Назад до змісту
Розв’яжіть задачі та виконайте вправи
5.1. Виконайте множення:
1 ) 4 x a ⋅ b 3 m = 4 x ⋅ b a ⋅ 3 m = 4 x b 3 a m 1) \dfrac{4x}{a} \cdot \dfrac{b}{3m} = \dfrac{4x \cdot b}{a \cdot 3m} = \dfrac{4xb}{3am} 1 ) a 4 x ⋅ 3 m b = a ⋅ 3 m 4 x ⋅ b = 3 am 4 x b
2 ) 2 a ⋅ a 5 = 2 ⋅ a a ⋅ 5 = 2 5 2) \dfrac{2}{a} \cdot \dfrac{a}{5} = \dfrac{2 \cdot a}{a \cdot 5} = \dfrac{2}{5} 2 ) a 2 ⋅ 5 a = a ⋅ 5 2 ⋅ a = 5 2
3 ) 5 m 4 n ⋅ 3 p = 5 m ⋅ 3 4 n ⋅ p = 15 m 4 n p 3) \dfrac{5m}{4n} \cdot \dfrac{3}{p} = \dfrac{5m \cdot 3}{4n \cdot p} = \dfrac{15m}{4np} 3 ) 4 n 5 m ⋅ p 3 = 4 n ⋅ p 5 m ⋅ 3 = 4 n p 15 m
4 ) 3 x 8 ⋅ 1 x = 3 x ⋅ 1 8 ⋅ x = 3 8 4) \dfrac{3x}{8} \cdot \dfrac{1}{x} = \dfrac{3x \cdot 1}{8 \cdot x} = \dfrac{3}{8} 4 ) 8 3 x ⋅ x 1 = 8 ⋅ x 3 x ⋅ 1 = 8 3
5.2. Виконайте множення:
1 ) 5 p a ⋅ x 2 b = 5 p ⋅ x a ⋅ 2 b = 5 p x 2 a b 1) \dfrac{5p}{a} \cdot \dfrac{x}{2b} = \dfrac{5p \cdot x}{a \cdot 2b} = \dfrac{5px}{2ab} 1 ) a 5 p ⋅ 2 b x = a ⋅ 2 b 5 p ⋅ x = 2 ab 5 p x
2 ) b 9 ⋅ 7 b = b ⋅ 7 9 ⋅ b = 7 9 2) \dfrac{b}{9} \cdot \dfrac{7}{b} = \dfrac{b \cdot 7}{9 \cdot b} = \dfrac{7}{9} 2 ) 9 b ⋅ b 7 = 9 ⋅ b b ⋅ 7 = 9 7
3 ) 4 7 a ⋅ 5 b 3 = 4 ⋅ 5 b 7 a ⋅ 3 = 20 b 21 a 3) \dfrac{4}{7a} \cdot \dfrac{5b}{3} = \dfrac{4 \cdot 5b}{7a \cdot 3} = \dfrac{20b}{21a} 3 ) 7 a 4 ⋅ 3 5 b = 7 a ⋅ 3 4 ⋅ 5 b = 21 a 20 b
4 ) 1 m ⋅ m 8 = 1 ⋅ m m ⋅ 8 = 1 8 4) \dfrac{1}{m} \cdot \dfrac{m}{8} = \dfrac{1 \cdot m}{m \cdot 8} = \dfrac{1}{8} 4 ) m 1 ⋅ 8 m = m ⋅ 8 1 ⋅ m = 8 1
5.3. Перетворіть на дріб вираз:
1 ) a 2 5 ⋅ 7 a = a 2 ⋅ 7 5 ⋅ a = 7 a 5 1) \dfrac{a^2}{5} \cdot \dfrac{7}{a} = \dfrac{a^2 \cdot 7}{5 \cdot a} = \dfrac{7a}{5} 1 ) 5 a 2 ⋅ a 7 = 5 ⋅ a a 2 ⋅ 7 = 5 7 a
2 ) b 4 3 ⋅ 5 b 2 = b 4 ⋅ 5 3 ⋅ b 2 = 5 b 2 3 2) \dfrac{b^4}{3} \cdot \dfrac{5}{b^2} = \dfrac{b^4 \cdot 5}{3 \cdot b^2} = \dfrac{5b^2}{3} 2 ) 3 b 4 ⋅ b 2 5 = 3 ⋅ b 2 b 4 ⋅ 5 = 3 5 b 2
3 ) 1 a 2 ⋅ a 3 = 1 ⋅ a a 2 ⋅ 3 = 1 3 a 3) \dfrac{1}{a^2} \cdot \dfrac{a}{3} = \dfrac{1 \cdot a}{a^2 \cdot 3} = \dfrac{1}{3a} 3 ) a 2 1 ⋅ 3 a = a 2 ⋅ 3 1 ⋅ a = 3 a 1
4 ) 9 x 2 ⋅ x 3 = 9 ⋅ x x 2 ⋅ 3 = 3 x 4) \dfrac{9}{x^2} \cdot \dfrac{x}{3} = \dfrac{9 \cdot x}{x^2 \cdot 3} = \dfrac{3}{x} 4 ) x 2 9 ⋅ 3 x = x 2 ⋅ 3 9 ⋅ x = x 3
5.4. Перетворіть на дріб вираз:
1 ) 7 b ⋅ b 2 3 = 7 ⋅ b 2 b ⋅ 3 = 7 b 3 1) \dfrac{7}{b} \cdot \dfrac{b^2}{3} = \dfrac{7 \cdot b^2}{b \cdot 3} = \dfrac{7b}{3} 1 ) b 7 ⋅ 3 b 2 = b ⋅ 3 7 ⋅ b 2 = 3 7 b
2 ) 5 a 3 ⋅ a 5 2 = 5 ⋅ a 5 a 3 ⋅ 2 = 5 a 2 2 2) \dfrac{5}{a^3} \cdot \dfrac{a^5}{2} = \dfrac{5 \cdot a^5}{a^3 \cdot 2} = \dfrac{5a^2}{2} 2 ) a 3 5 ⋅ 2 a 5 = a 3 ⋅ 2 5 ⋅ a 5 = 2 5 a 2
3 ) m 8 ⋅ 1 m 2 = m ⋅ 1 8 ⋅ m 2 = 1 8 m 3) \dfrac{m}{8} \cdot \dfrac{1}{m^2} = \dfrac{m \cdot 1}{8 \cdot m^2} = \dfrac{1}{8m} 3 ) 8 m ⋅ m 2 1 = 8 ⋅ m 2 m ⋅ 1 = 8 m 1
4 ) a 2 12 ⋅ 4 a = a 2 ⋅ 4 12 ⋅ a = a 3 4) \dfrac{a^2}{12} \cdot \dfrac{4}{a} = \dfrac{a^2 \cdot 4}{12 \cdot a} = \dfrac{a}{3} 4 ) 12 a 2 ⋅ a 4 = 12 ⋅ a a 2 ⋅ 4 = 3 a
5.5. Виконайте дію:
1 ) 5 a 7 ⋅ 21 20 a 2 = 5 a ⋅ 21 7 ⋅ 20 a 2 = 1 ⋅ 3 4 a = 3 4 a 1) \dfrac{5a}{7} \cdot \dfrac{21}{20a^2} = \dfrac{5a \cdot 21}{7 \cdot 20a^2} = \dfrac{1 \cdot 3}{4a} = \dfrac{3}{4a} 1 ) 7 5 a ⋅ 20 a 2 21 = 7 ⋅ 20 a 2 5 a ⋅ 21 = 4 a 1 ⋅ 3 = 4 a 3
2 ) 3 , 5 14 a 2 ⋅ 4 a 3 5 b = 3 , 5 ⋅ 4 a 3 14 a 2 ⋅ 5 b = 14 a 70 b = a 5 b 2) \dfrac{3,5}{14a^2} \cdot \dfrac{4a^3}{5b} = \dfrac{3,5 \cdot 4a^3}{14a^2 \cdot 5b} = \dfrac{14a}{70b} = \dfrac{a}{5b} 2 ) 14 a 2 3 , 5 ⋅ 5 b 4 a 3 = 14 a 2 ⋅ 5 b 3 , 5 ⋅ 4 a 3 = 70 b 14 a = 5 b a
3 ) c 2 30 ⋅ 20 c m = 20 c 2 30 c m = 2 c 3 m 3) \dfrac{c^2}{30} \cdot \dfrac{20}{cm} = \dfrac{20c^2}{30cm} = \dfrac{2c}{3m} 3 ) 30 c 2 ⋅ c m 20 = 30 c m 20 c 2 = 3 m 2 c
4 ) − 3 m 5 a 2 ⋅ a 9 m 2 = − 3 a m 45 a 2 m 2 = − 1 15 a m 4) -\dfrac{3m}{5a^2} \cdot \dfrac{a}{9m^2} = -\dfrac{3am}{45a^2m^2} = -\dfrac{1}{15am} 4 ) − 5 a 2 3 m ⋅ 9 m 2 a = − 45 a 2 m 2 3 am = − 15 am 1
5 ) 4 x 2 7 p ⋅ ( − 21 p 8 x 3 ) = − 4 x 2 ⋅ 21 p 7 p ⋅ 8 x 3 = − 3 2 x 5) \dfrac{4x^2}{7p} \cdot \left(-\dfrac{21p}{8x^3}\right) = -\dfrac{4x^2 \cdot 21p}{7p \cdot 8x^3} = -\dfrac{3}{2x} 5 ) 7 p 4 x 2 ⋅ ( − 8 x 3 21 p ) = − 7 p ⋅ 8 x 3 4 x 2 ⋅ 21 p = − 2 x 3
6 ) − 5 x 2 7 y 3 ⋅ ( − 21 y 2 25 x ) = 5 x 2 ⋅ 21 y 2 7 y 3 ⋅ 25 x = 3 x 5 y 6) -\dfrac{5x^2}{7y^3} \cdot \left(-\dfrac{21y^2}{25x}\right) = \dfrac{5x^2 \cdot 21y^2}{7y^3 \cdot 25x} = \dfrac{3x}{5y} 6 ) − 7 y 3 5 x 2 ⋅ ( − 25 x 21 y 2 ) = 7 y 3 ⋅ 25 x 5 x 2 ⋅ 21 y 2 = 5 y 3 x
5.6. Перетворіть на дріб вираз:
1 ) 15 m 2 22 ⋅ 11 10 m = 15 m 2 ⋅ 11 22 ⋅ 10 m = 3 m 4 1) \dfrac{15m^2}{22} \cdot \dfrac{11}{10m} = \dfrac{15m^2 \cdot 11}{22 \cdot 10m} = \dfrac{3m}{4} 1 ) 22 15 m 2 ⋅ 10 m 11 = 22 ⋅ 10 m 15 m 2 ⋅ 11 = 4 3 m
2 ) 6 p 7 ⋅ 2 , 5 c 2 15 p 3 = 6 p ⋅ 2 , 5 c 2 7 ⋅ 15 p 3 = 15 p c 2 105 p 3 = c 2 7 p 2 2) \dfrac{6p}{7} \cdot \dfrac{2,5c^2}{15p^3} = \dfrac{6p \cdot 2,5c^2}{7 \cdot 15p^3} = \dfrac{15pc^2}{105p^3} = \dfrac{c^2}{7p^2} 2 ) 7 6 p ⋅ 15 p 3 2 , 5 c 2 = 7 ⋅ 15 p 3 6 p ⋅ 2 , 5 c 2 = 105 p 3 15 p c 2 = 7 p 2 c 2
3 ) 15 x p ⋅ x 2 45 = 15 x 2 45 x p = x 3 p 3) \dfrac{15}{xp} \cdot \dfrac{x^2}{45} = \dfrac{15x^2}{45xp} = \dfrac{x}{3p} 3 ) x p 15 ⋅ 45 x 2 = 45 x p 15 x 2 = 3 p x
4 ) 4 a p 2 ⋅ ( − p 8 a 2 ) = − 4 a p 8 a 2 p 2 = − 1 2 a p 4) \dfrac{4a}{p^2} \cdot \left(-\dfrac{p}{8a^2}\right) = -\dfrac{4ap}{8a^2p^2} = -\dfrac{1}{2ap} 4 ) p 2 4 a ⋅ ( − 8 a 2 p ) = − 8 a 2 p 2 4 a p = − 2 a p 1
5 ) − 5 c 2 7 y ⋅ 49 y 10 c 3 = − 5 c 2 ⋅ 49 y 7 y ⋅ 10 c 3 = − 7 2 c 5) -\dfrac{5c^2}{7y} \cdot \dfrac{49y}{10c^3} = -\dfrac{5c^2 \cdot 49y}{7y \cdot 10c^3} = -\dfrac{7}{2c} 5 ) − 7 y 5 c 2 ⋅ 10 c 3 49 y = − 7 y ⋅ 10 c 3 5 c 2 ⋅ 49 y = − 2 c 7
6 ) − 6 a 2 65 b 3 ⋅ ( − 13 b 30 a ) = 6 a 2 ⋅ 13 b 65 b 3 ⋅ 30 a = a 25 b 2 6) -\dfrac{6a^2}{65b^3} \cdot \left(-\dfrac{13b}{30a}\right) = \dfrac{6a^2 \cdot 13b}{65b^3 \cdot 30a} = \dfrac{a}{25b^2} 6 ) − 65 b 3 6 a 2 ⋅ ( − 30 a 13 b ) = 65 b 3 ⋅ 30 a 6 a 2 ⋅ 13 b = 25 b 2 a
5.7. Перетворіть на дріб вираз:
9 p ⋅ b 6 p 2 = 9 p b 6 p 2 = 3 b 2 p 9p \cdot \dfrac{b}{6p^2} = \dfrac{9pb}{6p^2} = \dfrac{3b}{2p} 9 p ⋅ 6 p 2 b = 6 p 2 9 p b = 2 p 3 b 4 m 3 x 2 ⋅ x 3 = 4 m 3 x 3 x 2 = 4 m 3 x \dfrac{4m^3}{x^2} \cdot x^3 = \dfrac{4m^3x^3}{x^2} = 4m^3x x 2 4 m 3 ⋅ x 3 = x 2 4 m 3 x 3 = 4 m 3 x 9 a b 2 ⋅ ( − 5 b 3 a 3 ) = − 9 a b 2 ⋅ 5 b 3 a 3 = − 45 a b 3 3 a 3 = − 15 b 3 a 2 9ab^2 \cdot (-\dfrac{5b}{3a^3}) = -\dfrac{9ab^2 \cdot 5b}{3a^3} = -\dfrac{45ab^3}{3a^3} = -\dfrac{15b^3}{a^2} 9 a b 2 ⋅ ( − 3 a 3 5 b ) = − 3 a 3 9 a b 2 ⋅ 5 b = − 3 a 3 45 a b 3 = − a 2 15 b 3 − 7 a b 3 ⋅ b 5 14 a = − 7 a b 8 14 a = − b 8 2 -7ab^3 \cdot \dfrac{b^5}{14a} = -\dfrac{7ab^8}{14a} = -\dfrac{b^8}{2} − 7 a b 3 ⋅ 14 a b 5 = − 14 a 7 a b 8 = − 2 b 8 − 4 m n 2 ⋅ 1 8 m n = − 4 m n 2 8 m n = − n 2 -4mn^2 \cdot \dfrac{1}{8mn} = -\dfrac{4mn^2}{8mn} = -\dfrac{n}{2} − 4 m n 2 ⋅ 8 mn 1 = − 8 mn 4 m n 2 = − 2 n − 11 a 2 b ⋅ ( − 5 22 a 3 b 2 ) = 11 a 2 b ⋅ 5 22 a 3 b 2 = 55 a 2 b 22 a 3 b 2 = 5 2 a b -11a^2b \cdot (-\dfrac{5}{22a^3b^2}) = \dfrac{11a^2b \cdot 5}{22a^3b^2} = \dfrac{55a^2b}{22a^3b^2} = \dfrac{5}{2ab} − 11 a 2 b ⋅ ( − 22 a 3 b 2 5 ) = 22 a 3 b 2 11 a 2 b ⋅ 5 = 22 a 3 b 2 55 a 2 b = 2 ab 5
5.8. Виконайте дію:
a 16 m 2 ⋅ 12 m = 12 a m 16 m 2 = 3 a 4 m \dfrac{a}{16m^2} \cdot 12m = \dfrac{12am}{16m^2} = \dfrac{3a}{4m} 16 m 2 a ⋅ 12 m = 16 m 2 12 am = 4 m 3 a a 3 ⋅ 7 x 3 a 2 = a 3 ⋅ 7 x 3 a 2 = 7 a x 3 a^3 \cdot \dfrac{7x^3}{a^2} = \dfrac{a^3 \cdot 7x^3}{a^2} = 7ax^3 a 3 ⋅ a 2 7 x 3 = a 2 a 3 ⋅ 7 x 3 = 7 a x 3 − 7 y 4 x 2 ⋅ 12 x y 3 = − 7 y ⋅ 12 x y 3 4 x 2 = − 84 x y 4 4 x 2 = − 21 y 4 x -\dfrac{7y}{4x^2} \cdot 12xy^3 = -\dfrac{7y \cdot 12xy^3}{4x^2} = -\dfrac{84xy^4}{4x^2} = -\dfrac{21y^4}{x} − 4 x 2 7 y ⋅ 12 x y 3 = − 4 x 2 7 y ⋅ 12 x y 3 = − 4 x 2 84 x y 4 = − x 21 y 4 5 c m 4 ⋅ ( − m 15 c ) = − 5 c m 5 15 c = − m 5 3 5cm^4 \cdot (-\dfrac{m}{15c}) = -\dfrac{5cm^5}{15c} = -\dfrac{m^5}{3} 5 c m 4 ⋅ ( − 15 c m ) = − 15 c 5 c m 5 = − 3 m 5 − 5 a b 2 ⋅ ( − 1 10 a b ) = 5 a b 2 10 a b = b 2 -5ab^2 \cdot (-\dfrac{1}{10ab}) = \dfrac{5ab^2}{10ab} = \dfrac{b}{2} − 5 a b 2 ⋅ ( − 10 ab 1 ) = 10 ab 5 a b 2 = 2 b 13 c 2 d ⋅ 7 26 c 3 d 2 = 13 c 2 d ⋅ 7 26 c 3 d 2 = 91 c 2 d 26 c 3 d 2 = 7 2 c d 13c^2d \cdot \dfrac{7}{26c^3d^2} = \dfrac{13c^2d \cdot 7}{26c^3d^2} = \dfrac{91c^2d}{26c^3d^2} = \dfrac{7}{2cd} 13 c 2 d ⋅ 26 c 3 d 2 7 = 26 c 3 d 2 13 c 2 d ⋅ 7 = 26 c 3 d 2 91 c 2 d = 2 c d 7
5.9. Спростіть вираз:
7 c 3 10 m 2 ⋅ 25 m 3 14 c 8 = 7 ⋅ 25 ⋅ c 3 m 3 10 ⋅ 14 ⋅ m 2 c 8 = 175 c 3 m 3 140 m 2 c 8 = 5 m 4 c 5 \dfrac{7c^3}{10m^2} \cdot \dfrac{25m^3}{14c^8} = \dfrac{7 \cdot 25 \cdot c^3m^3}{10 \cdot 14 \cdot m^2c^8} = \dfrac{175c^3m^3}{140m^2c^8} = \dfrac{5m}{4c^5} 10 m 2 7 c 3 ⋅ 14 c 8 25 m 3 = 10 ⋅ 14 ⋅ m 2 c 8 7 ⋅ 25 ⋅ c 3 m 3 = 140 m 2 c 8 175 c 3 m 3 = 4 c 5 5 m − 8 a 3 27 c 4 ⋅ 45 c 5 16 a 3 = − 8 ⋅ 45 ⋅ a 3 c 5 27 ⋅ 16 ⋅ c 4 a 3 = − 360 a 3 c 5 432 c 4 a 3 = − 5 c 6 -\dfrac{8a^3}{27c^4} \cdot \dfrac{45c^5}{16a^3} = -\dfrac{8 \cdot 45 \cdot a^3c^5}{27 \cdot 16 \cdot c^4a^3} = -\dfrac{360a^3c^5}{432c^4a^3} = -\dfrac{5c}{6} − 27 c 4 8 a 3 ⋅ 16 a 3 45 c 5 = − 27 ⋅ 16 ⋅ c 4 a 3 8 ⋅ 45 ⋅ a 3 c 5 = − 432 c 4 a 3 360 a 3 c 5 = − 6 5 c 4 c 3 15 a 8 ⋅ ( − 5 a 3 8 c 4 ) = − 4 ⋅ 5 ⋅ c 3 a 3 15 ⋅ 8 ⋅ a 8 c 4 = − 20 c 3 a 3 120 a 8 c 4 = − 1 6 a 5 c \dfrac{4c^3}{15a^8} \cdot (-\dfrac{5a^3}{8c^4}) = -\dfrac{4 \cdot 5 \cdot c^3a^3}{15 \cdot 8 \cdot a^8c^4} = -\dfrac{20c^3a^3}{120a^8c^4} = -\dfrac{1}{6a^5c} 15 a 8 4 c 3 ⋅ ( − 8 c 4 5 a 3 ) = − 15 ⋅ 8 ⋅ a 8 c 4 4 ⋅ 5 ⋅ c 3 a 3 = − 120 a 8 c 4 20 c 3 a 3 = − 6 a 5 c 1 − 1 25 p 2 q 7 ⋅ ( − 10 p 3 q 7 11 ) = 10 p 3 q 7 25 ⋅ 11 ⋅ p 2 q 7 = 10 p 3 q 7 275 p 2 q 7 = 2 p 55 -\dfrac{1}{25p^2q^7} \cdot (-\dfrac{10p^3q^7}{11}) = \dfrac{10p^3q^7}{25 \cdot 11 \cdot p^2q^7} = \dfrac{10p^3q^7}{275p^2q^7} = \dfrac{2p}{55} − 25 p 2 q 7 1 ⋅ ( − 11 10 p 3 q 7 ) = 25 ⋅ 11 ⋅ p 2 q 7 10 p 3 q 7 = 275 p 2 q 7 10 p 3 q 7 = 55 2 p
5.10. Спростіть вираз:
9 m 2 25 a 2 ⋅ 35 a 3 18 m 5 = 9 ⋅ 35 ⋅ m 2 a 3 25 ⋅ 18 ⋅ a 2 m 5 = 7 a 10 m 3 \dfrac{9m^2}{25a^2} \cdot \dfrac{35a^3}{18m^5} = \dfrac{9 \cdot 35 \cdot m^2a^3}{25 \cdot 18 \cdot a^2m^5} = \dfrac{7a}{10m^3} 25 a 2 9 m 2 ⋅ 18 m 5 35 a 3 = 25 ⋅ 18 ⋅ a 2 m 5 9 ⋅ 35 ⋅ m 2 a 3 = 10 m 3 7 a 7 p 3 18 a 3 ⋅ ( − 27 a 4 14 p 3 ) = − 7 ⋅ 27 ⋅ p 3 a 4 18 ⋅ 14 ⋅ a 3 p 3 = − 3 a 4 \dfrac{7p^3}{18a^3} \cdot (-\dfrac{27a^4}{14p^3}) = -\dfrac{7 \cdot 27 \cdot p^3a^4}{18 \cdot 14 \cdot a^3p^3} = -\dfrac{3a}{4} 18 a 3 7 p 3 ⋅ ( − 14 p 3 27 a 4 ) = − 18 ⋅ 14 ⋅ a 3 p 3 7 ⋅ 27 ⋅ p 3 a 4 = − 4 3 a − 5 m 3 21 n 7 ⋅ 7 n 2 10 m 4 = − 5 ⋅ 7 ⋅ m 3 n 2 21 ⋅ 10 ⋅ n 7 m 4 = − 1 6 m n 5 -\dfrac{5m^3}{21n^7} \cdot \dfrac{7n^2}{10m^4} = -\dfrac{5 \cdot 7 \cdot m^3n^2}{21 \cdot 10 \cdot n^7m^4} = -\dfrac{1}{6mn^5} − 21 n 7 5 m 3 ⋅ 10 m 4 7 n 2 = − 21 ⋅ 10 ⋅ n 7 m 4 5 ⋅ 7 ⋅ m 3 n 2 = − 6 m n 5 1 − 1 18 c 3 d 4 ⋅ ( − 12 c 4 d 4 7 ) = 12 c 4 d 4 18 ⋅ 7 ⋅ c 3 d 4 = 2 c 21 -\dfrac{1}{18c^3d^4} \cdot (-\dfrac{12c^4d^4}{7}) = \dfrac{12c^4d^4}{18 \cdot 7 \cdot c^3d^4} = \dfrac{2c}{21} − 18 c 3 d 4 1 ⋅ ( − 7 12 c 4 d 4 ) = 18 ⋅ 7 ⋅ c 3 d 4 12 c 4 d 4 = 21 2 c
5.11. Виконайте множення:
a 2 + 2 a 5 ⋅ a 4 a + 8 = a ( a + 2 ) 5 ⋅ a 4 ( a + 2 ) = a 2 20 \dfrac{a^2+2a}{5} \cdot \dfrac{a}{4a+8} = \dfrac{a(a+2)}{5} \cdot \dfrac{a}{4(a+2)} = \dfrac{a^2}{20} 5 a 2 + 2 a ⋅ 4 a + 8 a = 5 a ( a + 2 ) ⋅ 4 ( a + 2 ) a = 20 a 2 7 m a ⋅ a 2 − a b 21 = 7 m a ⋅ a ( a − b ) 21 = m ( a − b ) 3 \dfrac{7m}{a} \cdot \dfrac{a^2-ab}{21} = \dfrac{7m}{a} \cdot \dfrac{a(a-b)}{21} = \dfrac{m(a-b)}{3} a 7 m ⋅ 21 a 2 − ab = a 7 m ⋅ 21 a ( a − b ) = 3 m ( a − b ) 2 a − b 10 a ⋅ 15 a 2 b − 2 a = 2 a − b 10 a ⋅ 15 a 2 − ( 2 a − b ) = − 15 a 2 10 a = − 3 a 2 \dfrac{2a-b}{10a} \cdot \dfrac{15a^2}{b-2a} = \dfrac{2a-b}{10a} \cdot \dfrac{15a^2}{-(2a-b)} = -\dfrac{15a^2}{10a} = -\dfrac{3a}{2} 10 a 2 a − b ⋅ b − 2 a 15 a 2 = 10 a 2 a − b ⋅ − ( 2 a − b ) 15 a 2 = − 10 a 15 a 2 = − 2 3 a 10 a b x + y ⋅ x 2 − y 2 5 a b = 10 a b x + y ⋅ ( x − y ) ( x + y ) 5 a b = 2 ( x − y ) \dfrac{10ab}{x+y} \cdot \dfrac{x^2-y^2}{5ab} = \dfrac{10ab}{x+y} \cdot \dfrac{(x-y)(x+y)}{5ab} = 2(x-y) x + y 10 ab ⋅ 5 ab x 2 − y 2 = x + y 10 ab ⋅ 5 ab ( x − y ) ( x + y ) = 2 ( x − y ) − a b − a c 10 p ⋅ 25 p x c − x b = − a ( b − c ) 10 p ⋅ 25 p x ( c − b ) = − a ( b − c ) 10 p ⋅ 25 p − x ( b − c ) = 25 a p 10 p x = 5 a 2 x -\dfrac{ab-ac}{10p} \cdot \dfrac{25p}{xc-xb} = -\dfrac{a(b-c)}{10p} \cdot \dfrac{25p}{x(c-b)} = -\dfrac{a(b-c)}{10p} \cdot \dfrac{25p}{-x(b-c)} = \dfrac{25ap}{10px} = \dfrac{5a}{2x} − 10 p ab − a c ⋅ x c − x b 25 p = − 10 p a ( b − c ) ⋅ x ( c − b ) 25 p = − 10 p a ( b − c ) ⋅ − x ( b − c ) 25 p = 10 p x 25 a p = 2 x 5 a a 2 + a b x 2 ⋅ x y a 2 + 2 a b + b 2 = a ( a + b ) x 2 ⋅ x y ( a + b ) 2 = a y x ( a + b ) \dfrac{a^2+ab}{x^2} \cdot \dfrac{xy}{a^2+2ab+b^2} = \dfrac{a(a+b)}{x^2} \cdot \dfrac{xy}{(a+b)^2} = \dfrac{ay}{x(a+b)} x 2 a 2 + ab ⋅ a 2 + 2 ab + b 2 x y = x 2 a ( a + b ) ⋅ ( a + b ) 2 x y = x ( a + b ) a y
5.12. Виконайте множення:
m 2 − 3 m 7 ⋅ x 2 m − 6 = m ( m − 3 ) 7 ⋅ x 2 ( m − 3 ) = m x 14 \dfrac{m^2-3m}{7} \cdot \dfrac{x}{2m-6} = \dfrac{m(m-3)}{7} \cdot \dfrac{x}{2(m-3)} = \dfrac{mx}{14} 7 m 2 − 3 m ⋅ 2 m − 6 x = 7 m ( m − 3 ) ⋅ 2 ( m − 3 ) x = 14 m x 5 a x 2 + x y ⋅ x 15 = 5 a x ( x + y ) ⋅ x 15 = a 3 ( x + y ) \dfrac{5a}{x^2+xy} \cdot \dfrac{x}{15} = \dfrac{5a}{x(x+y)} \cdot \dfrac{x}{15} = \dfrac{a}{3(x+y)} x 2 + x y 5 a ⋅ 15 x = x ( x + y ) 5 a ⋅ 15 x = 3 ( x + y ) a a − b 16 m 2 ⋅ 24 m b − a = a − b 16 m 2 ⋅ 24 m − ( a − b ) = − 24 m 16 m 2 = − 3 2 m \dfrac{a-b}{16m^2} \cdot \dfrac{24m}{b-a} = \dfrac{a-b}{16m^2} \cdot \dfrac{24m}{-(a-b)} = -\dfrac{24m}{16m^2} = -\dfrac{3}{2m} 16 m 2 a − b ⋅ b − a 24 m = 16 m 2 a − b ⋅ − ( a − b ) 24 m = − 16 m 2 24 m = − 2 m 3 x 2 − y 2 5 p c ⋅ 20 p c x − y = ( x − y ) ( x + y ) 5 p c ⋅ 20 p c x − y = 4 ( x + y ) \dfrac{x^2-y^2}{5pc} \cdot \dfrac{20pc}{x-y} = \dfrac{(x-y)(x+y)}{5pc} \cdot \dfrac{20pc}{x-y} = 4(x+y) 5 p c x 2 − y 2 ⋅ x − y 20 p c = 5 p c ( x − y ) ( x + y ) ⋅ x − y 20 p c = 4 ( x + y ) 3 a − 3 b 12 x ⋅ ( − 18 x m b − m a ) = 3 ( a − b ) 12 x ⋅ ( − 18 x m ( b − a ) ) = 3 ( a − b ) 12 x ⋅ 18 x m ( a − b ) = 54 x 12 x m = 9 2 m \dfrac{3a-3b}{12x} \cdot (-\dfrac{18x}{mb-ma}) = \dfrac{3(a-b)}{12x} \cdot (-\dfrac{18x}{m(b-a)}) = \dfrac{3(a-b)}{12x} \cdot \dfrac{18x}{m(a-b)} = \dfrac{54x}{12xm} = \dfrac{9}{2m} 12 x 3 a − 3 b ⋅ ( − mb − ma 18 x ) = 12 x 3 ( a − b ) ⋅ ( − m ( b − a ) 18 x ) = 12 x 3 ( a − b ) ⋅ m ( a − b ) 18 x = 12 x m 54 x = 2 m 9 m 2 − 2 m n + n 2 p c ⋅ p 2 m 2 − m n = ( m − n ) 2 p c ⋅ p 2 m ( m − n ) = p ( m − n ) m c \dfrac{m^2-2mn+n^2}{pc} \cdot \dfrac{p^2}{m^2-mn} = \dfrac{(m-n)^2}{pc} \cdot \dfrac{p^2}{m(m-n)} = \dfrac{p(m-n)}{mc} p c m 2 − 2 mn + n 2 ⋅ m 2 − mn p 2 = p c ( m − n ) 2 ⋅ m ( m − n ) p 2 = m c p ( m − n )
5.13. Спростіть вираз та знайдіть його значення:
5 x 2 + 15 x y 2 − 2 y ⋅ 2 − y x + 3 = 5 x ( x + 3 ) y ( y − 2 ) ⋅ − ( y − 2 ) x + 3 = − 5 x y \dfrac{5x^2+15x}{y^2-2y} \cdot \dfrac{2-y}{x+3} = \dfrac{5x(x+3)}{y(y-2)} \cdot \dfrac{-(y-2)}{x+3} = -\dfrac{5x}{y} y 2 − 2 y 5 x 2 + 15 x ⋅ x + 3 2 − y = y ( y − 2 ) 5 x ( x + 3 ) ⋅ x + 3 − ( y − 2 ) = − y 5 x
Якщо x = 17 , y = − 1 23 x=17, y = -\dfrac{1}{23} x = 17 , y = − 23 1
− 5 ⋅ 17 − 1 23 = 85 1 23 = 85 ⋅ 23 = 1955 -\dfrac{5 \cdot 17}{-\dfrac{1}{23}} = \dfrac{85}{\dfrac{1}{23}} = 85 \cdot 23 = 1955 − − 23 1 5 ⋅ 17 = 23 1 85 = 85 ⋅ 23 = 1955
Відтак: у 1955 році українка Катерина Ющенко винайшла одну з перших у світі мов програмування високого рівня з назвою «Адресна мова».
5.14. Спростіть вираз та знайдіть його значення:
35 a − 5 a 2 b 2 + 4 b ⋅ 4 + b a − 7 = 5 a ( 7 − a ) b ( b + 4 ) ⋅ b + 4 a − 7 = − 5 a ( a − 7 ) b ( b + 4 ) ⋅ b + 4 a − 7 = − 5 a b \dfrac{35a-5a^2}{b^2+4b} \cdot \dfrac{4+b}{a-7} = \dfrac{5a(7-a)}{b(b+4)} \cdot \dfrac{b+4}{a-7} = \dfrac{-5a(a-7)}{b(b+4)} \cdot \dfrac{b+4}{a-7} = -\dfrac{5a}{b} b 2 + 4 b 35 a − 5 a 2 ⋅ a − 7 4 + b = b ( b + 4 ) 5 a ( 7 − a ) ⋅ a − 7 b + 4 = b ( b + 4 ) − 5 a ( a − 7 ) ⋅ a − 7 b + 4 = − b 5 a
Якщо a = 79 , b = − 0.2 a=79, b=-0.2 a = 79 , b = − 0.2
− 5 ⋅ 79 − 0.2 = 395 0.2 = 1975 -\dfrac{5 \cdot 79}{-0.2} = \dfrac{395}{0.2} = 1975 − − 0.2 5 ⋅ 79 = 0.2 395 = 1975
Відтак: у 1975 році футбольна команда «Динамо» (Київ) уперше виборола європейський Кубок кубків.
5.15. Піднесіть до степеня:
( p 4 m ) 3 = p 3 4 3 m 3 = p 3 64 m 3 (\dfrac{p}{4m})^3 = \dfrac{p^3}{4^3m^3} = \dfrac{p^3}{64m^3} ( 4 m p ) 3 = 4 3 m 3 p 3 = 64 m 3 p 3 ( 3 c 2 m ) 4 = 3 4 ( c 2 ) 4 m 4 = 81 c 8 m 4 (\dfrac{3c^2}{m})^4 = \dfrac{3^4(c^2)^4}{m^4} = \dfrac{81c^8}{m^4} ( m 3 c 2 ) 4 = m 4 3 4 ( c 2 ) 4 = m 4 81 c 8 ( − 3 m 2 n 7 ) 2 = ( − 3 ) 2 ( m 2 ) 2 n 2 7 2 = 9 m 4 n 2 49 (-\dfrac{3m^2n}{7})^2 = \dfrac{(-3)^2(m^2)^2n^2}{7^2} = \dfrac{9m^4n^2}{49} ( − 7 3 m 2 n ) 2 = 7 2 ( − 3 ) 2 ( m 2 ) 2 n 2 = 49 9 m 4 n 2 ( − 2 m 2 3 x 3 ) 3 = − 2 3 ( m 2 ) 3 3 3 ( x 3 ) 3 = − 8 m 6 27 x 9 (-\dfrac{2m^2}{3x^3})^3 = -\dfrac{2^3(m^2)^3}{3^3(x^3)^3} = -\dfrac{8m^6}{27x^9} ( − 3 x 3 2 m 2 ) 3 = − 3 3 ( x 3 ) 3 2 3 ( m 2 ) 3 = − 27 x 9 8 m 6 ( 2 a 3 b x 7 ) 5 = 2 5 ( a 3 ) 5 b 5 ( x 7 ) 5 = 32 a 15 b 5 x 35 (\dfrac{2a^3b}{x^7})^5 = \dfrac{2^5(a^3)^5b^5}{(x^7)^5} = \dfrac{32a^{15}b^5}{x^{35}} ( x 7 2 a 3 b ) 5 = ( x 7 ) 5 2 5 ( a 3 ) 5 b 5 = x 35 32 a 15 b 5 ( − c 2 m 3 p ) 10 = ( c 2 ) 10 ( m 3 ) 10 p 10 = c 20 m 30 p 10 (-\dfrac{c^2m^3}{p})^{10} = \dfrac{(c^2)^{10}(m^3)^{10}}{p^{10}} = \dfrac{c^{20}m^{30}}{p^{10}} ( − p c 2 m 3 ) 10 = p 10 ( c 2 ) 10 ( m 3 ) 10 = p 10 c 20 m 30
5.16. Подайте у вигляді дробу вираз:
( c 5 m ) 2 = c 2 5 2 m 2 = c 2 25 m 2 (\dfrac{c}{5m})^2 = \dfrac{c^2}{5^2m^2} = \dfrac{c^2}{25m^2} ( 5 m c ) 2 = 5 2 m 2 c 2 = 25 m 2 c 2 ( y 2 x 3 ) 4 = y 4 2 4 ( x 3 ) 4 = y 4 16 x 12 (\dfrac{y}{2x^3})^4 = \dfrac{y^4}{2^4(x^3)^4} = \dfrac{y^4}{16x^{12}} ( 2 x 3 y ) 4 = 2 4 ( x 3 ) 4 y 4 = 16 x 12 y 4 ( − 4 c 2 m 3 5 ) 2 = ( − 4 ) 2 ( c 2 ) 2 ( m 3 ) 2 5 2 = 16 c 4 m 6 25 (-\dfrac{4c^2m^3}{5})^2 = \dfrac{(-4)^2(c^2)^2(m^3)^2}{5^2} = \dfrac{16c^4m^6}{25} ( − 5 4 c 2 m 3 ) 2 = 5 2 ( − 4 ) 2 ( c 2 ) 2 ( m 3 ) 2 = 25 16 c 4 m 6 ( − 3 c 3 m 7 ) 3 = − 3 3 ( c 3 ) 3 ( m 7 ) 3 = − 27 c 9 m 21 (-\dfrac{3c^3}{m^7})^3 = -\dfrac{3^3(c^3)^3}{(m^7)^3} = -\dfrac{27c^9}{m^{21}} ( − m 7 3 c 3 ) 3 = − ( m 7 ) 3 3 3 ( c 3 ) 3 = − m 21 27 c 9 ( c 3 m 2 a 2 ) 6 = ( c 3 ) 6 m 6 2 6 ( a 2 ) 6 = c 18 m 6 64 a 12 (\dfrac{c^3m}{2a^2})^6 = \dfrac{(c^3)^6m^6}{2^6(a^2)^6} = \dfrac{c^{18}m^6}{64a^{12}} ( 2 a 2 c 3 m ) 6 = 2 6 ( a 2 ) 6 ( c 3 ) 6 m 6 = 64 a 12 c 18 m 6 ( − a b 3 c 2 ) 8 = a 8 ( b 3 ) 8 ( c 2 ) 8 = a 8 b 24 c 16 (-\dfrac{ab^3}{c^2})^8 = \dfrac{a^8(b^3)^8}{(c^2)^8} = \dfrac{a^8b^{24}}{c^{16}} ( − c 2 a b 3 ) 8 = ( c 2 ) 8 a 8 ( b 3 ) 8 = c 16 a 8 b 24
5.17. Спростіть вираз:
54 a 2 c 81 b 3 ⋅ 32 a b 13 c 3 ⋅ 52 b c 2 128 a 3 = 54 ⋅ 32 ⋅ 52 ⋅ a 2 c a b c 2 81 ⋅ 13 ⋅ 128 ⋅ b 3 c 3 a 3 = 2 ⋅ 1 ⋅ 4 ⋅ a 3 b 2 c 3 3 ⋅ 1 ⋅ 4 ⋅ a 3 b 3 c 3 = 2 3 b \dfrac{54a^2c}{81b^3} \cdot \dfrac{32ab}{13c^3} \cdot \dfrac{52bc^2}{128a^3} = \dfrac{54 \cdot 32 \cdot 52 \cdot a^2cab c^2}{81 \cdot 13 \cdot 128 \cdot b^3c^3a^3} = \dfrac{2 \cdot 1 \cdot 4 \cdot a^3b^2c^3}{3 \cdot 1 \cdot 4 \cdot a^3b^3c^3} = \dfrac{2}{3b} 81 b 3 54 a 2 c ⋅ 13 c 3 32 ab ⋅ 128 a 3 52 b c 2 = 81 ⋅ 13 ⋅ 128 ⋅ b 3 c 3 a 3 54 ⋅ 32 ⋅ 52 ⋅ a 2 c ab c 2 = 3 ⋅ 1 ⋅ 4 ⋅ a 3 b 3 c 3 2 ⋅ 1 ⋅ 4 ⋅ a 3 b 2 c 3 = 3 b 2 147 x 4 y 2 p 3 ⋅ 10 x p 2 ⋅ y 3 105 x 5 y = 147 ⋅ 10 ⋅ x 4 y 2 x p 2 y 3 105 ⋅ p 3 x 5 y = 14 ⋅ x 5 y 5 p 2 p 3 x 5 y = 14 y 4 p \dfrac{147x^4y^2}{p^3} \cdot 10xp^2 \cdot \dfrac{y^3}{105x^5y} = \dfrac{147 \cdot 10 \cdot x^4y^2xp^2y^3}{105 \cdot p^3x^5y} = \dfrac{14 \cdot x^5y^5p^2}{p^3x^5y} = \dfrac{14y^4}{p} p 3 147 x 4 y 2 ⋅ 10 x p 2 ⋅ 105 x 5 y y 3 = 105 ⋅ p 3 x 5 y 147 ⋅ 10 ⋅ x 4 y 2 x p 2 y 3 = p 3 x 5 y 14 ⋅ x 5 y 5 p 2 = p 14 y 4
5.18. Виконайте дії:
14 x z 3 81 y 2 ⋅ 27 y 3 5 x z ⋅ 45 x y 7 z 2 = 14 ⋅ 27 ⋅ 45 ⋅ x z 3 y 3 x y 81 ⋅ 5 ⋅ 7 ⋅ y 2 x z z 2 = 2 ⋅ 1 ⋅ 9 ⋅ x 2 y 4 z 3 3 ⋅ 1 ⋅ 1 ⋅ x y 2 z 3 = 6 x y 2 \dfrac{14xz^3}{81y^2} \cdot \dfrac{27y^3}{5xz} \cdot \dfrac{45xy}{7z^2} = \dfrac{14 \cdot 27 \cdot 45 \cdot xz^3y^3xy}{81 \cdot 5 \cdot 7 \cdot y^2xz z^2} = \dfrac{2 \cdot 1 \cdot 9 \cdot x^2y^4z^3}{3 \cdot 1 \cdot 1 \cdot xy^2z^3} = 6x y^2 81 y 2 14 x z 3 ⋅ 5 x z 27 y 3 ⋅ 7 z 2 45 x y = 81 ⋅ 5 ⋅ 7 ⋅ y 2 x z z 2 14 ⋅ 27 ⋅ 45 ⋅ x z 3 y 3 x y = 3 ⋅ 1 ⋅ 1 ⋅ x y 2 z 3 2 ⋅ 1 ⋅ 9 ⋅ x 2 y 4 z 3 = 6 x y 2 b 3 111 m 5 ⋅ 3 m c 3 ⋅ 74 m 3 b c 4 = 3 ⋅ 74 ⋅ b 3 m c 3 m 3 b 111 ⋅ m 5 c 4 = 2 ⋅ b 4 m 4 c 3 m 5 c 4 = 2 b 4 m c \dfrac{b^3}{111m^5} \cdot 3mc^3 \cdot \dfrac{74m^3b}{c^4} = \dfrac{3 \cdot 74 \cdot b^3mc^3m^3b}{111 \cdot m^5c^4} = \dfrac{2 \cdot b^4m^4c^3}{m^5c^4} = \dfrac{2b^4}{mc} 111 m 5 b 3 ⋅ 3 m c 3 ⋅ c 4 74 m 3 b = 111 ⋅ m 5 c 4 3 ⋅ 74 ⋅ b 3 m c 3 m 3 b = m 5 c 4 2 ⋅ b 4 m 4 c 3 = m c 2 b 4
5.19. Знайдіть добуток:
m 2 − 4 m + 4 m 2 + 6 m + 9 ⋅ m 2 − 9 3 m − 6 = ( m − 2 ) 2 ( m + 3 ) 2 ⋅ ( m − 3 ) ( m + 3 ) 3 ( m − 2 ) = ( m − 2 ) ( m − 3 ) 3 ( m + 3 ) \dfrac{m^2-4m+4}{m^2+6m+9} \cdot \dfrac{m^2-9}{3m-6} = \dfrac{(m-2)^2}{(m+3)^2} \cdot \dfrac{(m-3)(m+3)}{3(m-2)} = \dfrac{(m-2)(m-3)}{3(m+3)} m 2 + 6 m + 9 m 2 − 4 m + 4 ⋅ 3 m − 6 m 2 − 9 = ( m + 3 ) 2 ( m − 2 ) 2 ⋅ 3 ( m − 2 ) ( m − 3 ) ( m + 3 ) = 3 ( m + 3 ) ( m − 2 ) ( m − 3 ) − x 2 − 10 x + 25 x 2 − 3 x + 9 ⋅ x 3 + 27 25 − x 2 = − ( x − 5 ) 2 x 2 − 3 x + 9 ⋅ ( x + 3 ) ( x 2 − 3 x + 9 ) − ( x − 5 ) ( x + 5 ) = x − 5 1 ⋅ x + 3 x + 5 = ( x − 5 ) ( x + 3 ) x + 5 -\dfrac{x^2-10x+25}{x^2-3x+9} \cdot \dfrac{x^3+27}{25-x^2} = -\dfrac{(x-5)^2}{x^2-3x+9} \cdot \dfrac{(x+3)(x^2-3x+9)}{-(x-5)(x+5)} = \dfrac{x-5}{1} \cdot \dfrac{x+3}{x+5} = \dfrac{(x-5)(x+3)}{x+5} − x 2 − 3 x + 9 x 2 − 10 x + 25 ⋅ 25 − x 2 x 3 + 27 = − x 2 − 3 x + 9 ( x − 5 ) 2 ⋅ − ( x − 5 ) ( x + 5 ) ( x + 3 ) ( x 2 − 3 x + 9 ) = 1 x − 5 ⋅ x + 5 x + 3 = x + 5 ( x − 5 ) ( x + 3 )
5.20. Виконайте множення:
a 2 + 8 a + 16 a 2 − 2 a + 1 ⋅ 7 a − 7 a 2 − 16 = ( a + 4 ) 2 ( a − 1 ) 2 ⋅ 7 ( a − 1 ) ( a − 4 ) ( a + 4 ) = 7 ( a + 4 ) ( a − 1 ) ( a − 4 ) \dfrac{a^2+8a+16}{a^2-2a+1} \cdot \dfrac{7a-7}{a^2-16} = \dfrac{(a+4)^2}{(a-1)^2} \cdot \dfrac{7(a-1)}{(a-4)(a+4)} = \dfrac{7(a+4)}{(a-1)(a-4)} a 2 − 2 a + 1 a 2 + 8 a + 16 ⋅ a 2 − 16 7 a − 7 = ( a − 1 ) 2 ( a + 4 ) 2 ⋅ ( a − 4 ) ( a + 4 ) 7 ( a − 1 ) = ( a − 1 ) ( a − 4 ) 7 ( a + 4 ) y 3 − 8 9 − y 2 ⋅ y 2 − 6 y + 9 y 2 + 2 y + 4 = ( y − 2 ) ( y 2 + 2 y + 4 ) − ( y − 3 ) ( y + 3 ) ⋅ ( y − 3 ) 2 y 2 + 2 y + 4 = − ( y − 2 ) ( y − 3 ) y + 3 \dfrac{y^3-8}{9-y^2} \cdot \dfrac{y^2-6y+9}{y^2+2y+4} = \dfrac{(y-2)(y^2+2y+4)}{-(y-3)(y+3)} \cdot \dfrac{(y-3)^2}{y^2+2y+4} = \dfrac{-(y-2)(y-3)}{y+3} 9 − y 2 y 3 − 8 ⋅ y 2 + 2 y + 4 y 2 − 6 y + 9 = − ( y − 3 ) ( y + 3 ) ( y − 2 ) ( y 2 + 2 y + 4 ) ⋅ y 2 + 2 y + 4 ( y − 3 ) 2 = y + 3 − ( y − 2 ) ( y − 3 )
5.21. Перетворіть на дріб:
( 4 a + 20 b ) ⋅ 5 a 2 − 25 b 2 = 4 ( a + 5 b ) ⋅ 5 ( a − 5 b ) ( a + 5 b ) = 20 a − 5 b (4a+20b) \cdot \dfrac{5}{a^2-25b^2} = 4(a+5b) \cdot \dfrac{5}{(a-5b)(a+5b)} = \dfrac{20}{a-5b} ( 4 a + 20 b ) ⋅ a 2 − 25 b 2 5 = 4 ( a + 5 b ) ⋅ ( a − 5 b ) ( a + 5 b ) 5 = a − 5 b 20 ( m 2 − 4 ) ⋅ 2 m ( m − 2 ) 2 = ( m − 2 ) ( m + 2 ) ⋅ 2 m ( m − 2 ) 2 = 2 m ( m + 2 ) m − 2 (m^2-4) \cdot \dfrac{2m}{(m-2)^2} = (m-2)(m+2) \cdot \dfrac{2m}{(m-2)^2} = \dfrac{2m(m+2)}{m-2} ( m 2 − 4 ) ⋅ ( m − 2 ) 2 2 m = ( m − 2 ) ( m + 2 ) ⋅ ( m − 2 ) 2 2 m = m − 2 2 m ( m + 2 ) − a 2 a 2 − 18 ⋅ ( a 2 − 6 a + 9 ) = − a 2 ( a − 3 ) ( a + 3 ) ⋅ ( a − 3 ) 2 = − a ( a − 3 ) 2 ( a + 3 ) -\dfrac{a}{2a^2-18} \cdot (a^2-6a+9) = -\dfrac{a}{2(a-3)(a+3)} \cdot (a-3)^2 = -\dfrac{a(a-3)}{2(a+3)} − 2 a 2 − 18 a ⋅ ( a 2 − 6 a + 9 ) = − 2 ( a − 3 ) ( a + 3 ) a ⋅ ( a − 3 ) 2 = − 2 ( a + 3 ) a ( a − 3 ) ( x 3 + 27 y 3 ) ⋅ 5 3 x 2 − 9 x y + 27 y 2 = ( x + 3 y ) ( x 2 − 3 x y + 9 y 2 ) ⋅ 5 3 ( x 2 − 3 x y + 9 y 2 ) = 5 ( x + 3 y ) 3 (x^3+27y^3) \cdot \dfrac{5}{3x^2-9xy+27y^2} = (x+3y)(x^2-3xy+9y^2) \cdot \dfrac{5}{3(x^2-3xy+9y^2)} = \dfrac{5(x+3y)}{3} ( x 3 + 27 y 3 ) ⋅ 3 x 2 − 9 x y + 27 y 2 5 = ( x + 3 y ) ( x 2 − 3 x y + 9 y 2 ) ⋅ 3 ( x 2 − 3 x y + 9 y 2 ) 5 = 3 5 ( x + 3 y )
5.22. Перетворіть на дріб:
4 x 2 − 9 y 2 ⋅ ( 6 x + 18 y ) = 4 ( x − 3 y ) ( x + 3 y ) ⋅ 6 ( x + 3 y ) = 24 x − 3 y \dfrac{4}{x^2-9y^2} \cdot (6x+18y) = \dfrac{4}{(x-3y)(x+3y)} \cdot 6(x+3y) = \dfrac{24}{x-3y} x 2 − 9 y 2 4 ⋅ ( 6 x + 18 y ) = ( x − 3 y ) ( x + 3 y ) 4 ⋅ 6 ( x + 3 y ) = x − 3 y 24 ( c 2 + 4 c + 4 ) ⋅ ( − c 3 c 2 − 12 ) = ( c + 2 ) 2 ⋅ ( − c 3 ( c 2 − 4 ) ) = ( c + 2 ) 2 ⋅ ( − c 3 ( c − 2 ) ( c + 2 ) ) = − c ( c + 2 ) 3 ( c − 2 ) (c^2+4c+4) \cdot (-\dfrac{c}{3c^2-12}) = (c+2)^2 \cdot (-\dfrac{c}{3(c^2-4)}) = (c+2)^2 \cdot (-\dfrac{c}{3(c-2)(c+2)}) = -\dfrac{c(c+2)}{3(c-2)} ( c 2 + 4 c + 4 ) ⋅ ( − 3 c 2 − 12 c ) = ( c + 2 ) 2 ⋅ ( − 3 ( c 2 − 4 ) c ) = ( c + 2 ) 2 ⋅ ( − 3 ( c − 2 ) ( c + 2 ) c ) = − 3 ( c − 2 ) c ( c + 2 )
5.23. Виконайте дії:
( 25 x 2 8 y 3 ) 3 ⋅ ( − 16 y 5 125 x 3 ) 2 = 25 3 ( x 2 ) 3 8 3 ( y 3 ) 3 ⋅ 16 2 ( y 5 ) 2 125 2 ( x 3 ) 2 = 15625 x 6 512 y 9 ⋅ 256 y 10 15625 x 6 = y 2 (\dfrac{25x^2}{8y^3})^3 \cdot (-\dfrac{16y^5}{125x^3})^2 = \dfrac{25^3(x^2)^3}{8^3(y^3)^3} \cdot \dfrac{16^2(y^5)^2}{125^2(x^3)^2} = \dfrac{15625x^6}{512y^9} \cdot \dfrac{256y^{10}}{15625x^6} = \dfrac{y}{2} ( 8 y 3 25 x 2 ) 3 ⋅ ( − 125 x 3 16 y 5 ) 2 = 8 3 ( y 3 ) 3 2 5 3 ( x 2 ) 3 ⋅ 12 5 2 ( x 3 ) 2 1 6 2 ( y 5 ) 2 = 512 y 9 15625 x 6 ⋅ 15625 x 6 256 y 10 = 2 y x 2 − 2 x y + y 2 x 2 + 2 x y + y 2 ⋅ ( x + y x − y ) 3 = ( x − y ) 2 ( x + y ) 2 ⋅ ( x + y ) 3 ( x − y ) 3 = x + y x − y \dfrac{x^2-2xy+y^2}{x^2+2xy+y^2} \cdot (\dfrac{x+y}{x-y})^3 = \dfrac{(x-y)^2}{(x+y)^2} \cdot \dfrac{(x+y)^3}{(x-y)^3} = \dfrac{x+y}{x-y} x 2 + 2 x y + y 2 x 2 − 2 x y + y 2 ⋅ ( x − y x + y ) 3 = ( x + y ) 2 ( x − y ) 2 ⋅ ( x − y ) 3 ( x + y ) 3 = x − y x + y
5.24. Виконайте дії:
( − 16 m 3 27 n 5 ) 2 ⋅ ( 9 n 4 8 m 2 ) 3 = 16 2 ( m 3 ) 2 27 2 ( n 5 ) 2 ⋅ 9 3 ( n 4 ) 3 8 3 ( m 2 ) 3 = 256 m 6 729 n 10 ⋅ 729 n 12 512 m 6 = n 2 2 (-\dfrac{16m^3}{27n^5})^2 \cdot (\dfrac{9n^4}{8m^2})^3 = \dfrac{16^2(m^3)^2}{27^2(n^5)^2} \cdot \dfrac{9^3(n^4)^3}{8^3(m^2)^3} = \dfrac{256m^6}{729n^{10}} \cdot \dfrac{729n^{12}}{512m^6} = \dfrac{n^2}{2} ( − 27 n 5 16 m 3 ) 2 ⋅ ( 8 m 2 9 n 4 ) 3 = 2 7 2 ( n 5 ) 2 1 6 2 ( m 3 ) 2 ⋅ 8 3 ( m 2 ) 3 9 3 ( n 4 ) 3 = 729 n 10 256 m 6 ⋅ 512 m 6 729 n 12 = 2 n 2 ( m − n m + n ) 3 ⋅ m 2 + 2 m n + n 2 m 2 − 2 m n + n 2 = ( m − n ) 3 ( m + n ) 3 ⋅ ( m + n ) 2 ( m − n ) 2 = m − n m + n (\dfrac{m-n}{m+n})^3 \cdot \dfrac{m^2+2mn+n^2}{m^2-2mn+n^2} = \dfrac{(m-n)^3}{(m+n)^3} \cdot \dfrac{(m+n)^2}{(m-n)^2} = \dfrac{m-n}{m+n} ( m + n m − n ) 3 ⋅ m 2 − 2 mn + n 2 m 2 + 2 mn + n 2 = ( m + n ) 3 ( m − n ) 3 ⋅ ( m − n ) 2 ( m + n ) 2 = m + n m − n
5.25. Знайдіть значення виразу:
6 a b − b 5 a + b ⋅ 25 a 2 − b 2 6 a − 1 = b ( 6 a − 1 ) 5 a + b ⋅ ( 5 a − b ) ( 5 a + b ) 6 a − 1 = b ( 5 a − b ) \dfrac{6ab-b}{5a+b} \cdot \dfrac{25a^2-b^2}{6a-1} = \dfrac{b(6a-1)}{5a+b} \cdot \dfrac{(5a-b)(5a+b)}{6a-1} = b(5a-b) 5 a + b 6 ab − b ⋅ 6 a − 1 25 a 2 − b 2 = 5 a + b b ( 6 a − 1 ) ⋅ 6 a − 1 ( 5 a − b ) ( 5 a + b ) = b ( 5 a − b )
Якщо a = 1.2 , b = 6 a = 1.2, b = 6 a = 1.2 , b = 6
6 ⋅ ( 5 ⋅ 1.2 − 6 ) = 6 ⋅ ( 6 − 6 ) = 0 6 \cdot (5 \cdot 1.2 - 6) = 6 \cdot (6 - 6) = 0 6 ⋅ ( 5 ⋅ 1.2 − 6 ) = 6 ⋅ ( 6 − 6 ) = 0
a 3 + 8 a 2 − 1 ⋅ a 2 + a a 2 − 2 a + 4 = ( a + 2 ) ( a 2 − 2 a + 4 ) ( a − 1 ) ( a + 1 ) ⋅ a ( a + 1 ) a 2 − 2 a + 4 = a ( a + 2 ) a − 1 \dfrac{a^3+8}{a^2-1} \cdot \dfrac{a^2+a}{a^2-2a+4} = \dfrac{(a+2)(a^2-2a+4)}{(a-1)(a+1)} \cdot \dfrac{a(a+1)}{a^2-2a+4} = \dfrac{a(a+2)}{a-1} a 2 − 1 a 3 + 8 ⋅ a 2 − 2 a + 4 a 2 + a = ( a − 1 ) ( a + 1 ) ( a + 2 ) ( a 2 − 2 a + 4 ) ⋅ a 2 − 2 a + 4 a ( a + 1 ) = a − 1 a ( a + 2 )
Якщо a = 6 a = 6 a = 6
6 ( 6 + 2 ) 6 − 1 = 6 ⋅ 8 5 = 48 5 = 9.6 \dfrac{6(6+2)}{6-1} = \dfrac{6 \cdot 8}{5} = \dfrac{48}{5} = 9.6 6 − 1 6 ( 6 + 2 ) = 5 6 ⋅ 8 = 5 48 = 9.6
5.26. Виконайте множення:
x 2 + a x − c x − c a x 2 − a x + c x − a c ⋅ x 2 + a c + x c + x a x 2 + a c − x c − x a = ( x + a ) ( x − c ) ( x − a ) ( x + c ) ⋅ ( x + a ) ( x + c ) ( x − a ) ( x − c ) = ( x + a ) 2 ( x − a ) 2 \dfrac{x^2+ax-cx-ca}{x^2-ax+cx-ac} \cdot \dfrac{x^2+ac+xc+xa}{x^2+ac-xc-xa} = \dfrac{(x+a)(x-c)}{(x-a)(x+c)} \cdot \dfrac{(x+a)(x+c)}{(x-a)(x-c)} = \dfrac{(x+a)^2}{(x-a)^2} x 2 − a x + c x − a c x 2 + a x − c x − c a ⋅ x 2 + a c − x c − x a x 2 + a c + x c + x a = ( x − a ) ( x + c ) ( x + a ) ( x − c ) ⋅ ( x − a ) ( x − c ) ( x + a ) ( x + c ) = ( x − a ) 2 ( x + a ) 2 5 a − 5 b 3 c + 3 y ⋅ c 2 − y 2 − c − y a 2 − b 2 + a − b = 5 ( a − b ) 3 ( c + y ) ⋅ ( c + y ) ( c − y − 1 ) ( a − b ) ( a + b + 1 ) = 5 ( c − y − 1 ) 3 ( a + b + 1 ) \dfrac{5a-5b}{3c+3y} \cdot \dfrac{c^2-y^2-c-y}{a^2-b^2+a-b} = \dfrac{5(a-b)}{3(c+y)} \cdot \dfrac{(c+y)(c-y-1)}{(a-b)(a+b+1)} = \dfrac{5(c-y-1)}{3(a+b+1)} 3 c + 3 y 5 a − 5 b ⋅ a 2 − b 2 + a − b c 2 − y 2 − c − y = 3 ( c + y ) 5 ( a − b ) ⋅ ( a − b ) ( a + b + 1 ) ( c + y ) ( c − y − 1 ) = 3 ( a + b + 1 ) 5 ( c − y − 1 )
5.27. Обчисліть:
a 2 − b 2 + a + b a 2 − b 2 + a − b ⋅ 4 a − 4 b 8 a + 8 b = ( a + b ) ( a − b ) + ( a + b ) ( a − b ) ( a + b ) + ( a − b ) ⋅ 4 ( a − b ) 8 ( a + b ) = ( a + b ) ( a − b + 1 ) ( a − b ) ( a + b + 1 ) ⋅ a − b 2 ( a + b ) = a − b + 1 2 ( a + b + 1 ) \dfrac{a^2-b^2+a+b}{a^2-b^2+a-b} \cdot \dfrac{4a-4b}{8a+8b} = \dfrac{(a+b)(a-b)+(a+b)}{(a-b)(a+b)+(a-b)} \cdot \dfrac{4(a-b)}{8(a+b)} = \dfrac{(a+b)(a-b+1)}{(a-b)(a+b+1)} \cdot \dfrac{a-b}{2(a+b)} = \dfrac{a-b+1}{2(a+b+1)} a 2 − b 2 + a − b a 2 − b 2 + a + b ⋅ 8 a + 8 b 4 a − 4 b = ( a − b ) ( a + b ) + ( a − b ) ( a + b ) ( a − b ) + ( a + b ) ⋅ 8 ( a + b ) 4 ( a − b ) = ( a − b ) ( a + b + 1 ) ( a + b ) ( a − b + 1 ) ⋅ 2 ( a + b ) a − b = 2 ( a + b + 1 ) a − b + 1
Якщо a = 100 , b = 101 a = 100, b = 101 a = 100 , b = 101
100 − 101 + 1 2 ( 100 + 101 + 1 ) = 0 2 ( 202 ) = 0 \dfrac{100-101+1}{2(100+101+1)} = \dfrac{0}{2(202)} = 0 2 ( 100 + 101 + 1 ) 100 − 101 + 1 = 2 ( 202 ) 0 = 0
Вправи для повторення
5.28. Розв’яжіть систему рівнянь:
\cases 1 8 ( x + y ) = 3 \cr 1 3 ( x − y ) = 5 \cases{\dfrac{1}{8}(x+y) = 3 \cr \dfrac{1}{3}(x-y) = 5 } \cases 8 1 ( x + y ) = 3 \cr 3 1 ( x − y ) = 5
{ x + y = 24 x − y = 15 \begin{cases} x+y = 24 \cr x-y = 15 \end{cases} { x + y = 24 x − y = 15
( x + y ) + ( x − y ) = 24 + 15 (x+y)+(x-y) = 24+15 ( x + y ) + ( x − y ) = 24 + 15
2 x = 39 2x = 39 2 x = 39
x = 19.5 x = 19.5 x = 19.5
19.5 + y = 24 19.5+y=24 19.5 + y = 24
y = 4.5 y = 4.5 y = 4.5
Відповідь: ( 19.5 ; 4.5 ) (19.5; 4.5) ( 19.5 ; 4.5 )
{ x − 1 3 + y − 1 2 = 2 x − 1 2 − y − 1 12 = 4 3 \begin{cases} \dfrac{x-1}{3} + \dfrac{y-1}{2} = 2 \cr \dfrac{x-1}{2} - \dfrac{y-1}{12} = \dfrac{4}{3} \end{cases} ⎩ ⎨ ⎧ 3 x − 1 + 2 y − 1 = 2 2 x − 1 − 12 y − 1 = 3 4
Нехай a = x − 1 a = x-1 a = x − 1 b = y − 1 b = y-1 b = y − 1
{ a 3 + b 2 = 2 a 2 − b 12 = 4 3 \begin{cases} \dfrac{a}{3} + \dfrac{b}{2} = 2 \cr \dfrac{a}{2} - \dfrac{b}{12} = \dfrac{4}{3} \end{cases} ⎩ ⎨ ⎧ 3 a + 2 b = 2 2 a − 12 b = 3 4
{ 2 a + 3 b = 12 6 a − b = 16 \begin{cases} 2a+3b = 12 \cr 6a-b = 16 \end{cases} { 2 a + 3 b = 12 6 a − b = 16
{ 2 a + 3 b = 12 b = 6 a − 16 \begin{cases} 2a+3b = 12 \cr b = 6a-16 \end{cases} { 2 a + 3 b = 12 b = 6 a − 16
2 a + 3 ( 6 a − 16 ) = 12 2a+3(6a-16)=12 2 a + 3 ( 6 a − 16 ) = 12
2 a + 18 a − 48 = 12 2a+18a-48=12 2 a + 18 a − 48 = 12
20 a = 60 20a=60 20 a = 60
a = 3 a=3 a = 3
b = 6 ( 3 ) − 16 = 2 b=6(3)-16=2 b = 6 ( 3 ) − 16 = 2
x − 1 = 3 ⇒ x = 4 x-1=3 \Rightarrow x=4 x − 1 = 3 ⇒ x = 4
y − 1 = 2 ⇒ y = 3 y-1=2 \Rightarrow y=3 y − 1 = 2 ⇒ y = 3
Відповідь: ( 4 ; 3 ) (4; 3) ( 4 ; 3 )
5.29. Побудуйте графік функції:
y = x 3 − 8 x − 2 − x 2 y = \dfrac{x^3-8}{x-2}-x^2 y = x − 2 x 3 − 8 − x 2
Спростимо вираз:
y = ( x − 2 ) ( x 2 + 2 x + 4 ) x − 2 − x 2 y = \dfrac{(x-2)(x^2+2x+4)}{x-2}-x^2 y = x − 2 ( x − 2 ) ( x 2 + 2 x + 4 ) − x 2
Область визначення: x − 2 ≠ 0 ⇒ x ≠ 2 x-2 \neq 0 \Rightarrow x \neq 2 x − 2 = 0 ⇒ x = 2
y = x 2 + 2 x + 4 − x 2 y = x^2+2x+4-x^2 y = x 2 + 2 x + 4 − x 2
y = 2 x + 4 y = 2x+4 y = 2 x + 4
Графіком є пряма y = 2 x + 4 y = 2x+4 y = 2 x + 4 x = 2 x=2 x = 2
Знайдемо координати цієї точки: y = 2 ( 2 ) + 4 = 8 y = 2(2)+4 = 8 y = 2 ( 2 ) + 4 = 8
Отже, точка ( 2 ; 8 ) (2; 8) ( 2 ; 8 )
Пiдготуйтеся до вивчення нового матерiалу
5.30. Знайдіть число, взаємно обернене із числом:
4 = 4 1 ⇒ 1 4 4 = \dfrac{4}{1} \Rightarrow \dfrac{1}{4} 4 = 1 4 ⇒ 4 1 − 7 = − 7 1 ⇒ − 1 7 -7 = -\dfrac{7}{1} \Rightarrow -\dfrac{1}{7} − 7 = − 1 7 ⇒ − 7 1 1 3 ⇒ 3 \dfrac{1}{3} \Rightarrow 3 3 1 ⇒ 3 − 2 5 ⇒ − 5 2 -\dfrac{2}{5} \Rightarrow -\dfrac{5}{2} − 5 2 ⇒ − 2 5 0 , 16 = 16 100 = 4 25 ⇒ 25 4 = 6.25 0,16 = \dfrac{16}{100} = \dfrac{4}{25} \Rightarrow \dfrac{25}{4} = 6.25 0 , 16 = 100 16 = 25 4 ⇒ 4 25 = 6.25 1 , 2 = 12 10 = 6 5 ⇒ 5 6 1,2 = \dfrac{12}{10} = \dfrac{6}{5} \Rightarrow \dfrac{5}{6} 1 , 2 = 10 12 = 5 6 ⇒ 6 5
5.31. Обчисліть:
26 45 : 91 135 = 26 45 ⋅ 135 91 = 2 ⋅ 13 ⋅ 3 ⋅ 45 45 ⋅ 7 ⋅ 13 = 6 7 \dfrac{26}{45} : \dfrac{91}{135} = \dfrac{26}{45} \cdot \dfrac{135}{91} = \dfrac{2 \cdot 13 \cdot 3 \cdot 45}{45 \cdot 7 \cdot 13} = \dfrac{6}{7} 45 26 : 135 91 = 45 26 ⋅ 91 135 = 45 ⋅ 7 ⋅ 13 2 ⋅ 13 ⋅ 3 ⋅ 45 = 7 6 2 1 2 : 15 16 = 5 2 : 15 16 = 5 2 ⋅ 16 15 = 5 ⋅ 16 2 ⋅ 15 = 8 3 2\dfrac{1}{2} : \dfrac{15}{16} = \dfrac{5}{2} : \dfrac{15}{16} = \dfrac{5}{2} \cdot \dfrac{16}{15} = \dfrac{5 \cdot 16}{2 \cdot 15} = \dfrac{8}{3} 2 2 1 : 16 15 = 2 5 : 16 15 = 2 5 ⋅ 15 16 = 2 ⋅ 15 5 ⋅ 16 = 3 8 − 3 1 7 : 2 5 14 = − 22 7 : 33 14 = − 22 7 ⋅ 14 33 = − 2 ⋅ 2 3 = − 4 3 -3\dfrac{1}{7} : 2\dfrac{5}{14} = -\dfrac{22}{7} : \dfrac{33}{14} = -\dfrac{22}{7} \cdot \dfrac{14}{33} = -\dfrac{2 \cdot 2}{3} = -\dfrac{4}{3} − 3 7 1 : 2 14 5 = − 7 22 : 14 33 = − 7 22 ⋅ 33 14 = − 3 2 ⋅ 2 = − 3 4 − 5 13 15 : ( − 1 8 25 ) = − 88 15 : ( − 33 25 ) = 88 15 ⋅ 25 33 = 8 ⋅ 5 3 ⋅ 3 = 40 9 -5\dfrac{13}{15} : (-1\dfrac{8}{25}) = -\dfrac{88}{15} : (-\dfrac{33}{25}) = \dfrac{88}{15} \cdot \dfrac{25}{33} = \dfrac{8 \cdot 5}{3 \cdot 3} = \dfrac{40}{9} − 5 15 13 : ( − 1 25 8 ) = − 15 88 : ( − 25 33 ) = 15 88 ⋅ 33 25 = 3 ⋅ 3 8 ⋅ 5 = 9 40
Життєва математика
5.32. Родина витрачає 13% своїх доходів на оплату комірного, 45% – на продукти харчування, 17% – на побутові товари і послуги, а решту на відпочинок. Який річний бюджет родини, якщо на відпочинок вона витрачає 120 000 грн на рік?
Знайдемо, який відсоток доходів родина витрачає на відпочинок:
100% – (13% + 45% + 17%) = 100% – 75% = 25%
Знайдемо річний бюджет родини. Нехай x x x
0.25 x = 120000 0.25x = 120000 0.25 x = 120000
x = 120000 0.25 x = \dfrac{120000}{0.25} x = 0.25 120000
x = 480000 x = 480000 x = 480000
Відповідь: річний бюджет родини становить 480 000 грн.
Цікаві задачі – поміркуй одначе
5.33. На моніторі комп’ютера – число 2500. Щохвилини комп’ютерна програма множить або ділить це число на 2 або на 5, одержуючи при цьому натуральне число. Чи може на моніторі рівно через годину з’явитися число: 1) 10 000; 2) 20 000?
Розкладемо початкове число на прості множники:
2500 = 25 ⋅ 100 = 5 2 ⋅ 10 2 = 5 2 ⋅ ( 2 ⋅ 5 ) 2 = 2 2 ⋅ 5 4 2500 = 25 \cdot 100 = 5^2 \cdot 10^2 = 5^2 \cdot (2 \cdot 5)^2 = 2^2 \cdot 5^4 2500 = 25 ⋅ 100 = 5 2 ⋅ 1 0 2 = 5 2 ⋅ ( 2 ⋅ 5 ) 2 = 2 2 ⋅ 5 4
Сума показників степенів простих множників дорівнює 2 + 4 = 6 2+4 = 6 2 + 4 = 6
Кожна операція (множення або ділення на 2 або на 5) змінює суму показників степенів на 1 (або + 1 +1 + 1 − 1 -1 − 1
Початкова сума показників (6) є парною.
Після 1 хвилини сума стане непарною.
Після 2 хвилин – знову парною.
Через годину, тобто через 60 хвилин (парне число операцій), сума показників степенів у розкладі нового числа також має бути парною.
Перевіримо числа-кандидати:
1) 10 000
10000 = 10 4 = ( 2 ⋅ 5 ) 4 = 2 4 ⋅ 5 4 10000 = 10^4 = (2 \cdot 5)^4 = 2^4 \cdot 5^4 10000 = 1 0 4 = ( 2 ⋅ 5 ) 4 = 2 4 ⋅ 5 4
Сума показників степенів: 4 + 4 = 8 4+4 = 8 4 + 4 = 8
Оскільки парність суми показників (8) збігається з парністю початкової суми (6) після парної кількості кроків, то число 10 000 може з’явитися на моніторі.
2) 20 000
20000 = 2 ⋅ 10000 = 2 ⋅ ( 2 4 ⋅ 5 4 ) = 2 5 ⋅ 5 4 20000 = 2 \cdot 10000 = 2 \cdot (2^4 \cdot 5^4) = 2^5 \cdot 5^4 20000 = 2 ⋅ 10000 = 2 ⋅ ( 2 4 ⋅ 5 4 ) = 2 5 ⋅ 5 4
Сума показників степенів: 5 + 4 = 9 5+4 = 9 5 + 4 = 9
Оскільки парність суми показників (9) не збігається з необхідною парністю, число 20 000 не може з’явитися на моніторі.
Відповідь: 1) так, може; 2) ні, не може.
