1007 Упростите: a) 2 1 x+1 - - - x²-3x x²+3x x²-9 2y y+1 + 1 y+2 б) + - - y²+3y 3y-y² y' ; B) a²+16+12-22-30-22: a³-8 a²+2a+4 a-2 г) 2 + 46² + 18 1 - 462-66+9 863 +27 2b+3 ; 1008 Представьте в виде дроби: a) ab² - 16a 2065 ; 563 a²b+4a² 7xy 3x-6y; б) x²-4xy + 4y² 14y² 1009 Упростите: a) x²-4x24-6x. x²+7x 49-x² 6) y³-16y 4-y: 2y + 18 y² +9y 1010 Упростите выражение: a) (7(m-2)+2).2m² + 4m+8; m³-8 m²+2m+4 m-3 6) 05:(24429-2(+8)); a²-9a2-3a+9 a²+27 x+2 a x-14 3 x+2 1 B)(2-2-5 3x 1 г) +(

Ответ: Решения представлены ниже.

1007 Упростите:

a)

\[\frac{2}{x^2-3x} - \frac{1}{x^2+3x} - \frac{x+1}{x^2-9}\]

\[= \frac{2}{x(x-3)} - \frac{1}{x(x+3)} - \frac{x+1}{(x-3)(x+3)}\]

\[= \frac{2(x+3) - (x-3) - x(x+1)}{x(x-3)(x+3)}\]

\[= \frac{2x+6 - x+3 - x^2-x}{x(x-3)(x+3)}\]

\[= \frac{-x^2 + 9}{x(x-3)(x+3)}\]

\[= \frac{-(x^2 - 9)}{x(x-3)(x+3)}\]

\[= \frac{-(x-3)(x+3)}{x(x-3)(x+3)}\]

\[= -\frac{1}{x}\]

Ответ: \[-\frac{1}{x}\]

б)

\[\frac{2y+1}{y^2+3y} + \frac{y+2}{3y-y^2} - \frac{1}{y}\]

\[= \frac{2y+1}{y(y+3)} - \frac{y+2}{y(y-3)} - \frac{1}{y}\]

\[= \frac{(2y+1)(y-3) - (y+2)(y+3) - (y-3)(y+3)}{y(y+3)(y-3)}\]

\[= \frac{2y^2-6y+y-3 - (y^2+3y+2y+6) - (y^2-9)}{y(y+3)(y-3)}\]

\[= \frac{2y^2-5y-3 - y^2-5y-6 - y^2+9}{y(y+3)(y-3)}\]

\[= \frac{-10y-0}{y(y+3)(y-3)}\]

\[= \frac{-10}{y^2-9}\]

Ответ: \[\frac{-10}{y^2-9}\]

в)

\[\frac{a^2+16a+12}{a^3-8} - \frac{2-3a}{a^2+2a+4} - \frac{3}{a-2}\]

\[= \frac{a^2+16a+12}{(a-2)(a^2+2a+4)} - \frac{2-3a}{a^2+2a+4} - \frac{3}{a-2}\]

\[= \frac{a^2+16a+12 - (2-3a)(a-2) - 3(a^2+2a+4)}{(a-2)(a^2+2a+4)}\]

\[= \frac{a^2+16a+12 - (2a-4-3a^2+6a) - 3a^2-6a-12}{(a-2)(a^2+2a+4)}\]

\[= \frac{a^2+16a+12 - 2a+4+3a^2-6a - 3a^2-6a-12}{(a-2)(a^2+2a+4)}\]

\[= \frac{a^2+2a+4}{(a-2)(a^2+2a+4)}\]

\[= \frac{1}{a-2}\]

Ответ: \[\frac{1}{a-2}\]

г)

\[\frac{2}{4b^2-6b+9} + \frac{4b^2+18}{8b^3+27} - \frac{1}{2b+3}\]

\[= \frac{2}{(2b)^2-2(2b)( \frac{3}{2})+( \frac{3}{2})^2 + \frac{27}{4} } + \frac{4b^2+18}{(2b+3)(4b^2-6b+9)} - \frac{1}{2b+3}\]

\[= \frac{2}{4b^2-6b+9} + \frac{4b^2+18}{(2b+3)(4b^2-6b+9)} - \frac{1}{2b+3}\]

\[= \frac{2(2b+3) + 4b^2+18 - (4b^2-6b+9)}{(2b+3)(4b^2-6b+9)}\]

\[= \frac{4b+6 + 4b^2+18 - 4b^2+6b-9}{(2b+3)(4b^2-6b+9)}\]

\[= \frac{10b+15}{(2b+3)(4b^2-6b+9)}\]

\[= \frac{5(2b+3)}{(2b+3)(4b^2-6b+9)}\]

\[= \frac{5}{4b^2-6b+9}\]

Ответ: \[\frac{5}{4b^2-6b+9}\]

1008 Представьте в виде дроби:

a)

\[\frac{ab^2 - 16a}{5b^3} \cdot \frac{20b^5}{a^2b+4a^2}\]

\[= \frac{a(b^2 - 16)}{5b^3} \cdot \frac{20b^5}{a^2(b+4)}\]

\[= \frac{a(b - 4)(b + 4)}{5b^3} \cdot \frac{20b^5}{a^2(b+4)}\]

\[= \frac{a(b - 4)(b + 4) \cdot 20b^5}{5b^3 \cdot a^2(b+4)}\]

\[= \frac{4b^2(b - 4)}{a}\]

Ответ: \[\frac{4b^2(b - 4)}{a}\]

б)

\[\frac{7xy}{x^2-4xy+4y^2} \cdot \frac{3x-6y}{14y^2}\]

\[= \frac{7xy}{(x-2y)^2} \cdot \frac{3(x-2y)}{14y^2}\]

\[= \frac{7xy \cdot 3(x-2y)}{(x-2y)^2 \cdot 14y^2}\]

\[= \frac{3x}{2y(x-2y)}\]

Ответ: \[\frac{3x}{2y(x-2y)}\]

в)

\[\frac{p^3 - 125}{8p^2} \cdot \frac{4p}{p^2+5p+25}\]

\[= \frac{(p-5)(p^2+5p+25)}{8p^2} \cdot \frac{4p}{p^2+5p+25}\]

\[= \frac{(p-5)(p^2+5p+25) \cdot 4p}{8p^2 \cdot (p^2+5p+25)}\]

\[= \frac{p-5}{2p}\]

Ответ: \[\frac{p-5}{2p}\]

г)

\[\frac{9m^2 - 12mn+4n^2}{3m^3+24n^3} \cdot \frac{3m+6n}{2n-3m}\]

\[= \frac{(3m-2n)^2}{3(m^3+8n^3)} \cdot \frac{3(m+2n)}{2n-3m}\]

\[= \frac{(3m-2n)^2}{3(m+2n)(m^2-2mn+4n^2)} \cdot \frac{3(m+2n)}{2n-3m}\]

\[= \frac{(3m-2n)^2 \cdot 3(m+2n)}{3(m+2n)(m^2-2mn+4n^2) \cdot (2n-3m)}\]

\[= \frac{(3m-2n)}{-(m^2-2mn+4n^2)}\]

\[= \frac{-(3m-2n)}{(m^2-2mn+4n^2)}\]

\[= \frac{(2n-3m)}{(m^2-2mn+4n^2)}\]

Ответ: \[\frac{(2n-3m)}{(m^2-2mn+4n^2)}\]

1009 Упростите:

a)

\[\frac{x^2-4x}{x^2+7x} \cdot \frac{24-6x}{49-x^2}\]

\[= \frac{x(x-4)}{x(x+7)} \cdot \frac{-6(x-4)}{(7-x)(7+x)}\]

\[= \frac{x(x-4) \cdot -6(x-4)}{x(x+7) \cdot (7-x)(7+x)}\]

\[= \frac{-6(x-4)^2}{(x+7)(7-x)}\]

Ответ: \[\frac{-6(x-4)^2}{(x+7)(7-x)}\]

б)

\[\frac{y^3-16y}{2y+18} \cdot \frac{4-y}{y^2+9y}\]

\[= \frac{y(y^2-16)}{2(y+9)} \cdot \frac{-(y-4)}{y(y+9)}\]

\[= \frac{y(y-4)(y+4)}{2(y+9)} \cdot \frac{-(y-4)}{y(y+9)}\]

\[= \frac{y(y-4)(y+4) \cdot -(y-4)}{2(y+9) \cdot y(y+9)}\]

\[= \frac{-(y-4)^2(y+4)}{2(y+9)^2}\]

Ответ: \[\frac{-(y-4)^2(y+4)}{2(y+9)^2}\]

в)

\[\frac{(a+b)^2-2ab}{4a^2} : \frac{a^2 + b^2}{ab}\]

\[= \frac{a^2+2ab+b^2-2ab}{4a^2} : \frac{a^2 + b^2}{ab}\]

\[= \frac{a^2+b^2}{4a^2} \cdot \frac{ab}{a^2 + b^2}\]

\[= \frac{(a^2+b^2) \cdot ab}{4a^2 \cdot (a^2 + b^2)}\]

\[= \frac{b}{4a}\]

Ответ: \[\frac{b}{4a}\]

г)

\[\frac{5c^3-5}{c+2} \cdot \frac{(c + 1)^2 - c}{13c + 26}\]

\[= \frac{5(c^3-1)}{c+2} \cdot \frac{c^2+2c + 1 - c}{13(c + 2)}\]

\[= \frac{5(c-1)(c^2+c+1)}{c+2} \cdot \frac{c^2+c + 1}{13(c + 2)}\]

\[= \frac{5(c-1)(c^2+c+1) \cdot (c^2+c + 1)}{(c+2) \cdot 13(c + 2)}\]

\[= \frac{5(c-1)(c^2+c+1)^2}{13(c+2)^2}\]

Ответ: \[\frac{5(c-1)(c^2+c+1)^2}{13(c+2)^2}\]

1010 Упростите выражение:

a)

\[\left( \frac{7(m-2)}{m^3-8} - \frac{m+2}{m^2+2m+4} \right) \cdot \frac{2m^2 + 4m+8}{m-3}\]

\[= \left( \frac{7(m-2)}{(m-2)(m^2+2m+4)} - \frac{m+2}{m^2+2m+4} \right) \cdot \frac{2(m^2 + 2m+4)}{m-3}\]

\[= \left( \frac{7}{m^2+2m+4} - \frac{m+2}{m^2+2m+4} \right) \cdot \frac{2(m^2 + 2m+4)}{m-3}\]

\[= \left( \frac{7 - (m+2)}{m^2+2m+4} \right) \cdot \frac{2(m^2 + 2m+4)}{m-3}\]

\[= \left( \frac{5 - m}{m^2+2m+4} \right) \cdot \frac{2(m^2 + 2m+4)}{m-3}\]

\[= \frac{(5 - m) \cdot 2(m^2 + 2m+4)}{(m^2+2m+4) \cdot (m-3)}\]

\[= \frac{-2(m - 5)}{(m-3)}\]

Ответ: \[\frac{-2(m - 5)}{(m-3)}\]

б)

\[\left(\frac{a+5}{a^2-9} : \left(\frac{a+2}{a^2-3a+9} - \frac{2(a+8)}{a^3+27}\right)\right)\]

\[= \left(\frac{a+5}{(a-3)(a+3)} : \left(\frac{a+2}{a^2-3a+9} - \frac{2(a+8)}{(a+3)(a^2-3a+9)}\right)\right)\]

\[= \left(\frac{a+5}{(a-3)(a+3)} : \left(\frac{(a+2)(a+3) - 2(a+8)}{(a+3)(a^2-3a+9)}\right)\right)\]

\[= \left(\frac{a+5}{(a-3)(a+3)} : \left(\frac{a^2+5a+6 - 2a-16}{(a+3)(a^2-3a+9)}\right)\right)\]

\[= \left(\frac{a+5}{(a-3)(a+3)} : \left(\frac{a^2+3a-10}{(a+3)(a^2-3a+9)}\right)\right)\]

\[= \left(\frac{a+5}{(a-3)(a+3)} : \left(\frac{(a+5)(a-2)}{(a+3)(a^2-3a+9)}\right)\right)\]

\[= \frac{a+5}{(a-3)(a+3)} \cdot \frac{(a+3)(a^2-3a+9)}{(a+5)(a-2)}\]

\[= \frac{(a+5)(a+3)(a^2-3a+9)}{(a-3)(a+3)(a+5)(a-2)}\]

\[= \frac{a^2-3a+9}{(a-3)(a-2)}\]

Ответ: \[\frac{a^2-3a+9}{(a-3)(a-2)}\]

в)

\[\left(\frac{x+2}{3x} - \frac{2}{x-2} - \frac{x-14}{3x^2-6x}\right) : \frac{x+2}{6x} \cdot \frac{1}{x-5}\]

\[= \left(\frac{x+2}{3x} - \frac{2}{x-2} - \frac{x-14}{3x(x-2)}\right) : \frac{x+2}{6x} \cdot \frac{1}{x-5}\]

\[= \left(\frac{(x+2)(x-2) - 2 \cdot 3x - (x-14)}{3x(x-2)}\right) : \frac{x+2}{6x} \cdot \frac{1}{x-5}\]

\[= \left(\frac{x^2-4 - 6x - x+14}{3x(x-2)}\right) : \frac{x+2}{6x} \cdot \frac{1}{x-5}\]

\[= \left(\frac{x^2-7x+10}{3x(x-2)}\right) : \frac{x+2}{6x} \cdot \frac{1}{x-5}\]

\[= \left(\frac{(x-5)(x-2)}{3x(x-2)}\right) : \frac{x+2}{6x} \cdot \frac{1}{x-5}\]

\[= \frac{(x-5)}{3x} \cdot \frac{6x}{x+2} \cdot \frac{1}{x-5}\]

\[= \frac{(x-5) \cdot 6x}{3x \cdot (x+2) \cdot (x-5)}\]

\[= \frac{2}{x+2}\]

Ответ: \[\frac{2}{x+2}\]