Algebraliste murdude jagamine ja astendamine

Nagu algebraliste murdude korrutamisel, nii ka jagamisel ja astendamisel jäävad kehtima harilike murdude vastavad eeskirjad.

Kirjutame need lühidalt

$\frac{a}{b} : \frac{c}{d} = \frac{a d}{b c}$

Algebraliste murdude jagamiseks korrutatakse jagatav murruga, mis on saadud jagajast selle lugeja ja nimetaja vahetamise teel.

${(\frac{a}{b})}^{n} = \frac{a^{n}}{b^{n}}$

Algebralise murru n-es aste võrdub murruga, mille lugejaks on antud murru lugeja n-es aste ja nimetajaks antud murru nimetaja n-es aste.

Näide 1

$\frac{63 a^{2} x y^{3}}{20 b^{2} z^{4}} : (- \frac{21 a x^{2} y^{3}}{85 b^{3} z^{3}})$ = $- \frac{63 a^{2} x y^{3}}{20 b^{2} z^{4}} \cdot \frac{85 b^{3} z^{3}}{21 a x^{2} y^{3}}$ = $- \frac{51 a b}{4 x z}$

Näide 2

$\frac{x^{2} - 4}{2 x} : (x - 2)$ = $\frac{x^{2} - 4}{2 x} : \frac{x - 2}{1}$ = $\frac{x^{2} - 4}{2 x} \cdot \frac{1}{x - 2}$ = $\frac{x + 2}{2 x}$

Näide 3

${(a + x)}^{2} : \frac{2 a + 2 x}{6}$ = $\frac{{(a + x)}^{2}}{1} : \frac{2 a + 2 x}{6}$ = $\frac{{(a + x)}^{2} \cdot 6}{2 a + 2 x}$ = $\frac{\overset{a + x}{{(a + x)}^{2}} \cdot \overset{3}{6}}{\underset{1}{2} \underset{1}{(a + x)}}$ = $\frac{3 (a + x)}{1}$ = $3 (a + x)$

Näide 4

$\frac{2 b + 2 a}{m b - m a} : \frac{{(a + b)}^{2}}{a^{2} - b^{2}}$ = $\frac{(2 b + 2 a) (a^{2} - b^{2})}{(m b - m a) {(a + b)}^{2}}$ = $\frac{2 \overset{1}{(b + a)} \overset{1}{(a + b)} (a - b)}{m (b - a) (a + b) (a + b)}$ = $- \frac{2 \overset{1}{(a - b)}}{m \underset{1}{(a - b)}}$ = $- \frac{2}{m}$

${(a^{n})}^{m} = a^{n m}$

Näide 5

${(\frac{2 x^{3} y}{a b^{2}})}^{3}$ = $\frac{{(2 x^{3} y)}^{3}}{{(a b^{2})}^{3}}$ = $\frac{8 x^{9} y^{3}}{a^{3} b^{6}}$

${(- 1)}^{0} = {(- 1)}^{2} = {(- 1)}^{4} = \dots = 1$

${(- 1)}^{1} = {(- 1)}^{3} = {(- 1)}^{5} = \dots = - 1$

Näide 6

${(- \frac{u v^{3}}{2 x^{2}})}^{5}$ = $- {(\frac{u v^{3}}{2 x^{2}})}^{5}$ = $- \frac{{(u v^{3})}^{5}}{{(2 x^{2})}^{5}}$ = $- \frac{u^{5} v^{15}}{32 x^{10}}$

Näide 7

${(- \frac{a^{2} + 2 a b + b^{2}}{2 x - y})}^{4}$ = ${[\frac{{(a + b)}^{2}}{2 x - y}]}^{4}$ = $\frac{{(a + b)}^{8}}{{(2 x - y)}^{4}}$

Seotud sisu

Ülesanded A

$\frac{a}{b} : \frac{1}{b}$ = $\frac{}{}$

$\frac{3 p^{3}}{4 q} : \frac{15 p^{2}}{16 q^{3}}$ = $\frac{}{}$

$\frac{10 r}{27} : \frac{25 r^{3}}{81}$ = $\frac{}{}$

$\frac{3 m}{4 p} : \frac{q}{2 m}$ = $\frac{}{}$

$\frac{3 a^{2} b}{2 c^{2}} : \frac{6 a b}{2 c^{2}}$ = $\frac{}{}$

$\frac{14 x y}{9 z^{2}} : \frac{21 x^{2}}{- 2 c^{2}}$ = $\frac{}{}$

$\frac{x^{2}}{y} : (- \frac{x^{2}}{y^{2}})$ = $\frac{}{}$

$\frac{84 u^{2}}{v^{2}} : (- \frac{7 u^{5}}{v^{2}})$ = $\frac{}{}$

$- \frac{15 a^{4} b^{3}}{17 c} : (- \frac{12 a^{2} b}{17 c})$ = $\frac{}{}$

$\frac{3 x - 9}{x^{2}} : \frac{15 x}{2}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$\frac{2 y + 6}{3 y} : \frac{2 x}{y}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$\frac{2 a^{2} - 4 a}{x} : \frac{2 a}{a + 2}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$\frac{x + 5}{3 x} : \frac{2 a}{6 x^{3} + 3 x}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$\frac{2 u^{4} - 4 u}{12 v^{2}} : \frac{2 u^{3} - 4}{4 v}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$\frac{y^{3} + 2 y}{3 y^{2}} : \frac{4 y^{2} + y}{6 y}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$\frac{2 n^{2}}{7 m} : (m n)$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$b^{2} : \frac{a b}{c}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$\frac{42 p}{5 r} : (14 p r)$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$\frac{4 a x}{b} : (- 2 a x^{2})$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$\frac{a c^{2}}{5 x} : (- 14 c^{3} x)$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$- \frac{8 c^{2} d}{3 f} : (- 4 f)$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$\frac{2 x^{2} + 6}{3 x y} : (8 x y)$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$\frac{2 u^{2} - 4 v}{u v^{2}} : (4 u^{2} v)$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$(3 x^{2} + 12 x) : \frac{3 x}{2 x - 1}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$(6 a b + 15 b^{2}) : \frac{3 b}{a - b}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$\frac{a - b}{a b} : \frac{b - a}{b}$ = $\frac{}{}$

$\frac{m n}{a + b} : \frac{2 m n}{- a - b}$ = $\frac{}{}$

$\frac{3 a b}{a + b} : \frac{a b}{b + a}$ = $\frac{}{}$

$\frac{a^{2} b}{m - n} : \frac{a b}{n - m}$ = $\frac{}{}$

$\frac{2 u - 4 v}{m^{2} n^{3}} : \frac{4 v - 2 u}{m^{3} n^{4}}$ = $\frac{}{}$

$\frac{a^{2} - b^{2}}{x - y} : \frac{b - a}{y - x}$ = $\frac{}{}$

$\frac{u^{2} - v^{2}}{{(b - a)}^{2}} : \frac{v + u}{a^{2} - b^{2}}$ = $\frac{}{}$

$\frac{{(m - n)}^{2}}{5 a b} : \frac{{(n - m)}^{2}}{10 a b}$ = $\frac{}{}$

$\frac{{(m + n)}^{2}}{m - n} : \frac{{(- m - n)}^{2}}{n - m}$ = $\frac{}{}$

$\frac{{(- a - b)}^{2}}{a^{2} - b^{2}} : \frac{2 a + 2 b}{b - a}$ = $\frac{}{}$

$\frac{m + n}{a - b} : \frac{n + m}{a - b}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$\frac{3 m + 3 n}{a - b} : \frac{2 n + 2 m}{a + b}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$\frac{m n - m p}{a x + a y} : \frac{m^{2}}{a^{2} y + a^{2} x}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$\frac{a^{2} - b^{2}}{2 x^{2} y} : \frac{b + a}{a x - b x}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$\frac{4 a^{2} - 1}{x^{2} - a x} : \frac{2 a + 1}{x^{2} - a^{2}}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$\frac{a^{2} - x^{2}}{a x} : \frac{x + a}{a^{2} x^{2}}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$\frac{{(m + n)}^{2}}{4 m - 4 n} : \frac{6 n + 6 m}{n - m}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$\frac{p^{2} + p q}{r x + s x} : \frac{p^{3} + p^{2} q}{r x^{3} + s x^{3}}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$\frac{a + b}{a - b} : \frac{b + a}{b - a}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$\frac{2 a + 1}{x - 2} : \frac{4 a^{2} - 1}{2 - x}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$\frac{m^{2} + 2 m n + n^{2}}{2 m + 2 n} : \frac{- a m - a n}{a^{2} b}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$\frac{1}{9 x^{2} - 1} : \frac{a + b^{2}}{1 - 3 x}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$\frac{x^{2} - 9 x + 20}{x - 5} : \frac{4 - x}{3}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$\frac{a b^{2}}{a^{2} + 5 a + 6} : \frac{1}{3 + a}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$\frac{x^{2} - y^{2}}{3 x} : (x - y)$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$\frac{m n - m^{2}}{6} : (m^{2} - n^{2})$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$\frac{1 - 4 a^{2}}{2} : (4 a^{2} - 4 a + 1)$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$(b^{2} - 9 a^{2}) : \frac{3 a - b}{3 a + b}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$(a^{2} - 2 a m + m^{2}) : \frac{3 m - 3 a}{7}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

${(x + 2 y)}^{3} : \frac{2 y + x}{4 n}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$\frac{m a^{2} - m b^{2}}{b + a} : \frac{m^{2} b - m^{2} a}{m}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$\frac{2 m - 8 a^{2} m}{1 + 2 a} : \frac{4 a m^{2} - 2 m^{2}}{2 a m - m}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$(x^{2} - x - 6) : \frac{x^{2} + x - 2}{4 x^{2}}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$\frac{2 x^{2} - 9 x - 5}{3 x} : (2 x^{2} + 5 x + 2)$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

${(- \frac{b}{2})}^{5}$ = $\frac{}{}$

${(\frac{3 m^{2}}{4 n^{2}})}^{3}$ = $\frac{}{}$

${(- \frac{u^{2} v}{s t^{3}})}^{4}$ = $\frac{}{}$

${(\frac{x - 5}{3 x})}^{2}$ = $\frac{}{}$

${(- \frac{3}{c})}^{2}$ = $\frac{}{}$

${(- \frac{t^{3}}{v^{2}})}^{3}$ = $\frac{}{}$

${(\frac{3 x^{2}}{7 y^{4}})}^{2}$ = $\frac{}{}$

${(\frac{2 + u}{4 u})}^{2}$ = $\frac{}{}$

${(\frac{2 x - 5 y}{3 x + 2})}^{2}$ = $\frac{}{}$

${(\frac{4 a + 7 b}{3 a - b})}^{2}$ = $\frac{}{}$

${(\frac{5 u - 3 v}{2 v - u})}^{2}$ = $\frac{}{}$

Seotud sisu

Ülesanded B

${(\frac{a^{3} b}{c^{4}})}^{5} : {(\frac{a^{5} b^{2}}{c^{7}})}^{3}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

${(\frac{x^{6} y^{8}}{z^{5}})}^{5} \cdot {(\frac{z^{8}}{x^{10} y^{13}})}^{3}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

${(- \frac{2 a^{8} b^{3}}{c^{7}})}^{5} : {(- \frac{4 a^{10} b^{4}}{c^{9}})}^{4}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

${(- \frac{9 x^{7} y^{6}}{a^{12}})}^{4} \cdot {(- \frac{a^{8}}{3 x^{5} y^{4}})}^{3}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

${(\frac{a - b}{c - d})}^{2} \cdot \frac{c - d}{{(a - b)}^{2}}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

${(\frac{2 a^{2} b}{m - n})}^{3} : \frac{a^{6} b^{2}}{{(m - n)}^{2}}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

${(\frac{a^{2} - b^{2}}{x^{2} y})}^{2} \cdot \frac{x^{4}}{{(b - a)}^{2}}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

${[\frac{{(a + b)}^{2}}{a - b}]}^{3} : \frac{{(a + b)}^{5}}{{(b - a)}^{2}}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

${[\frac{b^{4} {(b - c)}^{2}}{a^{6} (c - a)}]}^{3} : {[\frac{b^{2} (b - c)}{a^{3} (a - c)}]}^{6}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

${[\frac{a^{2} (a - b)}{x^{4} {(a - x)}^{3}}]}^{6} \cdot {[\frac{x^{6} {(x - a)}^{5}}{a^{3} {(b - a)}^{2}}]}^{4}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

${(- \frac{a^{2} + a b}{a b^{2} - b^{3}})}^{4} \cdot {(\frac{b - a}{a^{2} + 2 a b + b^{2}})}^{3}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

$\frac{{(x - y)}^{2} - z^{2}}{{(x - z)}^{2} - y^{2}} : \frac{x - y - z}{x + y - z}$ = $\frac{\cdot}{\cdot}$ = $\frac{}{}$

Seotud sisu

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Algebraliste murdude jagamine ja astendamine

Näide 1

Näide 2

Näide 3

Näide 4

Näide 5

Näide 6

Näide 7

Seotud sisu

Ülesanded A

Seotud sisu

Ülesanded B

Seotud sisu

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