# 9.4: Rationalize sehemu za Aljebraic

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##### Ufafanuzi: Denominator ya Maneno ya busara

Ikiwa denominator ya kujieleza kwa busara ina kiasi au tofauti zinazohusisha radicals, ni fomu nzuri ya daima kupitisha denominator kwa kuzidisha namba na denominator kwa conjugate ya denominator.

Mchanganyiko wa denominator una maneno sawa, lakini shughuli za kinyume (kuongeza au kuondoa).

##### Mfano Template:index

Rationalize Denominator na Kurahisisha:

1. $$\dfrac{1}{1 − \sqrt{x}}$$
2. $$\dfrac{1}{\sqrt{x} − \sqrt{y}}$$
3. $$\dfrac{\sqrt{x} + \sqrt{y}}{\sqrt{x} − \sqrt{y}}$$

Suluhisho

1. $$\begin{array} &&\dfrac{1}{1 − \sqrt{x}} &\text{Example problem} \\ &\dfrac{(1)(1 + \sqrt{x})}{(1 − \sqrt{x})(1 + \sqrt{x})} &\text{Multiply both numerator and denominator by the conjugate, which is \((1+\sqrt{x})$$}\\ &\ dfrac {1 +\ sqrt {x} {1 -\ sqrt {x} +\ sqrt {x}} - (\ sqrt {x}) ^2} &\ Nakala {FOIL denominator.}\\ &\ drac {1 +\ sqrt {x}} {1}\\ sqrt {\ sqrt {\ sqrt {\ sqrt {\ sqrt {\ sqrt {\ sqrt {\ sqrt {\ sqrt {x} {x}}} - (\ sqrt {x}) ^2} &\ maandishi {Ondoa maneno kinyume kwamba jumla ya sifuri.}\\ &\ dfrac {1 +\ sqrt {x}} {1 合 x} &\ maandishi {mraba mzizi wa$$x$$, wingi squared ni$$x$$.}\\ &\ dfrac {1 +\ sqrt {x}} {1 合 x} &\ maandishi {Jibu la mwisho na denominator rationalized, maana yake ni kwamba hakuna maneno ya mizizi ya mraba katika denominator.} \ mwisho {safu}\)
1. $$\begin{array} &&\dfrac{1}{\sqrt{x} − \sqrt{y}} &\text{Example problem} \\ &\dfrac{(1)(\sqrt{x} + \sqrt{y})}{(\sqrt{x} − \sqrt{y})(\sqrt{x} + \sqrt{y})} &\text{Multiply both numerator and denominator by the conjugate, which is \((\sqrt{x} + \sqrt{y})$$}\\ &\ dfrac {(\ sqrt {x} +\ sqrt {y})} {(\ sqrt {x}) ^2 -\ sqrt {x}\ sqrt {y}\ sqrt {y}}} (\ sqrt {y}) ^2} &\ maandishi {FOIL denominator.}\\ &\ Drac ({\ sqrt {x} +\ sqrt {y})} {(\ sqrt {x}) ^2 -\ kufuta {\ sqrt {x}\ sqrt {y}} +\ kufuta {\ sqrt {x}\ sqrt {y}}} (\ sqrt {y}) ^2} &\ maandishi {Ondoa maneno kinyume ambayo jumla hadi sifuri.}\\ &\ dfrac {\ sqrt {x} +\ sqrt {y}} {x - y} &\ maandishi {Mzizi wa$$x$$ mraba wa$$x$$, wingi wa mraba ni$$y$$, na mizizi ya mraba ya, wingi wa mraba ni$$y$$.}\\ &\ &\ dfrac {x} {x} y} &\ maandishi {x} inator rationalized, maana kwamba hakuna masharti ya mizizi ya mraba katika denominator.} \ mwisho {safu}\)
1. $$\begin{array} &&\dfrac{\sqrt{x} + \sqrt{y}}{\sqrt{x} − \sqrt{y}} &\text{Example problem} \\ &\dfrac{(\sqrt{x} + \sqrt{y})(\sqrt{x} + \sqrt{y})}{(\sqrt{x} − \sqrt{y})(\sqrt{x} + \sqrt{y})} &\text{Multiply both numerator and denominator by the conjugate, which is \((\sqrt{x} + \sqrt{y})$$}\\ &\ dfrac {(\ sqrt {x}) ^2 + 2 (\ sqrt {x}\ sqrt {y}) + (\ sqrt {y}) ^2} {x}) ^2 -\ sqrt {x}\ sqrt {y}\ sqrt {y}}} ^2} &\ maandishi {FOIL nambari na denominator.}\\ &\ dfrac {x + 2\ sqrt {x}\ sqrt {y} + y} {(\ sqrt {x}) ^2}\ kufuta {\ sqrt {x}\ sqrt {y}}} (\ sqrt {y}) ^2} &\ maandishi {Ondoa maneno kinyume kwamba jumla ya sifuri.}\\ &\ dfrac {x + 2\ sqrt {x}\ sqrt {y} + y} {x - y} &\ maandishi {Mzizi wa mraba wa$$x$$, wingi wa mraba ni, na mizizi ya mraba ya$$x$$, kiasi cha$$y$$ mraba ni$$y$$.}\\ &\ dfrac {x + 2\ sqrt {x + 2\ sqrt {x}\ sqrt {y} + y} {x - y} &\ Nakala {Jibu la mwisho na denominator rationalized, maana yake ni kwamba hakuna masharti ya mizizi ya mraba katika denominator.} \ mwisho {safu}\)
##### Zoezi Template:index

Rationalize Denominator na Kurahisisha:

1. $$\dfrac{x}{1 − \sqrt{x}}$$
2. $$\dfrac{1}{1 − \sqrt{x}}$$
3. $$\dfrac{2 \sqrt{x}}{\sqrt{x} − 1}$$
4. $$\dfrac{x − 1}{\sqrt{x} − 1}$$