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Математика1. Выполните умножение или деление:
Ответ:
1. Выполните умножение или деление:
a) \( \frac{a}{3b} \cdot \frac{5b}{7a} = \frac{a \cdot 5b}{3b \cdot 7a} = \frac{5ab}{21ab} = \frac{5}{21} \)
б) \( \frac{x^7}{6y^{10}} : \frac{3y^9}{x^{11}} = \frac{x^7}{6y^{10}} \cdot \frac{x^{11}}{3y^9} = \frac{x^{7+11}}{6 3 y^{10+9}} = \frac{x^{18}}{18y^{19}} \)
в) \( \frac{3a^2b}{c} \cdot \frac{3c}{a^2b^2} = \frac{3a^2b \cdot 3c}{c \cdot a^2b^2} = \frac{9a^2bc}{a^2b^2c} = \frac{9}{b} \)
a) \( \frac{2m-n}{3p} : \frac{3}{2m-n} = \frac{2m-n}{3p} \cdot \frac{2m-n}{3} = \frac{(2m-n)^2}{9p} \)
б) \( \frac{m-n}{p+q} \cdot \frac{2p+2q}{3m-3n} = \frac{m-n}{p+q} \cdot \frac{2(p+q)}{3(m-n)} = \frac{2}{3} \)
г) \( \frac{mn-n^2}{pq+q^2} : \frac{3q+3p}{n^2-mn} = \frac{n(m-n)}{q(p+q)} \cdot \frac{n(n-m)}{3(q+p)} = \frac{n(m-n)}{q(p+q)} \cdot \frac{-n(m-n)}{3(p+q)} = \frac{-n^2(m-n)^2}{3q(p+q)^2} \)
a) \( \frac{5a}{3c} \cdot \frac{10a}{6c} = \frac{5a \cdot 10a}{3c \cdot 6c} = \frac{50a^2}{18c^2} = \frac{25a^2}{9c^2} \)
б) \( \frac{3a^{11}}{5b^{15}} \cdot \frac{21a^{10}}{10b^{14}} = \frac{3a^{11} \cdot 21a^{10}}{5b^{15} \cdot 10b^{14}} = \frac{63a^{21}}{50b^{29}} \)
a) \( \frac{5x+3y}{2c} \cdot \frac{3y+5x}{2a} = \frac{(5x+3y)^2}{4ac} \)
в) \( \frac{8a^2b}{c} \cdot \frac{a^2b}{8c} = \frac{8a^2b \cdot a^2b}{c \cdot 8c} = \frac{8a^4b^2}{8c^2} = \frac{a^4b^2}{c^2} \)
a) \( \frac{3a-5b}{2a} \cdot 3a = \frac{3a(3a-5b)}{2a} = \frac{3(3a-5b)}{2} = \frac{9a-15b}{2} \)
б) \( \frac{5b}{a^2-b^2} \cdot (a+b) = \frac{5b(a+b)}{(a-b)(a+b)} = \frac{5b}{a-b} \)
в) \( (3a-6b) \cdot \frac{a+b}{2a-4b} = 3(a-2b) \cdot \frac{a+b}{2(a-2b)} = \frac{3(a+b)}{2} \)
a) \( \frac{2a^2}{3a-b} : 5a = \frac{2a^2}{3a-b} \cdot \frac{1}{5a} = \frac{2a}{5(3a-b)} \)
б) \( 5a : \frac{3a^2}{3a+b} = 5a \cdot \frac{3a+b}{3a^2} = \frac{5(3a+b)}{3a} \)
в) \( \frac{a^2-b^2}{x+3y} : (a+b) = \frac{(a-b)(a+b)}{x+3y} \cdot \frac{1}{a+b} = \frac{a-b}{x+3y} \)
2. Упростите выражение:
a) \( \frac{a}{3b} \cdot \frac{b}{a^2} \cdot \frac{2a}{b^2} = \frac{a \cdot b \cdot 2a}{3b \cdot a^2 \cdot b^2} = \frac{2a^2b}{3a^2b^3} = \frac{2}{3b^2} \)
в) \( \frac{a^2}{3b} \cdot \left( \frac{b^2}{3a} \cdot \frac{b}{5a} \right) = \frac{a^2}{3b} \cdot \frac{b^3}{15a^2} = \frac{a^2b^3}{45a^2b} = \frac{b^2}{45} \)
б) \( \frac{a^2 b^2}{3b} \cdot \frac{b}{3a} \cdot \frac{b}{5a} = \frac{a^2 b^2 b b}{3b 3a 5a} = \frac{a^2 b^4}{45a^2} = \frac{b^4}{45} \)
г) \( \frac{a^2}{3b} \cdot \left( \frac{b^2}{3a} \cdot \frac{b}{5a} \right) = \frac{a^2}{3b} \cdot \frac{b^3}{15a^2} = \frac{a^2b^3}{45a^2b} = \frac{b^2}{45} \)
3. Упростите выражение:
a) \( \frac{a^2-9b^2}{c^2+8cd+16d^2} \cdot \frac{c^2-16d^2}{3b-a} = \frac{(a-3b)(a+3b)}{(c+4d)^2} \cdot \frac{(c-4d)(c+4d)}{-(a-3b)} = \frac{(a+3b)(c-4d)}{-(c+4d)} = \frac{(a+3b)(4d-c)}{c+4d} \)
б) \( \frac{a^2-b^2+a+b}{x^2-y^2+x-y} \cdot \frac{3a+3b}{2x-2y} = \frac{(a-b)(a+b)+(a+b)}{(x-y)(x+y)+(x-y)} \cdot \frac{3(a+b)}{2(x-y)} = \frac{(a+b)(a-b+1)}{(x-y)(x+y+1)} \cdot \frac{3(a+b)}{2(x-y)} = \frac{3(a+b)^2(a-b+1)}{2(x-y)^2(x+y+1)} \)
a) \( \frac{4a^2}{2a-b} \cdot \frac{12a^3}{4a^2-b^2} : \frac{2a^2}{6a^2-3ab} = \frac{4a^2}{2a-b} \cdot \frac{12a^3}{(2a-b)(2a+b)} \cdot \frac{3a(2a-b)}{2a^2} = \frac{4a^2 \cdot 12a^3 \cdot 3a(2a-b)}{(2a-b)(2a+b)(2a-b) \cdot 2a^2} = \frac{144a^6(2a-b)}{2a^2(2a-b)(2a+b)^2} = \frac{72a^4}{(2a+b)^2} \)
б) \( \frac{x^2-x}{2x+2} \cdot \frac{x^2+2x+1}{x^2+4x} \cdot \frac{3x-3}{x^2-16} = \frac{x(x-1)}{2(x+1)} \cdot \frac{(x+1)^2}{x(x+4)} \cdot \frac{3(x-1)}{(x-4)(x+4)} = \frac{x(x-1)(x+1)^2 3(x-1)}{2(x+1)x(x+4)(x-4)(x+4)} = \frac{3(x-1)^2(x+1)}{2(x+4)^2(x-4)} \)
