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8.5: Ongeza, Ondoa, na Kuzidisha Maneno makubwa

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    Malengo ya kujifunza

    Mwishoni mwa sehemu hii, utaweza:

    • Ongeza na uondoe maneno makubwa
    • Kuzidisha maneno makubwa
    • Tumia kuzidisha kwa polynomial kuzidisha maneno makubwa

    Kabla ya kuanza, fanya jaribio hili la utayari.

    1. Ongeza:\(3x^{2}+9x−5−(x^{2}−2x+3)\).
      Ikiwa umekosa tatizo hili, tathmini Mfano 5.5.
    2. Kurahisisha:\((2+a)(4−a)\).
      Kama amekosa tatizo hili, mapitio Mfano 5.28.
    3. Kurahisisha:\((9−5y)^{2}\).
      Kama amekosa tatizo hili, mapitio Mfano 5.31.

    Ongeza na Ondoa Maneno makubwa

    Kuongeza maneno makubwa na index sawa na radicand sawa ni kama kuongeza maneno kama. Tunaita radicals na index sawa na radicand sawa kama radicals kutukumbusha wanafanya kazi sawa na maneno kama.

    Ufafanuzi\(\PageIndex{1}\): Like Radicals

    Kama radicals ni maneno makubwa na index sawa na radicand sawa.

    Sisi kuongeza na Ondoa kama radicals kwa njia ile ile sisi kuongeza na Ondoa kama maneno. Tunajua kwamba\(3x+8x\)\(11x\) ni.Vivyo hivyo tunaongeza\(3 \sqrt{x}+8 \sqrt{x}\) na matokeo yake ni\(11 \sqrt{x}\).

    Fikiria juu ya kuongeza maneno kama na vigezo kama unavyofanya mifano michache ijayo. Wakati una kama radicals, wewe tu kuongeza au Ondoa coefficients. Wakati radicals si kama, huwezi kuchanganya maneno.

    Mfano\(\PageIndex{1}\)

    Kurahisisha:

    1. \(2 \sqrt{2}-7 \sqrt{2}\)
    2. \(5 \sqrt[3]{y}+4 \sqrt[3]{y}\)
    3. \(7 \sqrt[4]{x}-2 \sqrt[4]{y}\)

    Suluhisho:

    a.

    \(2 \sqrt{2}-7 \sqrt{2}\)

    Kwa kuwa radicals ni kama, sisi Ondoa coefficients.

    \(-5 \sqrt{2}\)

    b.

    \(5 \sqrt[3]{y}+4 \sqrt[3]{y}\)

    Kwa kuwa radicals ni kama, sisi kuongeza coefficients.

    \(9 \sqrt[3]{y}\)

    c.

    \(7 \sqrt[4]{x}-2 \sqrt[4]{y}\)

    Fahirisi ni sawa lakini radicals ni tofauti. Hizi si kama radicals. Kwa kuwa radicals si kama, hatuwezi kuondoa yao.

    Zoezi\(\PageIndex{1}\)

    Kurahisisha:

    1. \(8 \sqrt{2}-9 \sqrt{2}\)
    2. \(4 \sqrt[3]{x}+7 \sqrt[3]{x}\)
    3. \(3 \sqrt[4]{x}-5 \sqrt[4]{y}\)
    Jibu
    1. \(-\sqrt{2}\)
    2. \(11 \sqrt[3]{x}\)
    3. \(3 \sqrt[4]{x}-5 \sqrt[4]{y}\)
    Zoezi\(\PageIndex{2}\)

    Kurahisisha:

    1. \(5 \sqrt{3}-9 \sqrt{3}\)
    2. \(5 \sqrt[3]{y}+3 \sqrt[3]{y}\)
    3. \(5 \sqrt[4]{m}-2 \sqrt[3]{m}\)
    Jibu
    1. \(-4 \sqrt{3}\)
    2. \(8 \sqrt[3]{y}\)
    3. \(5 \sqrt[4]{m}-2 \sqrt[3]{m}\)

    Kwa radicals kuwa kama, lazima wawe na index sawa na radicand. Wakati radicands zina variable zaidi ya moja, kwa muda mrefu kama vigezo vyote na exponents yao ni sawa, radicands ni sawa.

    Mfano\(\PageIndex{2}\)

    Kurahisisha:

    1. \(2 \sqrt{5 n}-6 \sqrt{5 n}+4 \sqrt{5 n}\)
    2. \(\sqrt[4]{3 x y}+5 \sqrt[4]{3 x y}-4 \sqrt[4]{3 x y}\)

    Suluhisho:

    a.

    \(2 \sqrt{5 n}-6 \sqrt{5 n}+4 \sqrt{5 n}\)

    Kwa kuwa radicals ni kama, sisi kuchanganya yao.

    \(0 \sqrt{5 n}\)

    Kurahisisha.

    \(0\)

    b.

    \(\sqrt[4]{3 x y}+5 \sqrt[4]{3 x y}-4 \sqrt[4]{3 x y}\)

    Kwa kuwa radicals ni kama, sisi kuchanganya yao.

    \(2 \sqrt[4]{3 x y}\)

    Zoezi\(\PageIndex{3}\)

    Kurahisisha:

    1. \(\sqrt{7 x}-7 \sqrt{7 x}+4 \sqrt{7 x}\)
    2. \(4 \sqrt[4]{5 x y}+2 \sqrt[4]{5 x y}-7 \sqrt[4]{5 x y}\)
    Jibu
    1. \(-2 \sqrt{7 x}\)
    2. \(-\sqrt[4]{5 x y}\)
    Zoezi\(\PageIndex{4}\)

    Kurahisisha:

    1. \(4 \sqrt{3 y}-7 \sqrt{3 y}+2 \sqrt{3 y}\)
    2. \(6 \sqrt[3]{7 m n}+\sqrt[3]{7 m n}-4 \sqrt[3]{7 m n}\)
    Jibu
    1. \(-\sqrt{3 y}\)
    2. \(3 \sqrt[3]{7 m n}\)

    Kumbuka kwamba sisi daima kurahisisha radicals kwa kuondoa sababu kubwa kutoka radicana hiyo ni nguvu ya index. Mara baada ya kila radical ni rahisi, tunaweza kisha kuamua kama wao ni kama radicals.

    Mfano\(\PageIndex{3}\)

    Kurahisisha:

    1. \(\sqrt{20}+3 \sqrt{5}\)
    2. \(\sqrt[3]{24}-\sqrt[3]{375}\)
    3. \(\frac{1}{2} \sqrt[4]{48}-\frac{2}{3} \sqrt[4]{243}\)

    Suluhisho:

    a.

    \(\sqrt{20}+3 \sqrt{5}\)

    Kurahisisha radicals, wakati iwezekanavyo.

    \(\sqrt{4} \cdot \sqrt{5}+3 \sqrt{5}\)

    \(2 \sqrt{5}+3 \sqrt{5}\)

    Kuchanganya radicals kama.

    \(5 \sqrt{5}\)

    b.

    \(\sqrt[3]{24}-\sqrt[3]{375}\)

    Kurahisisha radicals.

    \(\sqrt[3]{8} \cdot \sqrt[3]{3}-\sqrt[3]{125} \cdot \sqrt[3]{3}\)

    \(2 \sqrt[3]{3}-5 \sqrt[3]{3}\)

    Kuchanganya radicals kama.

    \(-3 \sqrt[3]{3}\)

    c.

    \(\frac{1}{2} \sqrt[4]{48}-\frac{2}{3} \sqrt[4]{243}\)

    Kurahisisha radicals.

    \(\frac{1}{2} \sqrt[4]{16} \cdot \sqrt[4]{3}-\frac{2}{3} \sqrt[4]{81} \cdot \sqrt[4]{3}\)

    \(\frac{1}{2} \cdot 2 \cdot \sqrt[4]{3}-\frac{2}{3} \cdot 3 \cdot \sqrt[4]{3}\)

    \(\sqrt[4]{3}-2 \sqrt[4]{3}\)

    Kuchanganya radicals kama.

    \(-\sqrt[4]{3}\)

    Zoezi\(\PageIndex{5}\)

    Kurahisisha:

    1. \(\sqrt{18}+6 \sqrt{2}\)
    2. \(6 \sqrt[3]{16}-2 \sqrt[3]{250}\)
    3. \(\frac{2}{3} \sqrt[3]{81}-\frac{1}{2} \sqrt[3]{24}\)
    Jibu
    1. \(9 \sqrt{2}\)
    2. \(2 \sqrt[3]{2}\)
    3. \(\sqrt[3]{3}\)
    Zoezi\(\PageIndex{6}\)

    Kurahisisha:

    1. \(\sqrt{27}+4 \sqrt{3}\)
    2. \(4 \sqrt[3]{5}-7 \sqrt[3]{40}\)
    3. \(\frac{1}{2} \sqrt[3]{128}-\frac{5}{3} \sqrt[3]{54}\)
    Jibu
    1. \(7 \sqrt{3}\)
    2. \(-10 \sqrt[3]{5}\)
    3. \(-3 \sqrt[3]{2}\)

    Katika mfano unaofuata, tutaondoa mambo yote ya mara kwa mara na ya kutofautiana kutoka kwa radicals. Sasa kwa kuwa tumefanya mazoezi ya kuchukua mizizi hata na isiyo ya kawaida ya vigezo, ni kawaida mazoezi katika hatua hii kwa sisi kudhani vigezo vyote ni kubwa kuliko au sawa na sifuri ili maadili kamili hazihitajiki. Tutatumia dhana hii kwa kuzingatia sura hii yote.

    Mfano\(\PageIndex{4}\)

    Kurahisisha:

    1. \(9 \sqrt{50 m^{2}}-6 \sqrt{48 m^{2}}\)
    2. \(\sqrt[3]{54 n^{5}}-\sqrt[3]{16 n^{5}}\)

    Suluhisho:

    a.

    \(9 \sqrt{50 m^{2}}-6 \sqrt{48 m^{2}}\)

    Kurahisisha radicals.

    \(9 \sqrt{25 m^{2}} \cdot \sqrt{2}-6 \sqrt{16 m^{2}} \cdot \sqrt{3}\)

    \(9 \cdot 5 m \cdot \sqrt{2}-6 \cdot 4 m \cdot \sqrt{3}\)

    \(45 m \sqrt{2}-24 m \sqrt{3}\)

    Radicals si kama na hivyo haiwezi kuunganishwa.

    b.

    \(\sqrt[3]{54 n^{5}}-\sqrt[3]{16 n^{5}}\)

    Kurahisisha radicals.

    \(\sqrt[3]{27 n^{3}} \cdot \sqrt[3]{2 n^{2}}-\sqrt[3]{8 n^{3}} \cdot \sqrt[3]{2 n^{2}}\)

    \(3 n \sqrt[3]{2 n^{2}}-2 n \sqrt[3]{2 n^{2}}\)

    Kuchanganya radicals kama.

    \(n \sqrt[3]{2 n^{2}}\)

    Zoezi\(\PageIndex{7}\)

    Kurahisisha:

    1. \(\sqrt{32 m^{7}}-\sqrt{50 m^{7}}\)
    2. \(\sqrt[3]{135 x^{7}}-\sqrt[3]{40 x^{7}}\)
    Jibu
    1. \(-m^{3} \sqrt{2 m}\)
    2. \(x^{2} \sqrt[3]{5 x}\)
    Zoezi\(\PageIndex{8}\)

    Kurahisisha:

    1. \(\sqrt{27 p^{3}}-\sqrt{48 p^{3}}\)
    2. \(\sqrt[3]{256 y^{5}}-\sqrt[3]{32 n^{5}}\)
    Jibu
    1. \(-p \sqrt{3 p}\)
    2. \(4 y \sqrt[3]{4 y^{2}}-2 n \sqrt[3]{4 n^{2}}\)

    Kuzidisha maneno makubwa

    Tumetumia Mali ya Bidhaa ya Mizizi ili kurahisisha mizizi ya mraba kwa kuondoa mambo kamili ya mraba. Tunaweza kutumia Mali ya Bidhaa ya Mizizi 'katika reverse' kuzidisha mizizi ya mraba. Kumbuka, sisi kudhani vigezo vyote ni kubwa kuliko au sawa na sifuri.

    Tutaandika tena Mali ya Bidhaa ya Mizizi ili tuone njia zote mbili pamoja.

    Ufafanuzi\(\PageIndex{2}\): Product Property of Roots

    Kwa idadi yoyote halisi,\(\sqrt[n]{a}\) na\(\sqrt[b]{n}\), na kwa integer yoyote\(n≥2\)

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

    Wakati sisi kuzidisha radicals mbili lazima wawe na index sawa. Mara tu sisi kuzidisha radicals, sisi kisha kutafuta mambo ambayo ni nguvu ya index na kurahisisha radical wakati wowote iwezekanavyo.

    Kuzidisha radicals na coefficients ni sawa na kuzidisha vigezo na coefficients. Kuzidisha\(4x⋅3y\) sisi kuzidisha coefficients pamoja na kisha vigezo. Matokeo yake ni\(12xy\). Kuweka hii katika akili kama wewe kufanya mifano hii.

    Mfano\(\PageIndex{5}\)

    Kurahisisha:

    1. \((6 \sqrt{2})(3 \sqrt{10})\)
    2. \((-5 \sqrt[3]{4})(-4 \sqrt[3]{6})\)

    Suluhisho:

    a.

    \((6 \sqrt{2})(3 \sqrt{10})\)

    Kuzidisha kutumia Mali ya Bidhaa.

    \(18\sqrt{20}\)

    Kurahisisha radical.

    \(18 \sqrt{4} \cdot \sqrt{5}\)

    Kurahisisha.

    \(18 \cdot 2 \cdot \sqrt{5}\)

    \(36 \sqrt{5}\)

    b.

    \((-5 \sqrt[3]{4})(-4 \sqrt[3]{6})\)

    Kuzidisha kutumia Mali ya Bidhaa.

    \(20 \sqrt[3]{24}\)

    Kurahisisha radical.

    \(20 \sqrt[3]{8} \cdot \sqrt[3]{3}\)

    Kurahisisha.

    \(20 \cdot 2 \cdot \sqrt[3]{3}\)

    \(40 \sqrt[3]{3}\)

    Zoezi\(\PageIndex{9}\)

    Kurahisisha:

    1. \((3 \sqrt{2})(2 \sqrt{30})\)
    2. \((2 \sqrt[3]{18})(-3 \sqrt[3]{6})\)
    Jibu
    1. \(12 \sqrt{15}\)
    2. \(-18 \sqrt[3]{2}\)
    Zoezi\(\PageIndex{10}\)

    Kurahisisha:

    1. \((3 \sqrt{3})(3 \sqrt{6})\)
    2. \((-4 \sqrt[3]{9})(3 \sqrt[3]{6})\)
    Jibu
    1. \(27 \sqrt{2}\)
    2. \(-36 \sqrt[3]{2}\)

    Sisi kufuata taratibu sawa wakati kuna vigezo katika radicands.

    Mfano\(\PageIndex{6}\)

    Kurahisisha:

    1. \(\left(10 \sqrt{6 p^{3}}\right)(4 \sqrt{3 p})\)
    2. \(\left(2 \sqrt[4]{20 y^{2}}\right)\left(3 \sqrt[4]{28 y^{3}}\right)\)

    Suluhisho:

    a.

    \(\left(10 \sqrt{6 p^{3}}\right)(4 \sqrt{3 p})\)

    Kuzidisha.

    \(40 \sqrt{18 p^{4}}\)

    Kurahisisha radical.

    \(40 \sqrt{9 p^{4}} \cdot \sqrt{2}\)

    Kurahisisha.

    \(40 \cdot 3 p^{2} \cdot \sqrt{3}\)

    \(120 p^{2} \sqrt{3}\)

    b Wakati radicands kuhusisha idadi kubwa, mara nyingi ni faida kwa sababu yao ili kupata nguvu kamili.

    \(\left(2 \sqrt[4]{20 y^{2}}\right)\left(3 \sqrt[4]{28 y^{3}}\right)\)

    Kuzidisha.

    \(6 \sqrt[4]{4 \cdot 5 \cdot 4 \cdot 7 y^{5}}\)

    Kurahisisha radical.

    \(6 \sqrt[4]{16 y^{4}} \cdot \sqrt[4]{35 y}\)

    Kurahisisha.

    \(6 \cdot 2 y \sqrt[4]{35 y}\)

    Kuzidisha.

    \(12 y \sqrt[4]{35 y}\)

    Zoezi\(\PageIndex{11}\)

    Kurahisisha:

    1. \(\left(6 \sqrt{6 x^{2}}\right)\left(8 \sqrt{30 x^{4}}\right)\)
    2. \(\left(-4 \sqrt[4]{12 y^{3}}\right)\left(-\sqrt[4]{8 y^{3}}\right)\)
    Jibu
    1. \(36 x^{3} \sqrt{5}\)
    2. \(8 y \sqrt[4]{3 y^{2}}\)
    Zoezi\(\PageIndex{12}\)

    Kurahisisha:

    1. \(\left(2 \sqrt{6 y^{4}}\right)(12 \sqrt{30 y})\)
    2. \(\left(-4 \sqrt[4]{9 a^{3}}\right)\left(3 \sqrt[4]{27 a^{2}}\right)\)
    Jibu
    1. \(144 y^{2} \sqrt{5 y}\)
    2. \(-36 \sqrt[4]{3 a}\)

    Tumia Uzidishaji wa Polynomial ili kuzidisha Maneno makubwa

    Katika mifano michache ijayo, tutatumia Mali ya Distributive kuzidisha maneno na radicals. Kwanza tutasambaza na kisha kurahisisha radicals iwezekanavyo.

    Mfano\(\PageIndex{7}\)

    Kurahisisha:

    1. \(\sqrt{6}(\sqrt{2}+\sqrt{18})\)
    2. \(\sqrt[3]{9}(5-\sqrt[3]{18})\)

    Suluhisho:

    a.

    \(\sqrt{6}(\sqrt{2}+\sqrt{18})\)

    Kuzidisha.

    \(\sqrt{12}+\sqrt{108}\)

    Kurahisisha.

    \(\sqrt{4} \cdot \sqrt{3}+\sqrt{36} \cdot \sqrt{3}\)

    Kurahisisha.

    \(2 \sqrt{3}+6 \sqrt{3}\)

    Kuchanganya kama radicals.

    \(8\sqrt{3}\)

    b.

    \(\sqrt[3]{9}(5-\sqrt[3]{18})\)

    Kusambaza.

    \(5 \sqrt[3]{9}-\sqrt[3]{162}\)

    Kurahisisha.

    \(5 \sqrt[3]{9}-\sqrt[3]{27} \cdot \sqrt[3]{6}\)

    Kurahisisha.

    \(5 \sqrt[3]{9}-3 \sqrt[3]{6}\)

    Zoezi\(\PageIndex{13}\)

    Kurahisisha:

    1. \(\sqrt{6}(1+3 \sqrt{6})\)
    2. \(\sqrt[3]{4}(-2-\sqrt[3]{6})\)
    Jibu
    1. \(18+\sqrt{6}\)
    2. \(-2 \sqrt[3]{4}-2 \sqrt[3]{3}\)
    Zoezi\(\PageIndex{14}\)

    Kurahisisha:

    1. \(\sqrt{8}(2-5 \sqrt{8})\)
    2. \(\sqrt[3]{3}(-\sqrt[3]{9}-\sqrt[3]{6})\)
    Jibu
    1. \(-40+4 \sqrt{2}\)
    2. \(-3-\sqrt[3]{18}\)

    Tulipofanya kazi na polynomials, tuliongeza binomials na binomials. Kumbuka, hii ilitupa bidhaa nne kabla ya kuunganisha maneno kama hayo. Ili uhakikishe kupata bidhaa zote nne, tuliandaa kazi yetu—kwa kawaida kwa njia ya FOIL.

    Mfano\(\PageIndex{8}\)

    Kurahisisha:

    1. \((3-2 \sqrt{7})(4-2 \sqrt{7})\)
    2. \((\sqrt[3]{x}-2)(\sqrt[3]{x}+4)\)

    Suluhisho:

    a.

    \((3-2 \sqrt{7})(4-2 \sqrt{7})\)

    Kuzidisha.

    \(12-6 \sqrt{7}-8 \sqrt{7}+4 \cdot 7\)

    Kurahisisha.

    \(12-6 \sqrt{7}-8 \sqrt{7}+28\)

    Kuchanganya kama maneno.

    \(40-14 \sqrt{7}\)

    b.

    \((\sqrt[3]{x}-2)(\sqrt[3]{x}+4)\)

    Kuzidisha.

    \(\sqrt[3]{x^{2}}+4 \sqrt[3]{x}-2 \sqrt[3]{x}-8\)

    Kuchanganya kama maneno.

    \(\sqrt[3]{x^{2}}+2 \sqrt[3]{x}-8\)

    Zoezi\(\PageIndex{15}\)

    Kurahisisha:

    1. \((6-3 \sqrt{7})(3+4 \sqrt{7})\)
    2. \((\sqrt[3]{x}-2)(\sqrt[3]{x}-3)\)
    Jibu
    1. \(-66+15 \sqrt{7}\)
    2. \(\sqrt[3]{x^{2}}-5 \sqrt[3]{x}+6\)
    Zoezi\(\PageIndex{16}\)

    Kurahisisha:

    1. \((2-3 \sqrt{11})(4-\sqrt{11})\)
    2. \((\sqrt[3]{x}+1)(\sqrt[3]{x}+3)\)
    Jibu
    1. \(41-14 \sqrt{11}\)
    2. \(\sqrt[3]{x^{2}}+4 \sqrt[3]{x}+3\)
    Mfano\(\PageIndex{9}\)

    Kurahisisha:\((3 \sqrt{2}-\sqrt{5})(\sqrt{2}+4 \sqrt{5})\)

    Suluhisho:

    \((3 \sqrt{2}-\sqrt{5})(\sqrt{2}+4 \sqrt{5})\)

    Kuzidisha.

    \(3 \cdot 2+12 \sqrt{10}-\sqrt{10}-4 \cdot 5\)

    Kurahisisha.

    \(6+12 \sqrt{10}-\sqrt{10}-20\)

    Kuchanganya kama maneno.

    \(-14+11 \sqrt{10}\)

    Zoezi\(\PageIndex{17}\)

    Kurahisisha:\((5 \sqrt{3}-\sqrt{7})(\sqrt{3}+2 \sqrt{7})\)

    Jibu

    \(1+9 \sqrt{21}\)

    Zoezi\(\PageIndex{18}\)

    Kurahisisha:\((\sqrt{6}-3 \sqrt{8})(2 \sqrt{6}+\sqrt{8})\)

    Jibu

    \(-12-20 \sqrt{3}\)

    Kutambua baadhi ya bidhaa maalum ilifanya kazi yetu iwe rahisi wakati tulizidisha binomials mapema. Hii ni kweli wakati sisi kuzidisha radicals, pia. Fomu maalum za bidhaa tulizotumia zinaonyeshwa hapa.

    Bidhaa Maalum

    Mraba ya Binomial

    \(\begin{array}{l}{(a+b)^{2}=a^{2}+2 a b+b^{2}} \\ {(a-b)^{2}=a^{2}-2 a b+b^{2}}\end{array}\)

    Bidhaa ya Conjugates

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

    Tutatumia fomu maalum za bidhaa katika mifano michache ijayo. Tutaanza na Bidhaa ya Mraba ya Binomial Pattern.

    Mfano\(\PageIndex{10}\)

    Kurahisisha:

    1. \(2+\sqrt{3})^{2}\)
    2. \((4-2 \sqrt{5})^{2}\)

    Suluhisho:

    a.

      .
    Kuzidisha kutumia Bidhaa ya Mraba ya Binomial Pattern. .
    Kurahisisha. .
    Kuchanganya kama maneno. .
    Jedwali 8.4.1

    b.

    Jedwali 8.4.2
     

    .
    Nyingi, kwa kutumia Bidhaa ya Mraba ya Binomial Pattern.

    .
    Kurahisisha.

    .
     

    .
    Kuchanganya kama maneno.

    .
    Zoezi\(\PageIndex{19}\)

    Kurahisisha:

    1. \((10+\sqrt{2})^{2}\)
    2. \((1+3 \sqrt{6})^{2}\)
    Jibu
    1. \(102+20 \sqrt{2}\)
    2. \(55+6 \sqrt{6}\)
    Zoezi\(\PageIndex{20}\)

    Kurahisisha:

    1. \((6-\sqrt{5})^{2}\)
    2. \((9-2 \sqrt{10})^{2}\)
    Jibu
    1. \(41-12 \sqrt{5}\)
    2. \(121-36 \sqrt{10}\)

    Katika mfano unaofuata, tutatumia Bidhaa ya Conjugates Pattern. Angalia kwamba bidhaa ya mwisho haina radical.

    Mfano\(\PageIndex{11}\)

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

    Suluhisho:

     

    .
    Kuzidisha kutumia Bidhaa ya Conjugates Pattern.

    .
    Kurahisisha.

    .
     

    .
    Jedwali 8.4.3
    Zoezi\(\PageIndex{21}\)

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

    Jibu

    \(-11\)

    Zoezi\(\PageIndex{22}\)

    Kurahisisha:\((4+5 \sqrt{7})(4-5 \sqrt{7})\)

    Jibu

    \(-159\)

    Fikia rasilimali hizi za mtandaoni kwa maelekezo ya ziada na ufanyie mazoezi kwa kuongeza, kuondoa, na kuzidisha maneno makubwa.

    • Kuzidisha Kuongeza Kutoa Radicals
    • Kuzidisha Bidhaa maalum: Binomials za Mraba Zenye Mizizi ya Mraba
    • Kuzidisha conjugates

    Dhana muhimu

    • Bidhaa Mali ya Mizizi
      • Kwa idadi yoyote halisi,\(\sqrt[n]{a}\) na\(\sqrt[n]{b}\), na kwa integer yoyote\(n≥2\)\(\sqrt[n]{a b}=\sqrt[n]{a} \cdot \sqrt[n]{b}\) na\(\sqrt[n]{a} \cdot \sqrt[n]{b}=\sqrt[n]{a b}\)
    • Bidhaa Maalum

    \(\begin{array}{c c}{\text { Binomial Squares }}& {\text{Product of Conjugates}} \\ {(a+b)^{2}=a^{2}+2 a b+b^{2}} & {(a+b)(a-b)=a^{2}-b^{2}} \\ {(a-b)^{2}=a^{2}-2 a b+b^{2}}\end{array}\)

    faharasa

    kama radicals
    Kama radicals ni maneno makubwa na index sawa na radicand sawa.