Задание 41.
a) $\sqrt{3^6 \cdot 2^4} = \sqrt{(3^3)^2 \cdot (2^2)^2} = 3^3 \cdot 2^2 = 27 \cdot 4 = 108$
б) $\sqrt{4^4 \cdot 5^6} = \sqrt{(4^2)^2 \cdot (5^3)^2} = 4^2 \cdot 5^3 = 16 \cdot 125 = 2000$
в) $\sqrt{7^2 \cdot 2^8} = \sqrt{7^2 \cdot (2^4)^2} = 7 \cdot 2^4 = 7 \cdot 16 = 112$
г) $\sqrt{5^2 \cdot 12^4} = \sqrt{5^2 \cdot (12^2)^2} = 5 \cdot 12^2 = 5 \cdot 144 = 720$
д) $\sqrt{(-5)^4 \cdot 11^2} = \sqrt{(5^2)^2 \cdot 11^2} = 5^2 \cdot 11 = 25 \cdot 11 = 275$
е) $\sqrt{7^4 \cdot (-2)^8} = \sqrt{(7^2)^2 \cdot ((-2)^4)^2} = 7^2 \cdot (-2)^4 = 49 \cdot 16 = 784$
ж) $\sqrt{6^8 \cdot (-25)^2 \cdot (-3)^4} = \sqrt{(6^4)^2 \cdot (-25)^2 \cdot ((-3)^2)^2} = 6^4 \cdot (-25) \cdot (-3)^2 = 1296 \cdot 25 \cdot 9 = 291600$
з) $\sqrt{(-23)^2 \cdot (-5)^2 \cdot (-2)^4} = \sqrt{(-23)^2 \cdot (-5)^2 \cdot ((-2)^2)^2} = -23 \cdot (-5) \cdot (-2)^2 = 23 \cdot 5 \cdot 4 = 460$
Задание 42.
a) $\frac{\sqrt{363}}{\sqrt{3}} = \sqrt{\frac{363}{3}} = \sqrt{121} = 11$
б) $\frac{\sqrt{512}}{\sqrt{32}} = \sqrt{\frac{512}{32}} = \sqrt{16} = 4$
в) $\frac{\sqrt{252}}{\sqrt{700}} = \sqrt{\frac{252}{700}} = \sqrt{\frac{9}{25}} = \frac{3}{5} = 0.6$
г) $\frac{\sqrt{405}}{\sqrt{500}} = \sqrt{\frac{405}{500}} = \sqrt{\frac{81}{100}} = \frac{9}{10} = 0.9$
д) $\frac{\sqrt{75} \cdot \sqrt{32}}{\sqrt{24}} = \frac{\sqrt{75 \cdot 32}}{\sqrt{24}} = \sqrt{\frac{75 \cdot 32}{24}} = \sqrt{\frac{75 \cdot 4}{3}} = \sqrt{25 \cdot 4} = \sqrt{100} = 10$
е) $\frac{\sqrt{98} \cdot \sqrt{125}}{\sqrt{40}} = \sqrt{\frac{98 \cdot 125}{40}} = \sqrt{\frac{49 \cdot 25}{4}} = \frac{7 \cdot 5}{2} = \frac{35}{2} = 17.5$
ж) $\frac{\sqrt{2100}}{\sqrt{48} \cdot \sqrt{28}} = \frac{\sqrt{2100}}{\sqrt{48 \cdot 28}} = \sqrt{\frac{2100}{48 \cdot 28}} = \sqrt{\frac{25}{16}} = \frac{5}{4} = 1.25$
з) $\frac{\sqrt{10800}}{\sqrt{15} \cdot \sqrt{80}} = \frac{\sqrt{10800}}{\sqrt{15 \cdot 80}} = \sqrt{\frac{10800}{15 \cdot 80}} = \sqrt{\frac{10800}{1200}} = \sqrt{9} = 3$
Задание 43.
a) $\frac{(5\sqrt{7})^2}{14} = \frac{5^2 \cdot (\sqrt{7})^2}{14} = \frac{25 \cdot 7}{14} = \frac{25}{2} = 12.5$
б) $\frac{(9\sqrt{3})^2}{12} = \frac{9^2 \cdot (\sqrt{3})^2}{12} = \frac{81 \cdot 3}{12} = \frac{27 \cdot 3}{4} = \frac{81}{4} = 20.25$
в) $\frac{33}{(2\sqrt{11})^2} = \frac{33}{2^2 \cdot (\sqrt{11})^2} = \frac{33}{4 \cdot 11} = \frac{3}{4} = 0.75$
г) $\frac{34}{(5\sqrt{17})^2} = \frac{34}{5^2 \cdot (\sqrt{17})^2} = \frac{34}{25 \cdot 17} = \frac{2}{25} = 0.08$
Задание 44.
a) $\sqrt{145^2 - 144^2} = \sqrt{(145-144)(145+144)} = \sqrt{1 \cdot 289} = \sqrt{289} = 17$
б) $\sqrt{82^2 - 80^2} = \sqrt{(82-80)(82+80)} = \sqrt{2 \cdot 162} = \sqrt{324} = 18$
в) $\sqrt{2.6^2 - 2.4^2} = \sqrt{(2.6-2.4)(2.6+2.4)} = \sqrt{0.2 \cdot 5} = \sqrt{1} = 1$
г) $\sqrt{2.5^2 - 0.7^2} = \sqrt{(2.5-0.7)(2.5+0.7)} = \sqrt{1.8 \cdot 3.2} = \sqrt{5.76} = 2.4$
