Найдите значения выражения: 1) √5⁶; 2) √8⋅√192/√24; 3) √13⋅18⋅√26; 4) (√125−√5)⋅√5; 5) (√48+√3)⋅√3; 6) 9√7⋅2√2⋅√14; 7) (√37−5)(√37+5); 8) (√19−√2)(√19+√2); 9) (√15−2)² + 4√15; 10) (4√3)²/60; 11) 72/(2√3)²; 12) a²²⋅(b³)^6/(a⋅b)¹⁸ при a = 2 и b = √2; 13) √(−a)⁴⋅a² при a = 5; 14) √(16a¹⁴)/a⁸ при a = 3; 15) √(1/16⋅x⁶y⁴) при x = 2 и y = 5; 16) √(25x⁴)/y⁶ при x = 10 и y = 5; 17) √25a²+10ab + b² при a = 4/9 и b = 3 7/9; 18) √a²−6ab +9b² при a = 3 и b = 6.

Выполняю задания по порядку. 1) √5⁶ $$√5^6 = 5^{6/2} = 5^3 = 125$$ 2) √8⋅√192/√24 $$ \frac{\sqrt{8} \cdot \sqrt{192}}{\sqrt{24}} = \frac{\sqrt{8 \cdot 192}}{\sqrt{24}} = \sqrt{\frac{8 \cdot 192}{24}} = \sqrt{\frac{8 \cdot 8 \cdot 24}{24}} = \sqrt{64} = 8$$ 3) √13⋅18⋅√26 $$ \sqrt{13} \cdot 18 \cdot \sqrt{26} = 18 \sqrt{13 \cdot 26} = 18 \sqrt{13 \cdot 13 \cdot 2} = 18 \cdot 13 \sqrt{2} = 234\sqrt{2}$$ 4) (√125−√5)⋅√5 $$(\sqrt{125} - \sqrt{5}) \cdot \sqrt{5} = (\sqrt{25 \cdot 5} - \sqrt{5}) \cdot \sqrt{5} = (5\sqrt{5} - \sqrt{5}) \cdot \sqrt{5} = 4\sqrt{5} \cdot \sqrt{5} = 4 \cdot 5 = 20$$ 5) (√48+√3)⋅√3 $$(\sqrt{48} + \sqrt{3}) \cdot \sqrt{3} = (\sqrt{16 \cdot 3} + \sqrt{3}) \cdot \sqrt{3} = (4\sqrt{3} + \sqrt{3}) \cdot \sqrt{3} = 5\sqrt{3} \cdot \sqrt{3} = 5 \cdot 3 = 15$$ 6) 9√7⋅2√2⋅√14 $$9\sqrt{7} \cdot 2\sqrt{2} \cdot \sqrt{14} = 18 \sqrt{7 \cdot 2 \cdot 14} = 18 \sqrt{7 \cdot 2 \cdot 7 \cdot 2} = 18 \sqrt{7^2 \cdot 2^2} = 18 \cdot 7 \cdot 2 = 18 \cdot 14 = 252$$ 7) (√37−5)(√37+5) $$(√37 - 5)(√37 + 5) = (\sqrt{37})^2 - 5^2 = 37 - 25 = 12$$ 8) (√19−√2)(√19+√2) $$(\sqrt{19} - \sqrt{2})(\sqrt{19} + \sqrt{2}) = (\sqrt{19})^2 - (\sqrt{2})^2 = 19 - 2 = 17$$ 9) (√15−2)² + 4√15 $$(\sqrt{15} - 2)^2 + 4\sqrt{15} = (\sqrt{15})^2 - 2 \cdot 2 \cdot \sqrt{15} + 2^2 + 4\sqrt{15} = 15 - 4\sqrt{15} + 4 + 4\sqrt{15} = 19$$ 10) (4√3)²/60 $$\frac{(4\sqrt{3})^2}{60} = \frac{16 \cdot 3}{60} = \frac{48}{60} = \frac{4}{5} = 0.8$$ 11) 72/(2√3)² $$\frac{72}{(2\sqrt{3})^2} = \frac{72}{4 \cdot 3} = \frac{72}{12} = 6$$ 12) a²²⋅(b³)^6/(a⋅b)¹⁸ при a = 2 и b = √2 $$\frac{a^{22} \cdot (b^3)^6}{(a \cdot b)^{18}} = \frac{a^{22} \cdot b^{18}}{a^{18} \cdot b^{18}} = a^{22-18} = a^4$$ Подставим $$a = 2$$: $$2^4 = 16$$ 13) √(−a)⁴⋅a² при a = 5 $$\sqrt{(-a)^4 \cdot a^2} = \sqrt{a^4 \cdot a^2} = \sqrt{a^6} = a^{6/2} = a^3$$ Подставим $$a = 5$$: $$5^3 = 125$$ 14) √(16a¹⁴)/a⁸ при a = 3 $$\sqrt{\frac{16a^{14}}{a^8}} = \sqrt{16a^{14-8}} = \sqrt{16a^6} = 4a^3$$ Подставим $$a = 3$$: $$4 \cdot 3^3 = 4 \cdot 27 = 108$$ 15) √(1/16⋅x⁶y⁴) при x = 2 и y = 5 $$\sqrt{\frac{1}{16} \cdot x^6 y^4} = \frac{1}{4} x^3 y^2$$ Подставим $$x = 2$$ и $$y = 5$$: $$\frac{1}{4} \cdot 2^3 \cdot 5^2 = \frac{1}{4} \cdot 8 \cdot 25 = 2 \cdot 25 = 50$$ 16) √(25x⁴)/y⁶ при x = 10 и y = 5 $$\sqrt{\frac{25x^4}{y^6}} = \frac{5x^2}{y^3}$$ Подставим $$x = 10$$ и $$y = 5$$: $$\frac{5 \cdot 10^2}{5^3} = \frac{5 \cdot 100}{125} = \frac{500}{125} = 4$$ 17) √25a²+10ab + b² при a = 4/9 и b = 3 7/9 $$ \sqrt{25a^2 + 10ab + b^2} = \sqrt{(5a + b)^2} = |5a + b|$$ Подставим $$a = \frac{4}{9}$$ и $$b = 3\frac{7}{9} = \frac{34}{9}$$: $$|5 \cdot \frac{4}{9} + \frac{34}{9}| = |\frac{20}{9} + \frac{34}{9}| = |\frac{54}{9}| = 6$$ 18) √a²−6ab +9b² при a = 3 и b = 6 $$\sqrt{a^2 - 6ab + 9b^2} = \sqrt{(a - 3b)^2} = |a - 3b|$$ Подставим $$a = 3$$ и $$b = 6$$: $$|3 - 3 \cdot 6| = |3 - 18| = |-15| = 15$$