№ 7.21 Алгебра = № 12.21 Математика
Спростіть вираз:
1. $\frac{1-\frac{1}{x}}{1+\frac{1}{x}};$
2. $\frac{\frac{7x-a}{a}+1}{\frac{7x+a}{a}-1};$
3. $\frac{\frac{1}{p}-\frac{1}{2p}}{\frac{1}{p^2}-\frac{1}{2p^2}};$
4. $\frac{c-\frac{6c-9}{c}}{\frac{3}{c}-1};$
5. $\frac{\frac{x}{x-1}-\frac{x+1}{x}}{\frac{x}{x+1}-\frac{x-1}{x}};$
6. $\frac{\frac{1}{n-m}-\frac{1}{n+m}}{\frac{1}{n-m}+\frac{1}{n+m}}.$
Розв'язок:
1. $\frac{1-\frac{1}{x}}{1+\frac{1}{x}}=\frac{x-1}{x}:\frac{x+1}{x}=$
$= \frac{(x-1)\cdot x}{x\cdot(x+1)}=\frac{x-1}{x+1};$
2. $\frac{\frac{7x-a}{a}+1}{\frac{7x+a}{a}-1}=\frac{7x-a+a}{a}:\frac{7x+a-a}{a}=$
$= \frac{7x}{a}\cdot\frac{a}{7x}=1;$
3. $\frac{\frac{1}{p}-\frac{1}{2p}}{\frac{1}{p^2}-\frac{1}{2p^2}}=\frac{\frac{2-1}{2p}}{\frac{2-1}{2p^2}}=$
$= \frac{1}{2p}\cdot\frac{2p^2}{1}=\frac{2p^2}{2p}=p;$
4. $\frac{c-\frac{6c-9}{c}}{\frac{3}{c}-1}=\frac{\frac{c^2-\left(6c-9\right)}{c}}{\frac{3-c}{c}}=$
$= \frac{c^2-6c-9}{c}\cdot\frac{c}{3-c}=$
$= \frac{\left(c-3\right)^2\cdot c}{c\cdot\left(3-c\right)}=3-c;$
5. $\frac{\frac{x}{x-1}-\frac{x+1}{x}}{\frac{x}{x+1}-\frac{x-1}{x}}=\frac{\frac{x^2-\left(x-1\right)\left(x+1\right)}{x\left(x-1\right)}}{\frac{x^2-\left(x-1\right)\left(x+1\right)}{x\left(x+1\right)}}=$
$= \frac{x^2-\left(x^2-1\right)}{x\left(x-1\right)}\cdot\frac{x\left(x+1\right)}{x^2-\left(x^2-1\right)}=$
$= \frac{x(x+1)}{x(x-1)}=\frac{x+1}{x-1};$
6. $\frac{\frac{1}{n-m}-\frac{1}{n+m}}{\frac{1}{n-m}+\frac{1}{n+m}}=\frac{\frac{\left(n+m\right)-\left(n-m\right)}{\left(n-m\right)\left(n+m\right)}}{\frac{\left(n+m\right)+\left(n-m\right)}{\left(n-m\right)\left(n+m\right)}}=$
$= \frac{n+m-n+m}{n^2-m^2}\cdot\frac{n^2-m^2}{m+n+n-m}=$
$= \frac{2m\left(n^2-m^2\right)}{2n\left(n^2-m^2\right)}=\frac{m}{n}.$