Soalan 8:
Permudah.
$$ \text { (a) } \frac{m}{x+2} \times \frac{2(x+2)}{m^2(x-a)} $$
$$ \text { (b) } \frac{2 r^2}{r s-s^2} \times \frac{5 r-5 s}{2 r-4 r^2} $$
$$ \text { (c) } \frac{x}{x+2} \times \frac{x^2+5 x+6}{5 x^2} $$
$$ \text { (d) } \frac{e+2 f}{5 e-2 f} \times \frac{4 f^2-10 e f}{3 e^2-9 e f} $$
Penyelesaian:
(a)
$$ \frac{m}{x+2} \times \frac{2(x+2)}{m^2(x-a)}=\frac{2}{m(x-a)} $$
(b)
$$ \begin{aligned} \frac{2 r^2}{r s-s^2} \times \frac{5 r-5 s}{2 r-4 r^2} & =\frac{2 r^2}{s(r-s)} \times \frac{5(r-s)}{2 r(1-2 r)} \\ & =\frac{r}{s} \times \frac{5}{1-2 r} \\ & =\frac{5 r}{s(1-2 r)} \end{aligned} $$
(c)
$$ \begin{aligned} & \frac{x}{x+2} \times \frac{x^2+5 x+6}{5 x^2} \\ & =\frac{x}{x+2} \times \frac{(x+3)(x+2)}{5 x^2} \\ & =\frac{x+3}{5 x} \end{aligned} $$
(d)
$$ \begin{aligned} \frac{e+2 f}{5 e-2 f} \times \frac{4 f^2-10 e f}{3 e^2-9 e f} & =\frac{e+2 f}{5 e-2 f} \times \frac{2 f(2 f-5 e)}{3 e(e-3 f)} \\ & =\frac{e+2 f}{5 e-2 f} \times \frac{2 f(-1)(5 e-2 f)}{3 e(e-3 f)} \\ & =\frac{-2 f(e+2 f)}{3 e(e-3 f)} \end{aligned} $$
Permudah.
$$ \text { (a) } \frac{m}{x+2} \times \frac{2(x+2)}{m^2(x-a)} $$
$$ \text { (b) } \frac{2 r^2}{r s-s^2} \times \frac{5 r-5 s}{2 r-4 r^2} $$
$$ \text { (c) } \frac{x}{x+2} \times \frac{x^2+5 x+6}{5 x^2} $$
$$ \text { (d) } \frac{e+2 f}{5 e-2 f} \times \frac{4 f^2-10 e f}{3 e^2-9 e f} $$
Penyelesaian:
(a)
$$ \frac{m}{x+2} \times \frac{2(x+2)}{m^2(x-a)}=\frac{2}{m(x-a)} $$
(b)
$$ \begin{aligned} \frac{2 r^2}{r s-s^2} \times \frac{5 r-5 s}{2 r-4 r^2} & =\frac{2 r^2}{s(r-s)} \times \frac{5(r-s)}{2 r(1-2 r)} \\ & =\frac{r}{s} \times \frac{5}{1-2 r} \\ & =\frac{5 r}{s(1-2 r)} \end{aligned} $$
(c)
$$ \begin{aligned} & \frac{x}{x+2} \times \frac{x^2+5 x+6}{5 x^2} \\ & =\frac{x}{x+2} \times \frac{(x+3)(x+2)}{5 x^2} \\ & =\frac{x+3}{5 x} \end{aligned} $$
(d)
$$ \begin{aligned} \frac{e+2 f}{5 e-2 f} \times \frac{4 f^2-10 e f}{3 e^2-9 e f} & =\frac{e+2 f}{5 e-2 f} \times \frac{2 f(2 f-5 e)}{3 e(e-3 f)} \\ & =\frac{e+2 f}{5 e-2 f} \times \frac{2 f(-1)(5 e-2 f)}{3 e(e-3 f)} \\ & =\frac{-2 f(e+2 f)}{3 e(e-3 f)} \end{aligned} $$
Soalan 9:
Permudah.
$$ \text { (a) } \frac{5 a}{2 a+3} \div \frac{3 b}{a+b} $$
$$ \text { (b) } \frac{4}{n-3} \div \frac{8 a}{3 n-9} $$
$$ \text { (c) } \frac{6 y^2}{x^2+x y} \div \frac{18 x y}{x+y} $$
$$ \text { (d) } \frac{f-1}{e g+2 e} \div \frac{f g-g}{g+2} $$
Penyelesaian:
(a)
$$ \begin{aligned} \frac{5 a}{2 a+3} \div \frac{3 b}{a+b} & =\frac{5 a}{2 a+3} \times \frac{a+b}{3 b} \\ & =\frac{5 a(a+b)}{3 b(2 a+3)} \end{aligned} $$
(b)
$$ \begin{aligned} \frac{4}{n-3} \div \frac{8 a}{3 n-9} & =\frac{4}{n-3} \times \frac{3 n-9}{-8 a} \\ & =\frac{1}{n-3} \times \frac{3(n-3)}{2 a} \\ & =\frac{3}{2 a} \end{aligned} $$
(c)
$$ \begin{aligned} \frac{6 y^2}{x^2+x y} \div \frac{18 x y}{x+y} & =\frac{6 y^2}{x(x+y)} \times \frac{x+y}{18 x y} \\ & =\frac{y}{x} \times \frac{1}{3 x} \\ & =\frac{y}{3 x^2} \end{aligned} $$
(d)
$$ \begin{aligned} \frac{f-1}{e g+2 e} \div \frac{f g-g}{g+2} & =\frac{f-1}{e(g+2)} \times \frac{g+2}{g(f-1)} \\ & =\frac{1}{e g} \end{aligned} $$
Permudah.
$$ \text { (a) } \frac{5 a}{2 a+3} \div \frac{3 b}{a+b} $$
$$ \text { (b) } \frac{4}{n-3} \div \frac{8 a}{3 n-9} $$
$$ \text { (c) } \frac{6 y^2}{x^2+x y} \div \frac{18 x y}{x+y} $$
$$ \text { (d) } \frac{f-1}{e g+2 e} \div \frac{f g-g}{g+2} $$
Penyelesaian:
(a)
$$ \begin{aligned} \frac{5 a}{2 a+3} \div \frac{3 b}{a+b} & =\frac{5 a}{2 a+3} \times \frac{a+b}{3 b} \\ & =\frac{5 a(a+b)}{3 b(2 a+3)} \end{aligned} $$
(b)
$$ \begin{aligned} \frac{4}{n-3} \div \frac{8 a}{3 n-9} & =\frac{4}{n-3} \times \frac{3 n-9}{-8 a} \\ & =\frac{1}{n-3} \times \frac{3(n-3)}{2 a} \\ & =\frac{3}{2 a} \end{aligned} $$
(c)
$$ \begin{aligned} \frac{6 y^2}{x^2+x y} \div \frac{18 x y}{x+y} & =\frac{6 y^2}{x(x+y)} \times \frac{x+y}{18 x y} \\ & =\frac{y}{x} \times \frac{1}{3 x} \\ & =\frac{y}{3 x^2} \end{aligned} $$
(d)
$$ \begin{aligned} \frac{f-1}{e g+2 e} \div \frac{f g-g}{g+2} & =\frac{f-1}{e(g+2)} \times \frac{g+2}{g(f-1)} \\ & =\frac{1}{e g} \end{aligned} $$
Soalan 10:
Selesaikan gabungan operasi berikut.
$$ \text { (a) } \frac{x^2+x}{x^2-y^2} \times \frac{x y-y^2}{x+y} $$
$$ \text { (b) } \frac{4 p^2-1}{p^2-1} \times \frac{p q+q}{4 p-2} $$
$$ \text { (c) } \frac{p q-p r}{r^2-1} \div \frac{q^2-r^2}{r^2+r} $$
$$ \text { (d) } \frac{s t+t u}{4 t^2-1} \div \frac{s^2-u^2}{4 t^2+4 t+1} $$
Penyelesaian:
(a)
$$ \begin{aligned} \frac{x^2+x}{x^2-y^2} \times \frac{x y-y^2}{x+y} & =\frac{x(x+1)}{(x+y)(x-y)} \times \frac{y(x-y)}{x+y} \\ & =\frac{x y(x+1)}{(x+y)(x+y)} \\ & =\frac{x y(x+1)}{(x+y)^2} \end{aligned} $$
(b)
$$ \begin{aligned} \frac{4 p^2-1}{p^2-1} \times \frac{p q+q}{4 p-2} & =\frac{(2 p)^2-1^2}{p^2-1^2} \times \frac{q(p+1)}{2(2 p-1)} \\ & =\frac{(2 p+1)(2 p-1)}{(p+1)(p-1)} \times \frac{q(p+1)}{2(2 p-1)} \\ & =\frac{2 p+1}{p-1} \times \frac{q}{2} \\ & =\frac{q(2 p+1)}{2(p-1)} \end{aligned} $$
(c)
$$ \begin{aligned} \frac{p q-p r}{r^2-1} \div \frac{q^2-r^2}{r^2+r} & =\frac{p(q-r)}{r^2-1^2} \times \frac{r^2+r}{q^2-r^2} \\ & =\frac{p(q-r)}{(q+1)(r-1)} \times \frac{r(x+1)}{(q+r)(q-r)} \\ & =\frac{p}{r-1} \times \frac{r}{q+r} \\ & =\frac{p r}{(r-1)(q+r)} \end{aligned} $$
(d)
$$ \begin{aligned} \frac{s t+t u}{4 t^2-1} \div \frac{s^2-u^2}{4 t^2+4 t+1} & =\frac{t(s+u)}{(2 t)^2-1^2} \times \frac{4 t^2+4 t+1}{s^2-u^2} \\ & =\frac{t(s+u)}{(2 t+1)(2 t-1)} \times \frac{(2 t+1)(2 t+1)}{(s+u)(s-u)} \\ & =\frac{t}{2 t-1} \times \frac{2 t+1}{s-u} \\ & =\frac{t(2 t+1)}{(2 t-1)(s-u)} \end{aligned} $$
Selesaikan gabungan operasi berikut.
$$ \text { (a) } \frac{x^2+x}{x^2-y^2} \times \frac{x y-y^2}{x+y} $$
$$ \text { (b) } \frac{4 p^2-1}{p^2-1} \times \frac{p q+q}{4 p-2} $$
$$ \text { (c) } \frac{p q-p r}{r^2-1} \div \frac{q^2-r^2}{r^2+r} $$
$$ \text { (d) } \frac{s t+t u}{4 t^2-1} \div \frac{s^2-u^2}{4 t^2+4 t+1} $$
Penyelesaian:
(a)
$$ \begin{aligned} \frac{x^2+x}{x^2-y^2} \times \frac{x y-y^2}{x+y} & =\frac{x(x+1)}{(x+y)(x-y)} \times \frac{y(x-y)}{x+y} \\ & =\frac{x y(x+1)}{(x+y)(x+y)} \\ & =\frac{x y(x+1)}{(x+y)^2} \end{aligned} $$
(b)
$$ \begin{aligned} \frac{4 p^2-1}{p^2-1} \times \frac{p q+q}{4 p-2} & =\frac{(2 p)^2-1^2}{p^2-1^2} \times \frac{q(p+1)}{2(2 p-1)} \\ & =\frac{(2 p+1)(2 p-1)}{(p+1)(p-1)} \times \frac{q(p+1)}{2(2 p-1)} \\ & =\frac{2 p+1}{p-1} \times \frac{q}{2} \\ & =\frac{q(2 p+1)}{2(p-1)} \end{aligned} $$
(c)
$$ \begin{aligned} \frac{p q-p r}{r^2-1} \div \frac{q^2-r^2}{r^2+r} & =\frac{p(q-r)}{r^2-1^2} \times \frac{r^2+r}{q^2-r^2} \\ & =\frac{p(q-r)}{(q+1)(r-1)} \times \frac{r(x+1)}{(q+r)(q-r)} \\ & =\frac{p}{r-1} \times \frac{r}{q+r} \\ & =\frac{p r}{(r-1)(q+r)} \end{aligned} $$
(d)
$$ \begin{aligned} \frac{s t+t u}{4 t^2-1} \div \frac{s^2-u^2}{4 t^2+4 t+1} & =\frac{t(s+u)}{(2 t)^2-1^2} \times \frac{4 t^2+4 t+1}{s^2-u^2} \\ & =\frac{t(s+u)}{(2 t+1)(2 t-1)} \times \frac{(2 t+1)(2 t+1)}{(s+u)(s-u)} \\ & =\frac{t}{2 t-1} \times \frac{2 t+1}{s-u} \\ & =\frac{t(2 t+1)}{(2 t-1)(s-u)} \end{aligned} $$