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8.6: Gawanya Maneno makubwa

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    Malengo ya kujifunza

    Mwishoni mwa sehemu hii, utaweza:

    • Gawanya maneno makubwa
    • Rationalize denominator neno moja
    • Rationalize mbili denominator mrefu

    Kabla ya kuanza, fanya jaribio hili la utayari.

    1. Kurahisisha:\(\dfrac{30}{48}\).
      Kama amekosa tatizo hili, mapitio Mfano 1.24.
    2. Kurahisisha:\(x^{2}⋅x^{4}\).
      Kama amekosa tatizo hili, mapitio Mfano 5.12.
    3. Kuzidisha:\((7+3x)(7−3x)\).
      Kama amekosa tatizo hili, mapitio Mfano 5.32.

    Gawanya maneno makubwa

    Tumetumia Mali ya Quotient ya Maneno ya Radical ili kurahisisha mizizi ya sehemu ndogo. Tutahitaji kutumia mali hii 'katika reverse' ili kurahisisha sehemu na radicals. Sisi kutoa Quotient Mali ya maneno Radical tena kwa ajili ya kumbukumbu rahisi. Kumbuka, sisi kudhani vigezo vyote ni kubwa kuliko au sawa na sifuri ili hakuna baa thamani kabisa zinahitajika.

    Ufafanuzi\(\PageIndex{1}\): Quotient Property of Radical Expressions

    Ikiwa\(\sqrt[n]{a}\) na\(\sqrt[n]{b}\) ni namba halisi,\(b≠0\), na kwa integer yoyote\(n≥2\) basi,

    \(\sqrt[n]{\dfrac{a}{b}}=\dfrac{\sqrt[n]{a}}{\sqrt[n]{b}} \quad \text { and } \quad \dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\dfrac{a}{b}}\)

    Tutatumia Mali ya Quotient ya maneno makubwa wakati sehemu tunayoanza na ni quotient ya radicals mbili, na wala radicand ni nguvu kamili ya index. Tunapoandika sehemu katika radical moja, tunaweza kupata mambo ya kawaida katika nambari na denominator.

    Mfano\(\PageIndex{1}\)

    Kurahisisha:

    1. \(\dfrac{\sqrt{72 x^{3}}}{\sqrt{162 x}}\)
    2. \(\dfrac{\sqrt[3]{32 x^{2}}}{\sqrt[3]{4 x^{5}}}\)

    Suluhisho:

    a.

    \(\dfrac{\sqrt{72 x^{3}}}{\sqrt{162 x}}\)

    Andika upya kwa kutumia mali ya quotient,

    \(\sqrt{\dfrac{72 x^{3}}{162 x}}\)

    Ondoa mambo ya kawaida.

    \(\sqrt{\dfrac{\cancel{18} \cdot 4 \cdot x^{2} \cdot \cancel{x}}{\cancel{18} \cdot 9 \cdot \cancel{x}}}\)

    Kurahisisha.

    \(\sqrt{\dfrac{4 x^{2}}{9}}\)

    Kurahisisha radical.

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

    b.

    \(\dfrac{\sqrt[3]{32 x^{2}}}{\sqrt[3]{4 x^{5}}}\)

    Andika upya kwa kutumia mali ya quotient,\(\dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\dfrac{a}{b}}\).

    \(\sqrt[3]{\dfrac{32 x^{2}}{4 x^{5}}}\)

    Kurahisisha sehemu chini ya radical.

    \(\sqrt[3]{\dfrac{8}{x^{3}}}\)

    Kurahisisha radical.

    \(\dfrac{2}{x}\)

    Zoezi\(\PageIndex{1}\)

    Kurahisisha:

    1. \(\dfrac{\sqrt{50 s^{3}}}{\sqrt{128 s}}\)
    2. \(\dfrac{\sqrt[3]{56 a}}{\sqrt[3]{7 a^{4}}}\)
    Jibu
    1. \(\dfrac{5s}{8}\)
    2. \(\dfrac{2}{a}\)
    Zoezi\(\PageIndex{2}\)

    Kurahisisha:

    1. \(\dfrac{\sqrt{75 q^{5}}}{\sqrt{108 q}}\)
    2. \(\dfrac{\sqrt[3]{72 b^{2}}}{\sqrt[3]{9 b^{5}}}\)
    Jibu
    1. \(\dfrac{5 q^{2}}{6}\)
    2. \(\dfrac{2}{b}\)
    Mfano\(\PageIndex{2}\)

    Kurahisisha:

    1. \(\dfrac{\sqrt{147 a b^{8}}}{\sqrt{3 a^{3} b^{4}}}\)
    2. \(\dfrac{\sqrt[3]{-250 m n^{-2}}}{\sqrt[3]{2 m^{-2} n^{4}}}\)

    Suluhisho:

    a.

    \(\dfrac{\sqrt{147 a b^{8}}}{\sqrt{3 a^{3} b^{4}}}\)

    Andika upya kwa kutumia mali ya quotient.

    \(\sqrt{\dfrac{147 a b^{8}}{3 a^{3} b^{4}}}\)

    Ondoa mambo ya kawaida katika sehemu.

    \(\sqrt{\dfrac{49 b^{4}}{a^{2}}}\)

    Kurahisisha radical.

    \(\dfrac{7 b^{2}}{a}\)

    b.

    \(\dfrac{\sqrt[3]{-250 m n^{-2}}}{\sqrt[3]{2 m^{-2} n^{4}}}\)

    Andika upya kwa kutumia mali ya quotient.

    \(\sqrt[3]{\dfrac{-250 m n^{-2}}{2 m^{-2} n^{4}}}\)

    Kurahisisha sehemu chini ya radical.

    \(\sqrt[3]{\dfrac{-125 m^{3}}{n^{6}}}\)

    Kurahisisha radical.

    \(-\dfrac{5 m}{n^{2}}\)

    Zoezi\(\PageIndex{3}\)

    Kurahisisha:

    1. \(\dfrac{\sqrt{162 x^{10} y^{2}}}{\sqrt{2 x^{6} y^{6}}}\)
    2. \(\dfrac{\sqrt[3]{-128 x^{2} y^{-1}}}{\sqrt[3]{2 x^{-1} y^{2}}}\)
    Jibu
    1. \(\dfrac{9 x^{2}}{y^{2}}\)
    2. \(\dfrac{-4 x}{y}\)
    Zoezi\(\PageIndex{4}\)

    Kurahisisha:

    1. \(\dfrac{\sqrt{300 m^{3} n^{7}}}{\sqrt{3 m^{5} n}}\)
    2. \(\dfrac{\sqrt[3]{-81 p q^{-1}}}{\sqrt[3]{3 p^{-2} q^{5}}}\)
    Jibu
    1. \(\dfrac{10 n^{3}}{m}\)
    2. \(\dfrac{-3 p}{q^{2}}\)
    Mfano\(\PageIndex{3}\)

    Kurahisisha:\(\dfrac{\sqrt{54 x^{5} y^{3}}}{\sqrt{3 x^{2} y}}\)

    Suluhisho:

    \(\dfrac{\sqrt{54 x^{5} y^{3}}}{\sqrt{3 x^{2} y}}\)

    Andika upya kwa kutumia mali ya quotient.

    \(\sqrt{\dfrac{54 x^{5} y^{3}}{3 x^{2} y}}\)

    Ondoa mambo ya kawaida katika sehemu.

    \(\sqrt{18 x^{3} y^{2}}\)

    Andika upya radicna kama bidhaa kwa kutumia sababu kubwa zaidi ya mraba.

    \(\sqrt{9 x^{2} y^{2} \cdot 2 x}\)

    Andika upya radical kama bidhaa ya radicals mbili.

    \(\sqrt{9 x^{2} y^{2}} \cdot \sqrt{2 x}\)

    Kurahisisha.

    \(3 x y \sqrt{2 x}\)

    Zoezi\(\PageIndex{5}\)

    Kurahisisha:\(\dfrac{\sqrt{64 x^{4} y^{5}}}{\sqrt{2 x y^{3}}}\)

    Jibu

    \(4 x y \sqrt{2 x}\)

    Zoezi\(\PageIndex{6}\)

    Kurahisisha:\(\dfrac{\sqrt{96 a^{5} b^{4}}}{\sqrt{2 a^{3} b}}\)

    Jibu

    \(4 a b \sqrt{3 b}\)

    Tambua Denominator ya Muda Mmoja

    Kabla ya calculator kuwa chombo cha maisha ya kila siku, kukadiria thamani ya sehemu na radical katika denominator ilikuwa mchakato mbaya sana!

    Kwa sababu hii, mchakato unaoitwa rationalizing denominator ulianzishwa. Sehemu yenye radical katika denominator inabadilishwa kwa sehemu sawa ambayo denominator ni integer. Mizizi ya mraba ya namba ambazo si mraba kamili ni namba zisizo na maana. Wakati sisi rationalize denominator, sisi kuandika sehemu sawa na idadi ya busara katika denominator. Utaratibu huu bado unatumika leo, na ni muhimu katika maeneo mengine ya hisabati, pia.

    Ufafanuzi\(\PageIndex{2}\): Rationalizing the Denominator

    Kutambua denominator ni mchakato wa kubadili sehemu na radical katika denominator kwa sehemu sawa ambayo denominator ni integer.

    Ingawa tuna calculators inapatikana karibu kila mahali, sehemu na radical katika denominator bado lazima rationalized. Haichukuliwi kuwa rahisi kama denominator ina radical.

    Vile vile, kujieleza kwa kiasi kikubwa haukuzingatiwa kuwa rahisi kama radicand ina sehemu.

    Rahisi Radical Maneno

    Maneno makubwa yanachukuliwa kuwa rahisi ikiwa kuna

    • hakuna sababu katika radicand kuwa na nguvu kamili ya index
    • hakuna sehemu ndogo katika radicand
    • hakuna radicals katika denominator ya sehemu

    Ili kurekebisha denominator na mizizi ya mraba, tunatumia mali hiyo\((\sqrt{a})^{2}=a\). Ikiwa sisi mraba mizizi ya mraba isiyo na maana, tunapata namba ya busara.

    Tutatumia mali hii kwa rationalize denominator katika mfano unaofuata.

    Mfano\(\PageIndex{4}\)

    Kurahisisha:

    1. \(\dfrac{4}{\sqrt{3}}\)
    2. \(\sqrt{\dfrac{3}{20}}\)
    3. \(\dfrac{3}{\sqrt{6 x}}\)

    Suluhisho:

    Ili kurekebisha denominator kwa muda mmoja, tunaweza kuzidisha mizizi ya mraba yenyewe. Ili kuweka sehemu sawa, tunazidisha nambari zote na denominator kwa sababu sawa.

    a.

     

    .
    Kuzidisha wote namba na denominator na\(\sqrt{3}\).

    .
    Kurahisisha.

    .
    Jedwali 8.5.1

    b Sisi daima kurahisisha radical katika denominator kwanza, kabla ya kuifanya. Kwa njia hii idadi hukaa ndogo na rahisi kufanya kazi na.

     

    .
    Sehemu sio mraba kamili, hivyo uandike upya kwa kutumia Mali ya Quotient.

    .
    Kurahisisha denominator.

    .
    Panua nambari na denominator na\(\sqrt{5}\).

    .
    Kurahisisha.

    .
    Kurahisisha.

    .
    Jedwali 8.5.2

    c.

     

    .
    Panua nambari na denominator na\(\sqrt{6x}\).

    .
    Kurahisisha.

    .
    Kurahisisha.

    .
    Jedwali 8.5.3
    Zoezi\(\PageIndex{7}\)

    Kurahisisha:

    1. \(\dfrac{5}{\sqrt{3}}\)
    2. \(\sqrt{\dfrac{3}{32}}\)
    3. \(\dfrac{2}{\sqrt{2 x}}\)
    Jibu
    1. \(\dfrac{5 \sqrt{3}}{3}\)
    2. \(\dfrac{\sqrt{6}}{8}\)
    3. \(\dfrac{\sqrt{2 x}}{x}\)
    Zoezi\(\PageIndex{8}\)

    Kurahisisha:

    1. \(\dfrac{6}{\sqrt{5}}\)
    2. \(\sqrt{\dfrac{7}{18}}\)
    3. \(\dfrac{5}{\sqrt{5 x}}\)
    Jibu
    1. \(\dfrac{6 \sqrt{5}}{5}\)
    2. \(\dfrac{\sqrt{14}}{6}\)
    3. \(\dfrac{\sqrt{5 x}}{x}\)

    Tulipotambua mizizi ya mraba, tuliongeza namba na denominator kwa mizizi ya mraba ambayo itatupa mraba kamili chini ya radical katika denominator. Tulipochukua mizizi ya mraba, denominator hakuwa na radical.

    Tutafuata mchakato sawa wa kupatanisha mizizi ya juu. Ili kurekebisha denominator na index ya juu ya radical, tunazidisha nambari na denominator kwa radical ambayo itatupa radicana hiyo ni nguvu kamili ya index. Wakati sisi kurahisisha radical mpya, denominator tena kuwa radical.

    Kwa mfano,

    Mifano miwili ya denominators ya rationalizing inavyoonyeshwa. Mfano wa kwanza ni 1 umegawanyika na mizizi ya mchemraba 2. Kumbuka inafanywa kuwa radicand katika denominator ni 1 nguvu ya 2 na kwamba tunahitaji 2 zaidi kupata mchemraba kamilifu. Tunazidisha nambari na denominator kwa mizizi ya mchemraba wa wingi 2 mraba. Matokeo yake ni mchemraba mizizi 4 imegawanywa na mizizi ya mchemraba wa wingi 2 cubed. Hii simplifies kwa mchemraba mizizi 4 kugawanywa na 2. Mfano wa pili ni 1 umegawanyika na mizizi ya nne 5. Kumbuka inafanywa kuwa radicand katika denominator ni 1 nguvu ya 5 na kwamba tunahitaji 3 zaidi kupata nne kamili. Tunazidisha nambari na denominator kwa mizizi ya nne ya wingi 5 cubed. Matokeo yake ni mizizi ya nne ya 125 imegawanywa na mizizi ya nne ya wingi 5 hadi ya nne. Hii simplifies kwa mizizi ya nne 125 kugawanywa na 5.
    Kielelezo 8.5.14

    Tutatumia mbinu hii katika mifano inayofuata.

    Mfano\(\PageIndex{5}\)

    Kurahisisha:

    1. \(\dfrac{1}{\sqrt[3]{6}}\)
    2. \(\sqrt[3]{\dfrac{7}{24}}\)
    3. \(\dfrac{3}{\sqrt[3]{4 x}}\)

    Suluhisho:

    Ili kurekebisha denominator na mizizi ya mchemraba, tunaweza kuzidisha na mizizi ya mchemraba ambayo itatupa mchemraba kamili katika radicand katika denominator. Ili kuweka sehemu sawa, tunazidisha nambari zote na denominator kwa sababu sawa.

    a.

     

    .
    radical katika denominator ina sababu moja ya\(6\). Kuzidisha wote nambari na denominator na\(\sqrt[3]{6^{2}}\), ambayo\(2\) inatupa mambo zaidi ya\(6\).

    .
    Kuzidisha. Angalia radicand katika denominator ina\(3\) mamlaka ya\(6\).

    .
    Kurahisisha mizizi ya mchemraba katika denominator.

    .
    Jedwali 8.5.4

    b Sisi daima kurahisisha radical katika denominator kwanza, kabla ya kuifanya. Kwa njia hii idadi hukaa ndogo na rahisi kufanya kazi na.

     

    .
    Sehemu sio mchemraba kamili, hivyo uandike upya kwa kutumia Mali ya Quotient.

    .
    Kurahisisha denominator.

    .
    Panua nambari na denominator na\(\sqrt[3]{3^{2}}\). Hii itatupa\(3\) sababu za\(3\).

    .
    Kurahisisha.

    .
    Kumbuka,\(\sqrt[3]{3^{3}}=3\).

    .
    Kurahisisha.

    .
    Jedwali 8.5.5

    c.

     

    .
    Andika upya radicna kuonyesha mambo.

    .
    Nyingi nambari na denominator na\(\sqrt[3]{2 \cdot x^{2}}\). Hii itapata sisi\(3\) sababu ya\(2\) na\(3\) mambo ya\(x\).

    .
    Kurahisisha.

    .
    Kurahisisha radical katika denominator.

    .
    Jedwali 8.5.6
    Zoezi\(\PageIndex{9}\)

    Kurahisisha:

    1. \(\dfrac{1}{\sqrt[3]{7}}\)
    2. \(\sqrt[3]{\dfrac{5}{12}}\)
    3. \(\dfrac{5}{\sqrt[3]{9 y}}\)
    Jibu
    1. \(\dfrac{\sqrt[3]{49}}{7}\)
    2. \(\dfrac{\sqrt[3]{90}}{6}\)
    3. \(\dfrac{5 \sqrt[3]{3 y^{2}}}{3 y}\)
    Zoezi\(\PageIndex{10}\)

    Kurahisisha:

    1. \(\dfrac{1}{\sqrt[3]{2}}\)
    2. \(\sqrt[3]{\dfrac{3}{20}}\)
    3. \(\dfrac{2}{\sqrt[3]{25 n}}\)
    Jibu
    1. \(\dfrac{\sqrt[3]{4}}{2}\)
    2. \(\dfrac{\sqrt[3]{150}}{10}\)
    3. \(\dfrac{2 \sqrt[3]{5 n^{2}}}{5 n}\)
    Mfano\(\PageIndex{6}\)

    Kurahisisha:

    1. \(\dfrac{1}{\sqrt[4]{2}}\)
    2. \(\sqrt[4]{\dfrac{5}{64}}\)
    3. \(\dfrac{2}{\sqrt[4]{8 x}}\)

    Suluhisho:

    Ili kurekebisha denominator na mizizi ya nne, tunaweza kuzidisha na mizizi ya nne ambayo itatupa nguvu kamili ya nne katika radicand katika denominator. Ili kuweka sehemu sawa, tunazidisha nambari zote na denominator kwa sababu sawa.

    a.

     

    .
    radical katika denominator ina sababu moja ya\(2\).
    Kuzidisha wote nambari na denominator na\(\sqrt[4]{2^{3}}\), ambayo\(3\) inatupa mambo zaidi ya\(2\).

    .
    Kuzidisha. Angalia radicand katika denominator ina\(4\) mamlaka ya\(2\).

    .
    Kurahisisha mizizi ya nne katika denominator.

    .
    Jedwali 8.5.7

    b Sisi daima kurahisisha radical katika denominator kwanza, kabla ya kuifanya. Kwa njia hii idadi hukaa ndogo na rahisi kufanya kazi na.

     

    .
    Sehemu sio nguvu kamili ya nne, hivyo uandike tena kwa kutumia Mali ya Quotient.

    .
    Andika upya radicand katika denominator ili kuonyesha mambo.

    .
    Kurahisisha denominator.

    .
    Panua nambari na denominator na\(\sqrt[4]{2^{2}}\). Hii itatupa\(4\) sababu za\(2\).

    .
    Kurahisisha.

    .
    Kumbuka,\(\sqrt[4]{2^{4}}=2\).

    .
    Kurahisisha.

    .
    Jedwali 8.5.8

    c.

     

    .
    Andika upya radicna kuonyesha mambo.

    .
    Panua nambari na denominator na\(\sqrt[4]{2 \cdot x^{3}}\). Hii itapata sisi\(4\) sababu ya\(2\) na\(4\) mambo ya\(x\).

    .
    Kurahisisha.

    .
    Kurahisisha radical katika denominator.

    .
    Kurahisisha sehemu.

    .
    Jedwali 8.5.9
    Zoezi\(\PageIndex{11}\)

    Kurahisisha:

    1. \(\dfrac{1}{\sqrt[4]{3}}\)
    2. \(\sqrt[4]{\dfrac{3}{64}}\)
    3. \(\dfrac{3}{\sqrt[4]{125 x}}\)
    Jibu
    1. \(\dfrac{\sqrt[4]{27}}{3}\)
    2. \(\dfrac{\sqrt[4]{12}}{4}\)
    3. \(\dfrac{3 \sqrt[4]{5 x^{3}}}{5 x}\)
    Zoezi\(\PageIndex{12}\)

    Kurahisisha:

    1. \(\dfrac{1}{\sqrt[4]{5}}\)
    2. \(\sqrt[4]{\dfrac{7}{128}}\)
    3. \(\dfrac{4}{\sqrt[4]{4 x}}\)
    Jibu
    1. \(\dfrac{\sqrt[4]{125}}{5}\)
    2. \(\dfrac{\sqrt[4]{224}}{8}\)
    3. \(\dfrac{\sqrt[4]{64 x^{3}}}{x}\)

    Rationalize Denominator ya Muda Mbili

    Wakati denominator ya sehemu ni jumla au tofauti na mizizi ya mraba, tunatumia Bidhaa ya Conjugates Pattern ili kurekebisha denominator.

    \(\begin{array}{c c}{(a-b)(a+b)} & {(2-\sqrt{5})(2+\sqrt{5})} \\ {a^{2}-b^{2}} &{ 2^{2}-(\sqrt{5})^{2}} \\ {}&{4-5} \\ {}&{-1}\end{array}\)

    Wakati sisi nyingi binomial ambayo ni pamoja na mizizi mraba na conjugate yake, bidhaa haina mizizi mraba.

    Mfano\(\PageIndex{7}\)

    Kurahisisha:\(\dfrac{5}{2-\sqrt{3}}\)

    Suluhisho:

      .
    Panua namba na denominator kwa conjugate ya denominator. .
    Kuzidisha conjugates katika denominator. .
    Kurahisisha denominator. .
    Kurahisisha denominator. .
    Kurahisisha. .
    Jedwali 8.5.10
    Zoezi\(\PageIndex{13}\)

    Kurahisisha:\(\dfrac{3}{1-\sqrt{5}}\).

    Jibu

    \(-\dfrac{3(1+\sqrt{5})}{4}\)

    Zoezi\(\PageIndex{14}\)

    Kurahisisha:\(\dfrac{2}{4-\sqrt{6}}\).

    Jibu

    \(\dfrac{4+\sqrt{6}}{5}\)

    Taarifa hatukuwa kusambaza\(5\) katika jibu la mfano wa mwisho. Kwa kuacha matokeo factored tunaweza kuona kama kuna mambo yoyote ambayo inaweza kuwa ya kawaida kwa wote namba na denominator.

    Mfano\(\PageIndex{8}\)

    Kurahisisha:\(\dfrac{\sqrt{3}}{\sqrt{u}-\sqrt{6}}\).

    Suluhisho:

      .
    Panua namba na denominator kwa conjugate ya denominator. .
    Kuzidisha conjugates katika denominator. .
    Kurahisisha denominator. .
    Jedwali 8.5.11
    Zoezi\(\PageIndex{15}\)

    Kurahisisha:\(\dfrac{\sqrt{5}}{\sqrt{x}+\sqrt{2}}\).

    Jibu

    \(\dfrac{\sqrt{5}(\sqrt{x}-\sqrt{2})}{x-2}\)

    Zoezi\(\PageIndex{16}\)

    Kurahisisha:\(\dfrac{\sqrt{10}}{\sqrt{y}-\sqrt{3}}\)

    Jibu

    \(\dfrac{\sqrt{10}(\sqrt{y}+\sqrt{3})}{y-3}\)

    Kuwa makini na ishara wakati unapozidi. Nambari na denominator huonekana sawa sana wakati unapozidisha na conjugate.

    Mfano\(\PageIndex{9}\)

    Kurahisisha:\(\dfrac{\sqrt{x}+\sqrt{7}}{\sqrt{x}-\sqrt{7}}\).

    Suluhisho:

      .
    Panua namba na denominator kwa conjugate ya denominator. .
    Kuzidisha conjugates katika denominator. .
    Kurahisisha denominator. .
    Jedwali 8.5.12

    Hatuna mraba namba. Kuiacha kwa fomu iliyosababishwa, tunaweza kuona hakuna sababu za kawaida za kuondoa kutoka kwa nambari na denominator.

    Zoezi\(\PageIndex{17}\)

    Kurahisisha:\(\dfrac{\sqrt{p}+\sqrt{2}}{\sqrt{p}-\sqrt{2}}\).

    Jibu

    \(\dfrac{(\sqrt{p}+\sqrt{2})^{2}}{p-2}\)

    Zoezi\(\PageIndex{18}\)

    Kurahisisha:\(\dfrac{\sqrt{q}-\sqrt{10}}{\sqrt{q}+\sqrt{10}}\)

    Jibu

    \(\dfrac{(\sqrt{q}-\sqrt{10})^{2}}{q-10}\)

    Dhana muhimu

    • Mali ya Quotient ya Maneno makubwa
      • Ikiwa\(\sqrt[n]{a}\) na\(\sqrt[n]{b}\) ni namba halisi,\(b≠0\), na kwa integer yoyote\(n≥2\) basi,\(\sqrt[n]{\dfrac{a}{b}}=\dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}\) na\(\dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\dfrac{a}{b}}\)
    • Rahisi Radical Maneno
      • Maneno makubwa yanachukuliwa kuwa rahisi ikiwa kuna:
        • hakuna sababu katika radicand kuwa na nguvu kamili ya index
        • hakuna sehemu ndogo katika radicand
        • hakuna radicals katika denominator ya sehemu

    faharasa

    rationalizing denominator
    Kutambua denominator ni mchakato wa kubadili sehemu na radical katika denominator kwa sehemu sawa ambayo denominator ni integer.