Упражнение 220

\({\largeа)}\ \frac{b^{14}-b^7+1}{b^{21}+1}=\frac{b^{14}-b^7+1}{ (b^7)^3+1}=\frac{b^{14}-b^7+1}{ (b^7+1)(b^{14}-b^7+1)}=\frac{1}{b^7+1};\)

\({\largeб)}\ \frac{x^{33}-1}{x^{33}+x^{22}+x^{11}}=\frac{ (x^{11})^3-1}{x^{33}+x^{22}+x^{11}}=\frac{ (x^{11}-1)(x^{22}+x^{11}+1)}{ x^{11}(x^{22}+x^{11}+1)}=\frac{x^{11}-1}{x^{11}};\)

\({\largeв)}\ \frac{ x(y-z)-y(x-z)}{ x(y-z)^2-y(x-z)^2}=\frac{xy-xz-xy+yz}{ x(y^2-2yz+z^2)-y(x^2-2xz+z^2)}=\frac{yz-xz}{xy^2-2xyz+xz^2-x^2y+2xyz-yz^2}=\frac{ z(y-x)}{xy^2-x^2y-yz^2+xz^2}=\frac{ z(y-x)}{ xy(y-x)-z^2(y-x)}=\frac{ z(y-x)}{ (y-x)(xy-z^2)}=\frac{z}{xy-z^2};\)

\({\largeг)}\ \frac{ a(b+1)^2-b(a+1)^2}{ a(b+1)-b(a+1)}=\frac{ a(b^2+2b+1)-b(a^2+2a+1)}{ab+a-ab-b}=\frac{ab^2+2ab+a-a^2b-2ab-b}{a-b}=\frac{ab^2-a^2b+a-b}{a-b}=\frac{ ab(b-a)+(a-b)}{a-b}=\frac{ {-}ab(a-b)+(a-b)}{a-b}=\frac{ (a-b)(1-ab)}{a-b}=1-ab.\)