Question

Выполнить сложение или вычитание дробей.

1)4а/7а+49 + 4/а+7=4а/7(а+7)+4/а+7=

2)а²-4ab/ab-2b² + 4b/a-2b=

3)6/2m-5 - 4m+5/2m²-5m=

4)1/c²-6c - 1/6c-36=1/c(c-6) - 1/6(c-6)=

5)y+3/xy-y² - x+3/x²-xy=

6)a²+64/2a²+16a + 8/a+8=

Answer 1

Ответ:

1)

$\dfrac{4a}{7a+49}+\dfrac{4}{a+7}=\dfrac{4a}{7(a+7)}+\dfrac{4}{a+7}^{\backslash7}=$

$=\dfrac{4a+28}{7(a+7)}=\dfrac{4(a+7)}{7(a+7)}=\boldsymbol{\dfrac{4}{7}}$

2)

$\dfrac{a^2-4ab}{ab-2b^2}+\dfrac{4b}{a-2b}=\dfrac{a^2-4ab}{b(a-2b)}+\dfrac{4b}{a-2b}^{\backslash b}=$

$=\dfrac{a^2-4ab+4b^2}{b(a-2b)}=\dfrac{(a-2b)^2}{b(a-2b)}=\boldsymbol{\dfrac{a-2b}{b}}$

Использована формула квадрата разности:

$a^2-4ab+4b^2=a^2-2\cdot a\cdot 2b+(2b)^2=(a-2b)^2$

3)

$\dfrac{6}{2m-5}-\dfrac{4m+5}{2m^2-5m}=\dfrac{6}{2m-5}^{\backslash m}-\dfrac{4m+5}{m(2m-5)}=\dfrac{6m-(4m+5)}{m(2m-5)}=$

$=\dfrac{6m-4m-5}{m(2m-5)}=\dfrac{2m-5}{m(2m-5)}=\boldsymbol{\dfrac{1}{m}}$

4)

$\dfrac{1}{c^2-6c}-\dfrac{1}{6c-36}=\dfrac{1}{c(c-6)}^{\backslash 6}-\dfrac{1}{6(c-6)}^{\backslash c}=\dfrac{6-c}{6c(c-6)}=$

$=-\dfrac{c-6}{6c(c-6)}=\boldsymbol{-\dfrac{1}{6c}}$

5)

$\dfrac{y+3}{xy-y^2}-\dfrac{x+3}{x^2-xy}=\dfrac{y+3}{y(x-y)}^{\backslash x}-\dfrac{x+3}{x(x-y)}^{\backslash y}=\dfrac{x(y+3)-y(x+3)}{xy(x-y)}=$

$=\dfrac{xy+3x-xy-3y}{xy(x-y)}=\dfrac{3x-3y}{xy(x-y)}=\dfrac{3(x-y)}{xy(x-y)}=\boldsymbol{\dfrac{3}{xy}}$

6)

$\dfrac{a^2+64}{2a^2+16a}+\dfrac{8}{a+8}=\dfrac{a^2+64}{2a(a+8)}+\dfrac{8}{a+8}^{\backslash 2a}=$

$=\dfrac{a^2+64+16a}{2a(a+8)}=\dfrac{(a+8)^2}{2a(a+8)}=\boldsymbol{\dfrac{a+8}{2a}}$

Использована формула квадрата суммы:

$a^2+64+16a=a^2+16a+64=a^2+2\cdot a\cdot 8+8^2=(a+8)^2$