1.

Дробь не имеет смысла, когда знаменатель [tex]0[/tex].

Тогда, [tex]\implies x+3=0\implies x=-3[/tex], а.

2.

[tex]\вfrac{x-1}{x+1}=0\implies x-1=0\implies x=1[/tex], в.

3.

а)

[tex]\dfrac{a}{a+b}+\dfrac{b}{a+b}=\dfrac{a+b}{a+b}[/tex], а не [tex]\dfrac{a+b}{2(a+b)}[/tex]. Нет.

б)

[tex]\dfrac{a}{a+b}+\dfrac{b}{a+b}=\dfrac{a+b}{a+b}=1[/tex], а не [tex]0[/tex]. Нет.

в)

[tex]\dfrac{a}{a+b}+\dfrac{b}{a+b}=\dfrac{a+b}{a+b}=1[/tex], да.

4.

а)

[tex]\dfrac{12m^3}{6m^4}\div\dfrac{6m^3}{6m^3}\overset{\sqrt}{=}\dfrac{2}{m}[/tex]

б)

[tex]\dfrac{3a^3b^4c}{24ab^6c}\div\dfrac{3ab^4c}{3ab^4c}=\dfrac{a^2}{8b^2}\ne\dfrac{1^ 3}{8b^2}[/tex]

в)

[tex]\dfrac{(a-b)^3}{(b-a)^5}=\dfrac{(a-b)^3}{(-(a-b))^5}=\dfrac{(a-b)^3}{(-1)^5(a-b)^5}\div\dfrac{(a-b)^3}{(a-b)^3}=-\dfrac{1}{(a-b)^2}\overset{\sqrt}{=}\dfrac{-1}{(b-a)^2}[/tex]

5.

а)

[tex]\dfrac{5a-b}{2a-b}+\dfrac{-3a}{2a-b}=\dfrac{5a-b+(-3a)}{2a-b}=\dfrac{2a-b}{2a-b}=1[/tex]

б)

[tex]\dfrac{15m+3n}{5m+n}\div\dfrac{5m+n}{5m+n}-2=3-2=1[/tex]

6.

а)

[tex]\dfrac{x^2-25y^2}{2x-10y}=\dfrac{(x-5y)(x+5y)}{2(x-5y)}\div\dfrac{x-5y}{x-5y}=\dfrac{x+5y}{2}[/tex]

б)

[tex]\dfrac{a^2-2a+1}{a^2-1}=\dfrac{(a-1)^2}{(a-1)(a+1)}=\dfrac{a-1}{a+1}[/tex]

7.

[tex]\dfrac{3}{x-5}-\dfrac{2}{x}+\dfrac{x-35}{x^2-25}=\dfrac{3}{x-5}\times\dfrac{x}{x}-\dfrac{2}{x}\times\dfrac{x-5}{x-5}+\dfrac{x-35}{(x-5)(x+5)}\\\\=\dfrac{3x-2(x-5)}{x(x-5)}+\dfrac{x-35}{(x-5)(x+5)}=\dfrac{x-10}{x(x-5)}\times\dfrac{x+5}{x+5}+\dfrac{x-35}{(x-5)(x+5)}\times\dfrac{x}{x}\\\\=\dfrac{(x-10)(x+5)+x(x-35)}{x(x-5)(x+5)}=\dfrac{2x^2-20x+50}{x(x-5)(x+5)}=\dfrac{2(x^2-10x+25)}{x(x-5)(x+5)}\\\\=\dfrac{2(x-5)^2}{x(x-5)(x+5)}\div\dfrac{x-5}{x-5}=\dfrac{2(x-5)}{x(x+5)}[/tex]

8.

[tex]\dfrac{4}{a-b}+\dfrac{9}{a+b}-\dfrac{8a}{a^2-b^2}=\dfrac{4}{a-b}\times\dfrac{a+b}{a+b}+\dfrac{9}{a+b}\times\dfrac{a-b}{a-b}-\dfrac{8a}{(a-b)(a+b)}\\\\=\dfrac{4(a+b)+9(a-b)-8a}{(a-b)(a+b)}=\dfrac{4a+4b+9a-9b-8a}{(a-b)(a+b)}=\dfrac{5a-5b}{(a-b)(a+b)}\\\\=\dfrac{5(a-b)}{(a-b)(a+b)}\div\dfrac{a-b}{a-b}\overset{\sqrt}{=}\dfrac{5}{a+b}[/tex]

9.

[tex]\dfrac{x^4-9x^2+54x-81}{x^3+27}=\dfrac{x^4-9x^2+54x-81}{(x+3)(x^2-3x+9)}[/tex]