1. Выполните умножение:
1) а) $\frac{3a}{b} \cdot \frac{b}{6a} = \frac{3ab}{6ab} = \frac{1}{2}$
б) $\frac{2x}{3y} \cdot \frac{y}{6x} = \frac{2xy}{18xy} = \frac{1}{9}$
в) $\frac{a^2b}{12c} \cdot \frac{4c}{ab^2} = \frac{4a^2bc}{12abc^2} = \frac{a}{3c}$
г) $\frac{18m^3}{5n^2} \cdot \frac{25n}{12m^2} = \frac{18 \cdot 25 \cdot m^3n}{60n^2m^2} = \frac{3m}{2n} \cdot \frac{5}{1} = \frac{15m}{2n}$
д) $6x \cdot \frac{a}{3x^2} = \frac{6ax}{3x^2} = \frac{2a}{x}$
е) $\frac{7y^3}{z^2} \cdot \frac{z^3}{y^2} = \frac{7y^3z^3}{z^2y^2} = 7yz$
2) а) $\frac{3}{x^2-2x} \cdot \frac{2x-4}{x} = \frac{3}{x(x-2)} \cdot \frac{2(x-2)}{x} = \frac{6}{x^2}$
б) $\frac{a-2b}{12c} \cdot \frac{18c^2}{2b-a} = \frac{a-2b}{12c} \cdot \frac{18c^2}{-(a-2b)} = -\frac{18c^2}{12c} = -1.5c$
в) $\frac{x^2-16}{8x^8} \cdot \frac{4x}{x+4} = \frac{(x-4)(x+4)}{8x^8} \cdot \frac{4x}{x+4} = \frac{4x(x-4)}{8x^8} = \frac{x-4}{2x^7}$
г) $\frac{5-y}{2y} \cdot \frac{3y^2}{y^2-25} = \frac{-(y-5)}{2y} \cdot \frac{3y^2}{(y-5)(y+5)} = -\frac{3y}{2(y+5)}$
д) $\frac{c^2+4c+4}{2c-6} \cdot \frac{c^2-9}{5c+10} = \frac{(c+2)^2}{2(c-3)} \cdot \frac{(c-3)(c+3)}{5(c+2)} = \frac{(c+2)(c+3)}{10} = 0.1(c^2+5c+6)$
е) $(3a-6b) \cdot \frac{a+2}{a^2-4b^2} = 3(a-2b) \cdot \frac{a+2}{(a-2b)(a+2b)} = \frac{3(a+2)}{a+2b}$
3) а) $\frac{x^2-6x+9}{x^2-3x+9} \cdot \frac{x^3+27}{3x-9} = \frac{(x-3)^2}{x^2-3x+9} \cdot \frac{(x+3)(x^2-3x+9)}{3(x-3)} = \frac{(x-3)(x+3)}{3} = \frac{x^2-9}{3}$
б) $\frac{1}{5a^2+10ab+20b^2} \cdot (a^3-8b^3) = \frac{a^3-8b^3}{5(a^2+2ab+4b^2)} = \frac{(a-2b)(a^2+2ab+4b^2)}{5(a^2+2ab+4b^2)} = \frac{a-2b}{5}$
в) $\frac{x^2-ax+bx-ab}{x^2+ax-bx-ab} \cdot \frac{x^2+2ax+a^2}{x^2-2bx+b^2} = \frac{x(x-a)+b(x-a)}{x(x+a)-b(x+a)} \cdot \frac{(x+a)^2}{(x-b)^2} = \frac{(x-a)(x+b)}{(x+a)(x-b)} \cdot \frac{(x+a)^2}{(x-b)^2} = \frac{(x-a)(x+a)(x+b)}{(x-b)^3}$
2. Представьте в виде дроби:
1) а) $(\frac{x}{y^2})^2 = \frac{x^2}{y^4}$
б) $(-\frac{2a^2}{c^2})^3 = -\frac{8a^6}{c^6}$
в) $(\frac{c^2}{b})^3 \cdot (\frac{b^2}{c^4})^2 = \frac{c^6}{b^3} \cdot \frac{b^4}{c^8} = \frac{b}{c^2}$
2) а) $(-\frac{16x^2}{27y^3})^2 \cdot (\frac{9y^2}{8x^2})^3 = \frac{256x^4}{729y^6} \cdot \frac{729y^6}{512x^6} = \frac{256}{512x^2} = \frac{1}{2x^2}$
б) $(\frac{a-b}{a+b})^3 \cdot \frac{a^2+2ab+b^2}{a^2-2ab+b^2} = \frac{(a-b)^3}{(a+b)^3} \cdot \frac{(a+b)^2}{(a-b)^2} = \frac{a-b}{a+b}$
3. Упростите выражение:
а) $\frac{45xy}{7z^2} \cdot \frac{14z^3}{81y^2} \cdot \frac{27y^3}{5xz} = \frac{45 \cdot 14 \cdot 27 \cdot xy \cdot z^3 \cdot y^3}{7 \cdot 81 \cdot 5 \cdot z^2 \cdot y^2 \cdot xz} = \frac{17010}{2835} \cdot \frac{xy^4z^3}{xy^2z^3} = 6y^2$
б) $\frac{74a^3b}{c^4} \cdot \frac{b^3}{111ab} \cdot \frac{1}{3ac^3} = \frac{74a^3b^4}{333a^2bc^7} = \frac{2ab^3}{9c^7}$
4. Докажите тождество:
$\frac{8-0.5y^4}{4+0.5y^3} \cdot \frac{0.5y^2-y+2}{0.5y^2+2} = \frac{0.5(16-y^4)}{0.5(8+y^3)} \cdot \frac{0.5y^2-y+2}{0.5(y^2+4)} = \frac{(4-y^2)(4+y^2)}{(2+y)(4-2y+y^2)} \cdot \frac{0.5y^2-y+2}{0.5(y^2+4)} \dots$
*Примечание: для доказательства нужно привести к общему виду.*
