7.4: Kurahisisha Maneno mazuri ya busara

Malengo ya kujifunza

Mwishoni mwa sehemu hii, utaweza:

• Kurahisisha tata kujieleza busara kwa kuandika kama mgawanyiko
• Kurahisisha kujieleza kwa busara kwa kutumia LCD

Kabla ya kuanza, fanya jaribio hili la utayari.

1. Kurahisisha:\(\dfrac{\dfrac{3}{5}}{\dfrac{9}{10}}\).
   Ikiwa umekosa tatizo hili, tathmini [kiungo].
2. Kurahisisha:\(\dfrac{1−\dfrac{1}{3}}{4^2+4·5}\).
   Ikiwa umekosa tatizo hili, tathmini [kiungo].
3. Kutatua:\(\dfrac{1}{2x}+\dfrac{1}{4}=\dfrac{1}{8}\).
   Ikiwa umekosa tatizo hili, tathmini [kiungo].

Complex Fraction

Sehemu ngumu ni sehemu ambayo namba na/au denominator ina sehemu.

Sisi hapo awali kilichorahisishwa sehemu ndogo kama hizi:

\[\dfrac{\dfrac{3}{4}}{\dfrac{5}{8}} \quad \quad \quad \dfrac{\dfrac{x}{2}}{\dfrac{x y}{6}} \nonumber \]

Katika sehemu hii, tutawezesha maneno mazuri ya busara, ambayo ni maneno ya busara na maneno ya busara katika namba au denominator.

Complex mantiki kujieleza

Maneno mazuri ya busara ni kujieleza kwa busara ambayo namba na/au denominator ina kujieleza kwa busara.

Hapa kuna maneno machache ya busara:

\[\dfrac{\dfrac{4}{y-3}}{\dfrac{8}{y^{2}-9}} \quad \quad \dfrac{\dfrac{1}{x}+\dfrac{1}{y}}{\dfrac{x}{y}-\dfrac{y}{x}} \quad \quad \dfrac{\dfrac{2}{x+6}}{\dfrac{4}{x-6}-\dfrac{4}{x^{2}-36}} \nonumber \]

Kumbuka, sisi daima hutenganisha maadili ambayo ingeweza kufanya zero yoyote ya denominator.

Tutatumia mbinu mbili ili kurahisisha maneno mazuri ya busara.

Tayari tumeona maneno haya mazuri ya busara mapema katika sura hii.

\[\dfrac{\dfrac{6 x^{2}-7 x+2}{4 x-8}}{\dfrac{2 x^{2}-8 x+3}{x^{2}-5 x+6}} \nonumber \]

Tulibainisha kuwa baa za sehemu zinatuambia kugawanya, hivyo upya upya kama tatizo la mgawanyiko:

\[\left(\dfrac{6 x^{2}-7 x+2}{4 x-8}\right) \div\left(\dfrac{2 x^{2}-8 x+3}{x^{2}-5 x+6}\right) \nonumber \]

Kisha, tuliongeza maneno ya kwanza ya busara kwa usawa wa pili, kama tunavyofanya tunapogawanya sehemu mbili.

Hii ni njia moja ya kurahisisha maneno mazuri ya busara. Tunahakikisha kujieleza kwa busara ni ya fomu ambapo sehemu moja iko juu ya sehemu moja. Sisi kisha kuandika kama sisi walikuwa kugawa sehemu mbili.

Mfano\(\PageIndex{1}\)

Kurahisisha kujieleza kwa busara kwa kuandika kama mgawanyiko:\[\dfrac{\dfrac{6}{x-4}}{\dfrac{3}{x^{2}-16}} \nonumber \]

Suluhisho

Andika upya sehemu tata kama mgawanyiko. \[\dfrac{6}{x-4} \div \dfrac{3}{x^{2}-16} \nonumber \]

Andika upya kama bidhaa ya mara ya kwanza ya kurudi kwa pili.Andika upya kama bidhaa ya mara ya kwanza ya kurudi kwa pili.Andika upya kama bidhaa ya mara ya kwanza ya kurudi kwa pili.

\[\dfrac{6}{x-4} \cdot \dfrac{x^{2}-16}{3} \nonumber \]

Factor.

\[\dfrac{3 \cdot 2}{x-4} \cdot \dfrac{(x-4)(x+4)}{3} \nonumber \]

Kuzidisha.

\[\dfrac{3 \cdot 2(x-4)(x+4)}{3(x-4)}\nonumber \]

Ondoa mambo ya kawaida.

\[\dfrac{\cancel{3} \cdot 2 \cancel {(x-4)}(x+4)}{\cancel{3} \cancel {(x-4)}} \nonumber \]

Kurahisisha.

\[2(x+4) \nonumber \]

Je, kuna thamani yoyote (s) ya\(x\) kwamba haipaswi kuruhusiwa? awali tata busara kujieleza alikuwa denominators ya\(x-4\) na\(x^2-16\)$.$Maneno haya yatakuwa haijulikani ikiwa\(x=4\) au\(x=-4\).

Jaribu\(\PageIndex{1}\)

Kurahisisha kujieleza kwa busara kwa kuandika kama mgawanyiko:\[\dfrac{\dfrac{2}{x^{2}-1}}{\dfrac{3}{x+1}} \nonumber \]

Jibu

\(\dfrac{2}{3(x-1)}\)

Jaribu\(\PageIndex{2}\)

Kurahisisha kujieleza kwa busara kwa kuandika kama mgawanyiko:\[\dfrac{\dfrac{1}{x^{2}-7 x+12}}{\dfrac{2}{x-4}} \nonumber \]

Jibu

\(\dfrac{1}{2(x-3)}\)

Sehemu za sehemu hufanya kama alama za makundi. Hivyo kufuata Amri ya Uendeshaji, sisi kurahisisha nambari na denominator iwezekanavyo kabla ya kufanya mgawanyiko.

Mfano\(\PageIndex{2}\)

Kurahisisha kujieleza kwa busara kwa kuandika kama mgawanyiko:\[\dfrac{\dfrac{1}{3}+\dfrac{1}{6}}{\dfrac{1}{2}-\dfrac{1}{3}} \nonumber \]

Suluhisho

Kurahisisha namba na denominator. Pata LCD na uongeze sehemu ndogo katika namba. Pata LCD na uondoe sehemu ndogo katika denominator.

\[\dfrac{\dfrac{1 \cdot {\color{red}2}}{3 \cdot {\color{red}2}}+\dfrac{1}{6}}{\dfrac{1 \cdot {\color{red}3}}{2 \cdot {\color{red}3}}-\dfrac{1 \cdot {\color{red}2}}{3 \cdot {\color{red}2}}} \nonumber \]

Kurahisisha namba na denominator.

\[\dfrac{\dfrac{2}{6}+\dfrac{1}{6}}{\dfrac{3}{6}-\dfrac{2}{6}} \nonumber \]

Andika upya maneno mazuri ya busara kama tatizo la mgawanyiko.

\[\dfrac{3}{6} \div \dfrac{1}{6} \nonumber \]

Panua kwanza kwa usawa wa pili.

\[\dfrac{3}{6} \cdot \dfrac{6}{1} \nonumber \]

Kurahisisha.

\[3 \nonumber \]

Jaribu\(\PageIndex{3}\)

Kurahisisha kujieleza kwa busara kwa kuandika kama mgawanyiko:\[\dfrac{\dfrac{1}{2}+\dfrac{2}{3}}{\dfrac{5}{6}+\dfrac{1}{12}} \nonumber \]

Jibu

\(\dfrac{14}{11}\)

Jaribu\(\PageIndex{4}\)

Kurahisisha kujieleza kwa busara kwa kuandika kama mgawanyiko:\[\dfrac{\dfrac{3}{4}-\dfrac{1}{3}}{\dfrac{1}{8}+\dfrac{5}{6}} \nonumber \]

Jibu

\(\dfrac{10}{23}\)

Sisi kufuata utaratibu huo wakati tata kujieleza busara ina vigezo.

Mfano\(\PageIndex{3}\): How to Simplify a Complex Rational Expression using Division

Kurahisisha kujieleza kwa busara kwa kuandika kama mgawanyiko:\[\dfrac{\dfrac{1}{x}+\dfrac{1}{y}}{\dfrac{x}{y}-\dfrac{y}{x}} \nonumber \]

Suluhisho

Hatua ya 1. Kurahisisha nambari.

Sisi kurahisisha jumla katika na denominator. nambari na tofauti katika denominator.

\[\dfrac{\dfrac{1}{x}+\dfrac{1}{y}}{\dfrac{x}{y}-\dfrac{y}{x}} \nonumber \]

Pata denominator ya kawaida na uongeze sehemu ndogo katika namba.

\[\dfrac{\dfrac{1 \cdot {\color{red}y}}{x \cdot {\color{red}y}}+\dfrac{1 \cdot {\color{red}x}}{y \cdot {\color{red}x}}}{\dfrac{x \cdot {\color{red}x}}{y \cdot {\color{red}x}}-\dfrac{y \cdot {\color{red}y}}{x \cdot {\color{red}y}}} \nonumber \]

\[\dfrac{\dfrac{y}{x y}+\dfrac{x}{x y}}{\dfrac{x^{2}}{x y}-\dfrac{y^{2}}{x y}} \nonumber \]

Pata denominator ya kawaida na uondoe sehemu ndogo katika denominator.

\[\dfrac{\dfrac{y+x}{x y}}{\dfrac{x^{2}-y^{2}}{x y}} \nonumber \]

Sasa tuna moja tu ya kujieleza kwa busara katika nambari na moja katika denominator.

Hatua ya 2. Andika upya maneno mazuri ya busara kama tatizo la mgawanyiko.

Tunaandika namba iliyogawanywa na denominator.

\[\left(\dfrac{y+x}{x y}\right) \div\left(\dfrac{x^{2}-y^{2}}{x y}\right) \nonumber \]

Hatua ya 3. Gawanya maneno.

Panua kwanza kwa usawa wa pili.

\[\left(\dfrac{y+x}{x y}\right) \cdot\left(\dfrac{x y}{x^{2}-y^{2}}\right) \nonumber \]

Fanya maneno yoyote ikiwa inawezekana.

\[\dfrac{x y(y+x)}{x y(x-y)(x+y)} \nonumber \]

Ondoa mambo ya kawaida.

\[\dfrac{\cancel {x y}\cancel {(y+x)}}{\cancel {x y}(x-y)\cancel {(x+y)}} \nonumber \]

Kurahisisha.

\[\dfrac{1}{x-y} \nonumber \]

Jaribu\(\PageIndex{5}\)

Kurahisisha kujieleza kwa busara kwa kuandika kama mgawanyiko:\[\dfrac{\dfrac{1}{x}+\dfrac{1}{y}}{\dfrac{1}{x}-\dfrac{1}{y}} \nonumber \]

Jibu

\(\dfrac{y+x}{y-x}\)

Jaribu\(\PageIndex{6}\)

Kurahisisha kujieleza kwa busara kwa kuandika kama mgawanyiko:\[\dfrac{\dfrac{1}{a}+\dfrac{1}{b}}{\dfrac{1}{a^{2}}-\dfrac{1}{b^{2}}} \nonumber \]

Jibu

\(\dfrac{a b}{b-a}\)

Sisi muhtasari hatua hapa.

Jinsi ya kurahisisha kujieleza kwa busara kwa kuandika kama mgawanyiko.

1. Andika upya maneno mazuri ya busara kama tatizo la mgawanyiko.
2. Gawanya maneno.

Mfano\(\PageIndex{4}\)

Kurahisisha kujieleza kwa busara kwa kuandika kama mgawanyiko:\[\dfrac{n-\dfrac{4 n}{n+5}}{\dfrac{1}{n+5}+\dfrac{1}{n-5}} \nonumber \]

Suluhisho

Kurahisisha namba na denominator. Pata denominators ya kawaida kwa nambari na denominator.

\[\dfrac{\dfrac{n{\color{red}(n+5)}}{1{\color{red}(n+5)}}-\dfrac{4 n}{n+5}}{\dfrac{1{\color{red}(n-5)}}{(n+5){\color{red}(n-5)}}+\dfrac{1{\color{red}(n+5)}}{(n-5){\color{red}(n+5)}}} \nonumber \]

Kurahisisha nambari.

\[\dfrac{\dfrac{n^{2}+5 n}{n+5}-\dfrac{4 n}{n+5}} {\dfrac{n-5}{(n+5)(n-5)}+\dfrac{n+5}{(n-5)(n+5)}} \nonumber \]

Ondoa maneno ya busara katika nambari na uongeze kwenye denominator.

\[\dfrac{\dfrac{n^{2}+5 n-4 n}{n+5}}{\dfrac{n-5+n+5}{(n+5)(n-5)}} \nonumber \]

Kurahisisha. (Sasa tuna kujieleza moja ya busara juu ya kujieleza moja ya busara.)

\[\dfrac{\dfrac{n^{2}+n}{n+5}}{\dfrac {2n}{(n+5)(n-5)}} \nonumber \]

Andika upya kama mgawanyiko wa sehemu.

\[\dfrac{n^{2}+n}{n+5} \div \dfrac{2 n}{(n+5)(n-5)} \nonumber \]

Panua mara ya kwanza usawa wa pili.

\[\dfrac{n^{2}+n}{n+5} \cdot \dfrac{(n+5)(n-5)}{2 n} \nonumber \]

Fanya maneno yoyote ikiwa inawezekana.

\[\dfrac{n(n+1)(n+5)(n-5)}{(n+5) 2 n} \nonumber \]

Ondoa mambo ya kawaida.

\[\dfrac{\cancel{n}(n+1)\cancel {(n+5)}(n-5)}{\cancel {(n+5)} 2 \cancel {n}} \nonumber \]

Kurahisisha.

\[\dfrac{(n+1)(n-5)}{2} \nonumber \]

Jaribu\(\PageIndex{7}\)

Kurahisisha kujieleza kwa busara kwa kuandika kama mgawanyiko:\[\dfrac{b-\dfrac{3 b}{b+5}}{\dfrac{2}{b+5}+\dfrac{1}{b-5}} \nonumber \]

Jibu

\(\dfrac{b(b+2)(b-5)}{3 b-5}\)

Jaribu\(\PageIndex{8}\)

Kurahisisha kujieleza kwa busara kwa kuandika kama mgawanyiko:\[\dfrac{1-\dfrac{3}{c+4}}{\dfrac{1}{c+4}+\dfrac{c}{3}} \nonumber \]

Jibu

\(\dfrac{3}{c+3}\)

Kurahisisha Ufafanuzi wa busara tata kwa kutumia LCD

Sisi “tulifuta” sehemu ndogo kwa kuzidisha na LCD wakati tulipotatua equations na sehemu ndogo. Tunaweza kutumia mkakati huo hapa ili kurahisisha maneno mazuri ya busara. Tutazidisha nambari na denominator na LCD ya maneno yote ya busara.

Hebu tuangalie maneno mazuri ya busara tuliyorahisisha njia moja katika Mfano 7.4.2. Tutaifanya rahisi hapa kwa kuzidisha nambari na denominator na LCD. Wakati sisi kuzidisha na\(\dfrac{LCD}{LCD}\) sisi ni kuzidisha kwa 1, hivyo thamani anakaa sawa.

Mfano\(\PageIndex{5}\)

Kurahisisha kujieleza kwa busara kwa kutumia LCD:\[\dfrac{\dfrac{1}{3}+\dfrac{1}{6}}{\dfrac{1}{2}-\dfrac{1}{3}} \nonumber \]

Suluhisho

LCD ya sehemu zote katika kujieleza nzima ni 6.

Futa sehemu ndogo kwa kuzidisha nambari na denominator kwa LCD hiyo.

\[\dfrac{{\color{red}6} \cdot\left(\dfrac{1}{3}+\dfrac{1}{6}\right)}{{\color{red}6} \cdot\left(\dfrac{1}{2}-\dfrac{1}{3}\right)} \nonumber \]

Kusambaza.

\[\dfrac{6 \cdot \dfrac{1}{3}+6 \cdot \dfrac{1}{6}}{6 \cdot \dfrac{1}{2}-6 \cdot \dfrac{1}{3}} \nonumber \]

Kurahisisha.

\[\dfrac{2+1}{3-2} \nonumber \]

\[\dfrac{3}{1}\nonumber \]

\[3\nonumber \]

Jaribu\(\PageIndex{9}\)

Kurahisisha kujieleza kwa busara kwa kutumia LCD:\[\dfrac{\dfrac{1}{2}+\dfrac{1}{5}}{\dfrac{1}{10}+\dfrac{1}{5}} \nonumber \]

Jibu

\(\dfrac{7}{3}\)

Jaribu\(\PageIndex{10}\)

Kurahisisha kujieleza kwa busara kwa kutumia LCD:\[\dfrac{\dfrac{1}{4}+\dfrac{3}{8}}{\dfrac{1}{2}-\dfrac{5}{16}} \nonumber \]

Jibu

\(\dfrac{10}{3}\)

Tutatumia mfano sawa na katika Mfano 7.4.3. Chagua njia ipi inayofanya kazi bora kwako.

Mfano\(\PageIndex{6}\): How to Simplify a Complex Rational Expressing using the LCD

Kurahisisha kujieleza kwa busara kwa kutumia LCD:\[\dfrac{\dfrac{1}{x}+\dfrac{1}{y}}{\dfrac{x}{y}-\dfrac{y}{x}} \nonumber \]

Suluhisho

Hatua ya 1. Pata LCD ya vipande vyote katika kujieleza ni ngumu ya busara.

LCD ya sehemu zote\(xy\).

\[\dfrac{\dfrac{1}{x}+\dfrac{1}{y}}{\dfrac{x}{y}-\dfrac{y}{x}} \nonumber \]

Hatua ya 2. Panua nambari na denominator na LCD.

Kuzidisha wote namba na denominator na\(xy\).

\[\dfrac{{\color{red}x y} \cdot\left(\dfrac{1}{x}+\dfrac{1}{y}\right)}{{\color{red}x y} \cdot\left(\dfrac{x}{y}-\dfrac{y}{x}\right)} \nonumber \]

Hatua ya 3. Kurahisisha usemi.

Kusambaza.

\[\dfrac{xy \cdot \dfrac{1}{x}+xy \cdot \dfrac{1}{y}}{xy \cdot \dfrac{x}{y}-xy \cdot \dfrac{y}{x}} \nonumber \]

\[\dfrac{y+x}{x^{2}-y^{2}} \nonumber \]

Kurahisisha.

\[\dfrac{\cancel{(y+x)}}{(x-y)\cancel{(x+y)}} \nonumber \]

Ondoa mambo ya kawaida.

\[\dfrac{1}{x-y} \nonumber \]

Jaribu\(\PageIndex{11}\)

Kurahisisha kujieleza kwa busara kwa kutumia LCD:\[\dfrac{\dfrac{1}{a}+\dfrac{1}{b}}{\dfrac{a}{b}+\dfrac{b}{a}} \nonumber \]

Jibu

\(\dfrac{b+a}{a^{2}+b^{2}}\)

Jaribu\(\PageIndex{12}\)

Kurahisisha kujieleza kwa busara kwa kutumia LCD:\[\dfrac{\dfrac{1}{x^{2}}-\dfrac{1}{y^{2}}}{\dfrac{1}{x}+\dfrac{1}{y}} \nonumber \]

Jibu

\(\dfrac{y-x}{x y}\)

Jinsi ya kurahisisha kujieleza kwa busara kwa kutumia LCD.

1. Panua nambari na denominator na LCD.
2. Kurahisisha usemi.

Hakikisha kuanza kwa kuzingatia madhehebu yote ili uweze kupata LCD.

Mfano\(\PageIndex{7}\)

Kurahisisha kujieleza kwa busara kwa kutumia LCD:\[\dfrac{\dfrac{2}{x+6}}{\dfrac{4}{x-6}-\dfrac{4}{x^{2}-36}} \nonumber \]

Suluhisho

Pata LCD ya vipande vyote katika kujieleza kwa busara. LCD ni:

\[x^{2}-36=(x+6)(x-6) \nonumber \]

Panua nambari na denominator na LCD.

\[\dfrac{(x+6)(x-6) \dfrac{2}{x+6}}{(x+6)(x-6)\left(\dfrac{4}{x-6}-\dfrac{4}{(x+6)(x-6)}\right)} \nonumber \]

Kurahisisha usemi.

Kusambaza katika denominator.

\[\dfrac{(x+6)(x-6) \dfrac{2}{x+6}}{{\color{red}(x+6)(x-6)}\left(\dfrac{4}{x-6}\right)-{\color{red}(x+6)(x-6)}\left(\dfrac{4}{(x+6)(x-6)}\right)} \nonumber \]

Kurahisisha.

\[\dfrac{\cancel{(x+6)}(x-6) \dfrac{2}{\cancel{x+6}}}{{\color{red}(x+6)\cancel{(x-6)}}\left(\dfrac{4}{x-6}\right)-{\color{red}\cancel{(x+6)(x-6)}}\left(\dfrac{4}{\cancel{(x+6)(x-6)}}\right)} \nonumber \]

Kurahisisha.

\[\dfrac{2(x-6)}{4(x+6)-4} \nonumber \]

Ili kurahisisha denominator, kusambaza na kuchanganya maneno kama hayo.

\[\dfrac{2(x-6)}{4 x+20} \nonumber \]

Sababu ya denominator.

\[\dfrac{2(x-6)}{4(x+5)} \nonumber \]

Ondoa mambo ya kawaida.

\[\dfrac{\cancel{2}(x-6)}{\cancel{2} \cdot 2(x+5)} \nonumber \]

Kurahisisha.

\[\dfrac{x-6}{2(x+5)} \nonumber \]

Angalia kwamba hakuna mambo zaidi ya kawaida kwa nambari na denominator.

Jaribu\(\PageIndex{13}\)

Kurahisisha kujieleza kwa busara kwa kutumia LCD:\[\dfrac{\dfrac{3}{x+2}}{\dfrac{5}{x-2}-\dfrac{3}{x^{2}-4}} \nonumber \]

Jibu

\(\dfrac{3(x-2)}{5 x+7}\)

Jaribu\(\PageIndex{14}\)

Kurahisisha kujieleza kwa busara kwa kutumia LCD:\[\dfrac{\dfrac{2}{x-7}-\dfrac{1}{x+7}}{\dfrac{6}{x+7}-\dfrac{1}{x^{2}-49}} \nonumber \]

Jibu

\(\dfrac{x+21}{6 x-43}\)

Hakikisha kuzingatia denominators kwanza. Kuendelea kwa makini kama hesabu unaweza kupata messy!

Mfano\(\PageIndex{8}\)

Kurahisisha kujieleza kwa busara kwa kutumia LCD:\[\dfrac{\dfrac{4}{m^{2}-7 m+12}}{\dfrac{3}{m-3}-\dfrac{2}{m-4}} \nonumber \]

Suluhisho

Pata LCD ya vipande vyote katika kujieleza kwa busara.

LCD ni\((m−3)(m−4)\).

Panua nambari na denominator na LCD.

\[\dfrac{(m-3)(m-4) \dfrac{4}{(m-3)(m-4)}}{(m-3)(m-4)\left(\dfrac{3}{m-3}-\dfrac{2}{m-4}\right)} \nonumber \]

Kurahisisha.

\[\dfrac{\cancel {(m-3)(m-4)}\dfrac{4}{\cancel {(m-3)(m-4)}}}{\cancel {(m-3)}(m-4)\left(\dfrac{3}{\cancel {m-3}}\right)-(m-3)\cancel {(m-4)}\left(\dfrac{2}{\cancel {m-4}}\right)} \nonumber\]

Kurahisisha.

\[\dfrac{4}{3(m-4)-2(m-3)} \nonumber \]

Kusambaza.

\[\dfrac{4}{3m-12-2m+6} \nonumber \]

Kuchanganya kama maneno.

\[\dfrac{4}{m-6} \nonumber \]

Jaribu\(\PageIndex{15}\)

Kurahisisha kujieleza kwa busara kwa kutumia LCD:\[\dfrac{\dfrac{3}{x^{2}+7 x+10}}{\dfrac{4}{x+2}+\dfrac{1}{x+5}} \nonumber \]

Jibu

\(\dfrac{3}{5 x+22}\)

Jaribu\(\PageIndex{16}\)

Kurahisisha kujieleza kwa busara kwa kutumia LCD:\[\dfrac{\dfrac{4 y}{y+5}+\dfrac{2}{y+6}}{\dfrac{3 y}{y^{2}+11 y+30}} \nonumber \]

Jibu

\(\dfrac{2\left(2 y^{2}+13 y+5\right)}{3 y}\)

Mfano\(\PageIndex{9}\)

Kurahisisha kujieleza kwa busara kwa kutumia LCD:\[\dfrac{\dfrac{y}{y+1}}{1+\dfrac{1}{y-1}} \nonumber \]

Suluhisho

Pata LCD ya vipande vyote katika kujieleza kwa busara.

LCD ni\((y+1)(y−1)\).

Panua nambari na denominator na LCD.

\[\dfrac{(y+1)(y-1) \dfrac{y}{y+1}}{(y+1)(y-1)\left(1+\dfrac{1}{y-1}\right)} \nonumber \]

Kusambaza katika denominator na kurahisisha.

\[\dfrac{\cancel{(y+1)}(y-1) \dfrac{y}{\cancel {y+1}}}{(y+1)(y-1)(1)+(y+1)\cancel{(y-1)}\left(\dfrac{1}{\cancel{(y-1)}}\right)} \nonumber \]

Kurahisisha.

\[\dfrac{(y-1) y}{(y+1)(y-1)+(y+1)} \nonumber \]

Kurahisisha denominator na uondoke namba ya hesabu.

\[\dfrac{y(y-1)}{y^{2}-1+y+1} \nonumber \]

\[\dfrac{y(y-1)}{y^{2}+y} \nonumber \]

Fanya denominator na uondoe mambo ya kawaida na namba.

\[\dfrac{\cancel {y}(y-1)}{\cancel {y}(y+1)} \nonumber \]

Kurahisisha.

\[\dfrac{y-1}{y+1} \nonumber \]

Jaribu\(\PageIndex{17}\)

Kurahisisha kujieleza kwa busara kwa kutumia LCD:\[\dfrac{\dfrac{x}{x+3}}{1+\dfrac{1}{x+3}} \nonumber \]

Jibu

\(\dfrac{x}{x+4}\)

Jaribu\(\PageIndex{18}\)

Kurahisisha kujieleza kwa busara kwa kutumia LCD:\[\dfrac{1+\dfrac{1}{x-1}}{\dfrac{3}{x+1}} \nonumber \]

Jibu

\(\dfrac{x(x+1)}{3(x-1)}\)

Fikia rasilimali hii ya mtandaoni kwa maelekezo ya ziada na mazoezi na sehemu ndogo.

• Fractions tata

Dhana muhimu

• Jinsi ya kurahisisha kujieleza kwa busara kwa kuandika kama mgawanyiko.
  1. Kurahisisha namba na denominator.
  2. Andika upya maneno mazuri ya busara kama tatizo la mgawanyiko.
  3. Gawanya maneno.
• Jinsi ya kurahisisha kujieleza kwa busara kwa kutumia LCD.
  1. Pata LCD ya vipande vyote katika kujieleza kwa busara.
  2. Panua nambari na denominator na LCD.
  3. Kurahisisha usemi.

faharasa

tata busara kujieleza

Maneno mazuri ya busara ni kujieleza kwa busara ambayo namba na/au denominator ina kujieleza kwa busara.