1.43. Разложите на множители числитель и знаменатель дроби и сократите дробь:

Решение:

1. \(\frac{4a-12b}{5a-15b} = \frac{4(a-3b)}{5(a-3b)} = \frac{4}{5}\)
2. \(\frac{2x-10y}{x^2-5xy} = \frac{2(x-5y)}{x(x-5y)} = \frac{2}{x}\)
3. \(\frac{x^2+xy}{y^2+xy} = \frac{x(x+y)}{y(y+x)} = \frac{x}{y}\)
4. \(\frac{m^2-16}{2m+8} = \frac{(m-4)(m+4)}{2(m+4)} = \frac{m-4}{2}\)
5. \(\frac{x^2+4x+4}{2x+4} = \frac{(x+2)^2}{2(x+2)} = \frac{x+2}{2}\)
6. \(\frac{m^2-6mn}{m^2-12mn+36m^2} = \frac{m(m-6n)}{m^2(1-12n+36n^2)} = \frac{m-6n}{m(1-6n)^2}\)
7. \(\frac{c^2-3c}{cd-3d} = \frac{c(c-3)}{d(c-3)} = \frac{c}{d}\)
8. \(\frac{a^2-5a}{a^2-25} = \frac{a(a-5)}{(a-5)(a+5)} = \frac{a}{a+5}\)
9. \(\frac{25c^2-1}{25c^2+10c+1} = \frac{(5c-1)(5c+1)}{(5c+1)^2} = \frac{5c-1}{5c+1}\)
10. \(\frac{b^2-6b+9}{b^2-9} = \frac{(b-3)^2}{(b-3)(b+3)} = \frac{b-3}{b+3}\)
11. \(\frac{x^2+8xy+16y^2}{x^2-16y^2} = \frac{(x+4y)^2}{(x-4y)(x+4y)} = \frac{x+4y}{x-4y}\)
12. \(\frac{12a^2-3}{4a^2-4a+1} = \frac{3(4a^2-1)}{(2a-1)^2} = \frac{3(2a-1)(2a+1)}{(2a-1)^2} = \frac{3(2a+1)}{2a-1}\)

Ответ: \(\frac{4}{5}\), \(\frac{2}{x}\), \(\frac{x}{y}\), \(\frac{m-4}{2}\), \(\frac{x+2}{2}\), \(\frac{m-6n}{m(1-6n)^2}\), \(\frac{c}{d}\), \(\frac{a}{a+5}\), \(\frac{5c-1}{5c+1}\), \(\frac{b-3}{b+3}\), \(\frac{x+4y}{x-4y}\), \(\frac{3(2a+1)}{2a-1}\).