Алгебра 8 класс учебник Макарычев, Миндюк ответы – номер 252

а) преобразуем выражение
$$ \frac{x-\frac{y z}{y-z}}{y-\frac{x z}{x-z}} $$
1) x − yz/y − z = x/1 − yz/y − z = x ⋅ (y − z)/1 ⋅ (y − z) − yz/y − z = xy − xz − yz/y − z
2) y − xz/x − z = y/1 − xz/x − z = y ⋅ (x − z)/1 ⋅ (x − z) − xz/x − z = xy − yz − xz/x − z
3) $$ \frac{\frac{x y-x z-y z}{y-z}}{\frac{x y-y z-x z}{x-z}} $$ = xy − xz − yz/y − z : xy − yz − xz/x − z = xy − xz − yz/y − z ⋅ x − z/xy − yz − xz = (xy − xz − yz) ⋅ (x − z)/(y − z) ⋅ (xy − xz − yz) = (xy − xz − yz) ⋅ (x − z) : (xy − xz − yz)/(y − z) ⋅ (xy − xz − yz) : (xy − xz − yz) = x − z/y − z
б) преобразуем выражение
$$ \frac{\frac{a-x}{a}+\frac{x}{a-x}}{\frac{a+x}{a}-\frac{x}{a+x}} $$
1) a − x/a + x/a − x = (a − x) ⋅ (a − x)/a ⋅ (a − x) + x ⋅ a/(a − x) ⋅ a = (a − x)² + ax/a(a − x) = a² − 2ax + x² + ax/a(a − x) = a² − ax + x²/a(a − x)
2) a + x/a + x/a + x = (a + x) ⋅ (a + x)/a ⋅ (a + x) + x ⋅ a/(a + x) ⋅ a = (a + x)² − ax/a(a + x) = a² + 2ax + x² − ax/a(a + x) = a² + ax + x²/a(a + x)
3) $$ \frac{\frac{a^2-a x+x^2}{a(a-x)}}{\frac{a^2+a x+x^2}{a(a+x)}} $$ = a² − ax + x²/a(a − x) : a² + ax + x²/a(a + x) = a² − ax + x²/a(a − x) ⋅ a(a + x)/a² + ax + x² = (a² − ax + x²) ⋅ a(a + x) : a/a(a − x) ⋅ (a² + ax + x²) : a = (a² − ax + x²)(a + x)/(a − x)(a² + ax + x²) = a³ + x³/a³ − x³
в) преобразуем выражение
$$ \frac{1}{1+\frac{1}{1+\frac{1}{x}}} $$
1) 1 + 1/x = x/x + 1/x = x + 1/x
2) $$ \frac{\frac{1}{x+1}}{\frac{x}{x}} $$ = 1 : x + 1/x = 1 ⋅ x/x + 1 = x/x + 1
3) 1 + x/x + 1 = x + 1/x + 1 + x/x + 1 = x + 1 + x/x + 1 = 2x + 1/x + 1
4) $$ \frac{\frac{1}{2 x+1}}{x+1} $$ = 1 : 2x + 1/x + 1 = 1 ⋅ x + 1/2x + 1 = x + 1/2x + 1
г) преобразуем выражение
$$ \frac{1}{1-\frac{1}{1+\frac{1}{x}}} $$
1) 1 + 1/x = x/x + 1/x = x + 1/x
2) $$ \frac{1}{\frac{x+1}{x}} $$ = 1 : x + 1/x = 1 ⋅ x/x + 1 = x/x + 1
3) 1 − x/x + 1 = x + 1/x + 1 − x/x + 1 = x + 1 − x/x + 1 = 1/x + 1
4) $$ \frac{1}{\frac{1}{x+1}} $$ = 1 : 1/x + 1 = 1 ⋅ x + 1/1 = x + 1