) 29.
13) $\sqrt{(3\sqrt{2}-5)^2} + 3\sqrt{2} = |3\sqrt{2}-5| + 3\sqrt{2} = 5-3\sqrt{2} + 3\sqrt{2} = 5$. (Так как $3\sqrt{2} \approx 4.2 < 5$).
14) $\sqrt{(5\sqrt{2}-8)^2} + 5\sqrt{2} = |5\sqrt{2}-8| + 5\sqrt{2} = 8-5\sqrt{2} + 5\sqrt{2} = 8$.
15) $\sqrt{(4\sqrt{2}-7)^2} + 4\sqrt{2} = |4\sqrt{2}-7| + 4\sqrt{2} = 7-4\sqrt{2} + 4\sqrt{2} = 7$.
16) $\sqrt{(6\sqrt{3}-11)^2} + 6\sqrt{3} = 11-6\sqrt{3} + 6\sqrt{3} = 11$.
17) $\sqrt{(2\sqrt{3}-5)^2} - 2\sqrt{3} = |2\sqrt{3}-5| - 2\sqrt{3} = 5-2\sqrt{3} - 2\sqrt{3} = 5-4\sqrt{3}$.
18) $\sqrt{(5\sqrt{3}-9)^2} + 5\sqrt{3} = |5\sqrt{3}-9| + 5\sqrt{3} = 9-5\sqrt{3} + 5\sqrt{3} = 9$.
**Задание 6**
1) $\frac{(2^2 \cdot 2^4)^7}{(2 \cdot 2^6)^6} = \frac{(2^6)^7}{(2^7)^6} = \frac{2^{42}}{2^{42}} = 1$.
2) $\frac{(3^3 \cdot 3^5)^6}{(3 \cdot 3^8)^5} = \frac{(3^8)^6}{(3^9)^5} = \frac{3^{48}}{3^{45}} = 3^3 = 27$.
3) $\frac{(5^2 \cdot 5^3)^4}{(5 \cdot 5^5)^3} = \frac{(5^5)^4}{(5^6)^3} = \frac{5^{20}}{5^{18}} = 5^2 = 25$.
4) $\frac{(7^2 \cdot 7^4)^5}{(7 \cdot 7^6)^4} = \frac{(7^6)^5}{(7^7)^4} = \frac{7^{30}}{7^{28}} = 7^2 = 49$.
5) $\frac{(2^2 \cdot 2^6)^5}{(2 \cdot 2^8)^4} = \frac{(2^8)^5}{(2^9)^4} = \frac{2^{40}}{2^{36}} = 2^4 = 16$.
6) $\frac{(3^2 \cdot 3^7)^9}{(3 \cdot 3^9)^8} = \frac{(3^9)^9}{(3^{10})^8} = \frac{3^{81}}{3^{80}} = 3$.
7) $16^4 / 8^6 = (2^4)^4 / (2^3)^6 = 2^{16} / 2^{18} = 2^{-2} = 0.25$.
8) $81^5 / 27^6 = (3^4)^5 / (3^3)^6 = 3^{20} / 3^{18} = 3^2 = 9$.
9) $125^3 / 25^5 = (5^3)^3 / (5^2)^5 = 5^9 / 5^{10} = 5^{-1} = 0.2$.
10) $64^2 / 16^3 = (4^3)^2 / (4^2)^3 = 4^6 / 4^6 = 1$.
11) $27^3 / 9^4 = (3^3)^3 / (3^2)^4 = 3^9 / 3^8 = 3$.
12) $8^3 / 4^5 = (2^3)^3 / (2^2)^5 = 2^9 / 2^{10} = 2^{-1} = 0.5$.
13) $2^{-7} \cdot 2^{-8} / 2^{-16} = 2^{-15} / 2^{-16} = 2^1 = 2$.
14) $9^{-5} \cdot 9^{-8} / 9^{-15} = 9^{-13} / 9^{-15} = 9^2 = 81$.
15) $3^{-4} \cdot 3^{-8} / 3^{-14} = 3^{-12} / 3^{-14} = 3^2 = 9$.
16) $7^{-3} \cdot 7^{-8} / 7^{-13} = 7^{-11} / 7^{-13} = 7^2 = 49$.
17) $11^{-5} \cdot 11^{-13} / 11^{-19} = 11^{-18} / 11^{-19} = 11^1 = 11$.
18) $5^{-3} \cdot 5^{-9} / 5^{-14} = 5^{-12} / 5^{-14} = 5^2 = 25$.
