8.3: Kurahisisha maneno makubwa

Malengo ya kujifunza

Mwishoni mwa sehemu hii, utaweza:

• Tumia Mali ya Bidhaa ili kurahisisha maneno makubwa
• Tumia Mali ya Quotient ili kurahisisha maneno makubwa

Kabla ya kuanza, fanya jaribio hili la utayari.

1. Kurahisisha:\(\dfrac{x^{9}}{x^{4}}\).
   Kama amekosa tatizo hili, mapitio Mfano 5.13.
2. Kurahisisha:\(\dfrac{y^{3}}{y^{11}}\).
   Kama amekosa tatizo hili, mapitio Mfano 5.13.
3. Kurahisisha:\(\left(n^{2}\right)^{6}\).
   Kama amekosa tatizo hili, mapitio Mfano 5.17.

Tumia Mali ya Bidhaa ili kurahisisha Maneno ya Radical

Sisi kurahisisha maneno radical kwa njia sawa na jinsi sisi kilichorahisishwa sehemu. Sehemu ni rahisi ikiwa hakuna mambo ya kawaida katika nambari na denominator. Ili kurahisisha sehemu, tunatafuta mambo yoyote ya kawaida katika nambari na denominator.

kujieleza radical\(\sqrt[n]{a}\), ni kuchukuliwa rahisi kama haina sababu ya\(m^{n}\). Kwa hiyo, ili kurahisisha kujieleza kwa kiasi kikubwa, tunatafuta sababu yoyote katika radicana ambayo ni nguvu za index.

Ufafanuzi\(\PageIndex{1}\): Simplified Radical Expression

Kwa idadi halisi\(a\) na\(m\), na\(n\geq 2\),

\(\sqrt[n]{a}\)ni kuchukuliwa kilichorahisishwa kama\(a\) hana sababu ya\(m^{n}\)

Kwa mfano,\(\sqrt{5}\) inachukuliwa kilichorahisishwa kwa sababu hakuna sababu kamili za mraba katika\(5\). Lakini\(\sqrt{12}\) si rahisi kwa sababu\(12\) ina kamili mraba sababu ya\(4\).

Vile vile,\(\sqrt[3]{4}\) ni rahisi kwa sababu hakuna sababu kamili mchemraba katika\(4\). Lakini\(\sqrt[3]{24}\) si rahisi kwa sababu\(24\) ina kamili mchemraba sababu ya\(8\).

Ili kurahisisha maneno makubwa, tutatumia pia mali fulani za mizizi. Mali ambayo tutatumia ili kurahisisha maneno makubwa yanafanana na mali ya watazamaji. Tunajua kwamba

\[(a b)^{n}=a^{n} b^{n}.\]

Sambamba ya Mali ya Bidhaa ya Mizizi anasema kuwa

\[\sqrt[n]{a b}=\sqrt[n]{a} \cdot \sqrt[n]{b}.\]

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

Ikiwa\(\sqrt[n]{a}\) na\(\sqrt[n]{b}\) ni namba halisi, na\(n\geq 2\) ni integer, basi

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

Tunatumia Mali ya Bidhaa ya Mizizi ili kuondoa mambo yote ya mraba kamili kutoka kwenye mizizi ya mraba.

Mfano\(\PageIndex{1}\): Simplify square roots using the product property of roots

Kurahisisha:\(\sqrt{98}\).

Suluhisho:

Hatua ya 1: Pata sababu kubwa zaidi katika radicana hiyo ni nguvu kamili ya index.	Tunaona kwamba\(49\) ni sababu kubwa ya\(98\) kuwa ina nguvu ya\(2\).	\(\sqrt{98}\)
Andika upya radicna kama bidhaa ya mambo mawili, kwa kutumia jambo hilo.	Kwa maneno mengine\(49\) ni kubwa kamili mraba sababu ya\(98\). \(98 = 49\cdot 2\) Daima kuandika sababu kamili ya mraba kwanza.	\(\sqrt{49\cdot 2}\)
Hatua ya 2: Tumia utawala wa bidhaa ili uandike upya radical kama bidhaa ya radicals mbili.		\(\sqrt{49} \cdot \sqrt{2}\)
Hatua ya 3: Kurahisisha mizizi ya nguvu kamili.		\(7\sqrt{2}\)

Jaribu\(\PageIndex{1}\)

Kurahisisha:\(\sqrt{48}\)

Jibu

\(4 \sqrt{3}\)

Jaribu\(\PageIndex{2}\)

Kurahisisha:\(\sqrt{45}\).

Jibu

\(3 \sqrt{5}\)

Angalia katika mfano uliopita kwamba fomu rahisi ya\(\sqrt{98}\) ni\(7\sqrt{2}\), ambayo ni bidhaa ya integer na mizizi ya mraba. Sisi daima kuandika integer mbele ya mizizi ya mraba.

Kuwa makini kuandika integer yako ili usiingizwe na index. Maneno\(7\sqrt{2}\) ni tofauti sana na\(\sqrt[7]{2}\).

Kurahisisha kujieleza Radical Kutumia Mali ya Bidhaa

1. Pata sababu kubwa zaidi katika radicana hiyo ni nguvu kamili ya index. Andika upya radicna kama bidhaa ya mambo mawili, kwa kutumia jambo hilo.
2. Tumia utawala wa bidhaa ili uandike upya radical kama bidhaa ya radicals mbili.
3. Kurahisisha mizizi ya nguvu kamili.

Tutatumia njia hii katika mfano unaofuata. Inaweza kuwa na manufaa kuwa na meza ya mraba kamili, cubes, na nguvu ya nne.

Mfano\(\PageIndex{2}\)

Kurahisisha:

1. \(\sqrt{500}\)
2. \(\sqrt[3]{16}\)
3. \(\sqrt[4]{243}\)

Suluhisho:

a.

\(\sqrt{500}\)

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

\(\sqrt{100 \cdot 5}\)

Andika upya radical kama bidhaa ya radicals mbili.

\(\sqrt{100} \cdot \sqrt{5}\)

Kurahisisha.

\(10\sqrt{5}\)

b.

\(\sqrt[3]{16}\)

Andika upya radicand kama bidhaa kwa kutumia sababu kubwa zaidi ya mchemraba. \(2^{3}=8\)

\(\sqrt[3]{8 \cdot 2}\)

Andika upya radical kama bidhaa ya radicals mbili.

\(\sqrt[3]{8} \cdot \sqrt[3]{2}\)

Kurahisisha.

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

c.

\(\sqrt[4]{243}\)

Andika upya radicna kama bidhaa kwa kutumia nguvu kubwa zaidi ya nne ya nguvu. \(3^{4}=81\)

\(\sqrt[4]{81 \cdot 3}\)

Andika upya radical kama bidhaa ya radicals mbili.

\(\sqrt[4]{81} \cdot \sqrt[4]{3}\)

Kurahisisha.

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

Jaribu\(\PageIndex{3}\)

Kurahisisha: a.\(\sqrt{288}\) b.\(\sqrt[3]{81}\) c.\(\sqrt[4]{64}\)

Jibu

a.\(12\sqrt{2}\) b.\(3 \sqrt[3]{3}\) c.\(2 \sqrt[4]{4}\)

Jaribu\(\PageIndex{4}\)

Kurahisisha: a.\(\sqrt{432}\) b.\(\sqrt[3]{625}\) c.\(\sqrt[4]{729}\)

Jibu

a.\(12\sqrt{3}\) b.\(5 \sqrt[3]{5}\) c.\(3 \sqrt[4]{9}\)

Mfano unaofuata ni sawa na mifano ya awali, lakini kwa vigezo. Usisahau kutumia ishara ya thamani kamili wakati wa kuchukua hata mizizi ya kujieleza na kutofautiana katika radical.

Mfano\(\PageIndex{3}\)

Kurahisisha:

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

Suluhisho:

a.

\(\sqrt{x^{3}}\)

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

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

Andika upya radical kama bidhaa ya radicals mbili.

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

Kurahisisha.

\(|x| \sqrt{x}\)

b.

\(\sqrt[3]{x^{4}}\)

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

\(\sqrt[3]{x^{3} \cdot x}\)

Andika upya radical kama bidhaa ya radicals mbili.

\(\sqrt[3]{x^{3}} \cdot \sqrt[3]{x}\)

Kurahisisha.

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

c.

\(\sqrt[4]{x^{7}}\)

Andika upya radicna kama bidhaa kwa kutumia nguvu kubwa zaidi ya nne ya nguvu.

\(\sqrt[4]{x^{4} \cdot x^{3}}\)

Andika upya radical kama bidhaa ya radicals mbili.

\(\sqrt[4]{x^{4}} \cdot \sqrt[4]{x^{3}}\)

Kurahisisha.

\(|x| \sqrt[4]{x^{3}}\)

Jaribu\(\PageIndex{5}\)

Kurahisisha: a.\(\sqrt{b^{5}}\) b.\(\sqrt[4]{y^{6}}\) c.\(\sqrt[3]{z^{5}}\)

Jibu

a.\(b^{2} \sqrt{b}\) b.\(|y| \sqrt[4]{y^{2}}\) c.\(z \sqrt[3]{z^{2}}\)

Jaribu\(\PageIndex{6}\)

Kurahisisha: a.\(\sqrt{p^{9}}\) b.\(\sqrt[5]{y^{8}}\) c.\(\sqrt[6]{q^{13}}\)

Jibu

a.\(p^{4} \sqrt{p}\) b.\(p \sqrt[5]{p^{3}}\) c.\(q^{2} \sqrt[6]{q}\)

Tunafuata utaratibu huo wakati kuna mgawo katika radicand. Katika mfano unaofuata, wote mara kwa mara na kutofautiana wana mambo kamili ya mraba.

Mfano\(\PageIndex{4}\)

Kurahisisha:

1. \(\sqrt{72 n^{7}}\)
2. \(\sqrt[3]{24 x^{7}}\)
3. \(\sqrt[4]{80 y^{14}}\)

Suluhisho:

a.

\(\sqrt{72 n^{7}}\)

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

\(\sqrt{36 n^{6} \cdot 2 n}\)

Andika upya radical kama bidhaa ya radicals mbili.

\(\sqrt{36 n^{6}} \cdot \sqrt{2 n}\)

Kurahisisha.

\(6\left|n^{3}\right| \sqrt{2 n}\)

b.

\(\sqrt[3]{24 x^{7}}\)

Andika upya radicand kama bidhaa kwa kutumia mambo kamili ya mchemraba.

\(\sqrt[3]{8 x^{6} \cdot 3 x}\)

Andika upya radical kama bidhaa ya radicals mbili.

\(\sqrt[3]{8 x^{6}} \cdot \sqrt[3]{3 x}\)

Andika upya radicand kwanza kama\(\left(2 x^{2}\right)^{3}\).

\(\sqrt[3]{\left(2 x^{2}\right)^{3}} \cdot \sqrt[3]{3 x}\)

Kurahisisha.

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

c.

\(\sqrt[4]{80 y^{14}}\)

Andika upya radicna kama bidhaa kwa kutumia mambo kamili ya nne ya nguvu.

\(\sqrt[4]{16 y^{12} \cdot 5 y^{2}}\)

Andika upya radical kama bidhaa ya radicals mbili.

\(\sqrt[4]{16 y^{12}} \cdot \sqrt[4]{5 y^{2}}\)

Andika upya radicand kwanza kama\(\left(2 y^{3}\right)^{4}\).

\(\sqrt[4]{\left(2 y^{3}\right)^{4}} \cdot \sqrt[4]{5 y^{2}}\)

Kurahisisha.

\(2\left|y^{3}\right| \sqrt[4]{5 y^{2}}\)

Jaribu\(\PageIndex{7}\)

Kurahisisha: a.\(\sqrt{32 y^{5}}\) b.\(\sqrt[3]{54 p^{10}}\) c.\(\sqrt[4]{64 q^{10}}\)

Jibu

a.\(4 y^{2} \sqrt{2 y}\) b.\(3 p^{3} \sqrt[3]{2 p}\) c.\(2 q^{2} \sqrt[4]{4 q^{2}}\)

Jaribu\(\PageIndex{8}\)

Kurahisisha: a.\(\sqrt{75 a^{9}}\) b.\(\sqrt[3]{128 m^{11}}\) c.\(\sqrt[4]{162 n^{7}}\)

Jibu

a.\(5 a^{4} \sqrt{3 a}\) b.\(4 m^{3} \sqrt[3]{2 m^{2}}\) c.\(3|n| \sqrt[4]{2 n^{3}}\)

Katika mfano unaofuata, tunaendelea kutumia mbinu sawa ingawa kuna variable zaidi ya moja chini ya radical.

Mfano\(\PageIndex{5}\)

Kurahisisha:

1. \(\sqrt{63 u^{3} v^{5}}\)
2. \(\sqrt[3]{40 x^{4} y^{5}}\)
3. \(\sqrt[4]{48 x^{4} y^{7}}\)

Suluhisho:

a.

\(\sqrt{63 u^{3} v^{5}}\)

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

\(\sqrt{9 u^{2} v^{4} \cdot 7 u v}\)

Andika upya radical kama bidhaa ya radicals mbili.

\(\sqrt{9 u^{2} v^{4}} \cdot \sqrt{7 u v}\)

Andika upya radicand kwanza kama\(\left(3 u v^{2}\right)^{2}\).

\(\sqrt{\left(3 u v^{2}\right)^{2}} \cdot \sqrt{7 u v}\)

Kurahisisha.

\(3|u| v^{2} \sqrt{7 u v}\)

b.

\(\sqrt[3]{40 x^{4} y^{5}}\)

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

\(\sqrt[3]{8 x^{3} y^{3} \cdot 5 x y^{2}}\)

Andika upya radical kama bidhaa ya radicals mbili.

\(\sqrt[3]{8 x^{3} y^{3}} \cdot \sqrt[3]{5 x y^{2}}\)

Andika upya radicand kwanza kama\((2xy)^{3}\).

\(\sqrt[3]{(2 x y)^{3}} \cdot \sqrt[3]{5 x y^{2}}\)

Kurahisisha.

\(2 x y \sqrt[3]{5 x y^{2}}\)

c.

\(\sqrt[4]{48 x^{4} y^{7}}\)

Andika upya radicna kama bidhaa kwa kutumia nguvu kubwa zaidi ya nne ya nguvu.

\(\sqrt[4]{16 x^{4} y^{4} \cdot 3 y^{3}}\)

Andika upya radical kama bidhaa ya radicals mbili.

\(\sqrt[4]{16 x^{4} y^{4}} \cdot \sqrt[4]{3 y^{3}}\)

Andika upya radicand kwanza kama\((2xy)^{4}\).

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

Kurahisisha.

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

Jaribu\(\PageIndex{9}\)

Kurahisisha:

1. \(\sqrt{98 a^{7} b^{5}}\)
2. \(\sqrt[3]{56 x^{5} y^{4}}\)
3. \(\sqrt[4]{32 x^{5} y^{8}}\)

Jibu

1. \(7\left|a^{3}\right| b^{2} \sqrt{2 a b}\)
2. \(2 x y \sqrt[3]{7 x^{2} y}\)
3. \(2|x| y^{2} \sqrt[4]{2 x}\)

Jaribu\(\PageIndex{10}\)

Kurahisisha:

1. \(\sqrt{180 m^{9} n^{11}}\)
2. \(\sqrt[3]{72 x^{6} y^{5}}\)
3. \(\sqrt[4]{80 x^{7} y^{4}}\)

Jibu

1. \(6 m^{4}\left|n^{5}\right| \sqrt{5 m n}\)
2. \(2 x^{2} y \sqrt[3]{9 y^{2}}\)
3. \(2|x y| \sqrt[4]{5 x^{3}}\)

Mfano\(\PageIndex{6}\)

Kurahisisha:

1. \(\sqrt[3]{-27}\)
2. \(\sqrt[4]{-16}\)

Suluhisho:

a.

\(\sqrt[3]{-27}\)

Andika upya radicand kama bidhaa kwa kutumia mambo kamili ya mchemraba.

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

Chukua mizizi ya mchemraba.

\(-3\)

b.

\(\sqrt[4]{-16}\)

Hakuna idadi halisi\(n\) ambapo\(n^{4}=-16\).

Si idadi halisi

Jaribu\(\PageIndex{11}\)

Kurahisisha:

1. \(\sqrt[3]{-64}\)
2. \(\sqrt[4]{-81}\)

Jibu

1. \(-4\)
2. hakuna idadi halisi

Jaribu\(\PageIndex{12}\)

Kurahisisha:

1. \(\sqrt[3]{-625}\)
2. \(\sqrt[4]{-324}\)

Jibu

1. \(-5 \sqrt[3]{5}\)
2. hakuna idadi halisi

Tumeona jinsi ya kutumia utaratibu wa shughuli ili kurahisisha baadhi ya maneno na radicals. Katika mfano unaofuata, tuna jumla ya integer na mizizi ya mraba. Sisi kurahisisha mizizi mraba lakini hawezi kuongeza kujieleza kusababisha integer tangu neno moja ina radical na nyingine haina. Mfano unaofuata pia unajumuisha sehemu yenye radical katika nambari. Kumbuka kwamba ili kurahisisha sehemu unahitaji jambo la kawaida katika nambari na denominator.

Mfano\(\PageIndex{7}\)

Kurahisisha:

1. \(3+\sqrt{32}\)
2. \(\dfrac{4-\sqrt{48}}{2}\)

Suluhisho:

a.

\(3+\sqrt{32}\)

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

\(3+\sqrt{16 \cdot 2}\)

Andika upya radical kama bidhaa ya radicals mbili.

\(3+\sqrt{16} \cdot \sqrt{2}\)

Kurahisisha.

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

Masharti hayawezi kuongezwa kwani moja ana radical na nyingine haina. Kujaribu kuongeza integer na radical ni kama kujaribu kuongeza integer na variable. Wao si kama maneno!

b.

\(\dfrac{4-\sqrt{48}}{2}\)

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

\(\dfrac{4-\sqrt{16 \cdot 3}}{2}\)

Andika upya radical kama bidhaa ya radicals mbili.

\(\dfrac{4-\sqrt{16} \cdot \sqrt{3}}{2}\)

Kurahisisha.

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

Fanya sababu ya kawaida kutoka kwa nambari.

\(\dfrac{4(1-\sqrt{3})}{2}\)

Ondoa jambo la kawaida, 2, kutoka kwa nambari na denominator.

\(\dfrac{\cancel{2} \cdot 2(1-\sqrt{3})}{\cancel{2}}\)

Kurahisisha.

\(2(1-\sqrt{3})\)

Jaribu\(\PageIndex{13}\)

Kurahisisha:

1. \(5+\sqrt{75}\)
2. \(\dfrac{10-\sqrt{75}}{5}\)

Jibu

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

Jaribu\(\PageIndex{14}\)

Kurahisisha:

1. \(2+\sqrt{98}\)
2. \(\dfrac{6-\sqrt{45}}{3}\)

Jibu

1. \(2+7 \sqrt{2}\)
2. \(2-\sqrt{5}\)

Tumia Mali ya Quotient ili kurahisisha maneno makubwa

Wakati wowote unapaswa kurahisisha kujieleza kwa kasi, hatua ya kwanza unapaswa kuchukua ni kuamua kama radicand ni nguvu kamili ya index. Ikiwa sio, angalia namba na denominator kwa sababu yoyote ya kawaida, na uondoe. Unaweza kupata sehemu ambayo namba zote na denominator ni nguvu kamili ya index.

Mfano\(\PageIndex{8}\)

Kurahisisha:

1. \(\sqrt{\dfrac{45}{80}}\)
2. \(\sqrt[3]{\dfrac{16}{54}}\)
3. \(\sqrt[4]{\dfrac{5}{80}}\)

Suluhisho:

a.

\(\sqrt{\dfrac{45}{80}}\)

Kurahisisha ndani ya radical kwanza. Andika upya kuonyesha mambo ya kawaida ya nambari na denominator.

\(\sqrt{\dfrac{5 \cdot 9}{5 \cdot 16}}\)

Kurahisisha sehemu kwa kuondoa mambo ya kawaida.

\(\sqrt{\dfrac{9}{16}}\)

Kurahisisha. Kumbuka\(\left(\dfrac{3}{4}\right)^{2}=\dfrac{9}{16}\).

\(\dfrac{3}{4}\)

b.

\(\sqrt[3]{\dfrac{16}{54}}\)

Kurahisisha ndani ya radical kwanza. Andika upya kuonyesha mambo ya kawaida ya nambari na denominator.

\(\sqrt[3]{\dfrac{2 \cdot 8}{2 \cdot 27}}\)

Kurahisisha sehemu kwa kuondoa mambo ya kawaida.

\(\sqrt[3]{\dfrac{8}{27}}\)

Kurahisisha. Kumbuka\(\left(\dfrac{2}{3}\right)^{3}=\dfrac{8}{27}\).

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

c.

\(\sqrt[4]{\dfrac{5}{80}}\)

Kurahisisha ndani ya radical kwanza. Andika upya kuonyesha mambo ya kawaida ya nambari na denominator.

\(\sqrt[4]{\dfrac{5 \cdot 1}{5 \cdot 16}}\)

Kurahisisha sehemu kwa kuondoa mambo ya kawaida.

\(\sqrt[4]{\dfrac{1}{16}}\)

Kurahisisha. Kumbuka\(\left(\dfrac{1}{2}\right)^{4}=\dfrac{1}{16}\).

\(\dfrac{1}{2}\)

Jaribu\(\PageIndex{15}\)

Kurahisisha:

1. \(\sqrt{\dfrac{75}{48}}\)
2. \(\sqrt[3]{\dfrac{54}{250}}\)
3. \(\sqrt[4]{\dfrac{32}{162}}\)

Jibu

1. \(\dfrac{5}{4}\)
2. \(\dfrac{3}{5}\)
3. \(\dfrac{2}{3}\)

Jaribu\(\PageIndex{16}\)

Kurahisisha:

1. \(\sqrt{\dfrac{98}{162}}\)
2. \(\sqrt[3]{\dfrac{24}{375}}\)
3. \(\sqrt[4]{\dfrac{4}{324}}\)

Jibu

1. \(\dfrac{7}{9}\)
2. \(\dfrac{2}{5}\)
3. \(\dfrac{1}{3}\)

Katika mfano wa mwisho, hatua yetu ya kwanza ilikuwa kurahisisha sehemu chini ya radical kwa kuondoa mambo ya kawaida. Katika mfano unaofuata tutatumia Mali ya Quotient ili kurahisisha chini ya radical. Sisi kugawanya misingi kama hiyo kwa kutoa exponents yao,

\(\dfrac{a^{m}}{a^{n}}=a^{m-n}, \quad a \neq 0\)

Mfano\(\PageIndex{9}\)

Kurahisisha:

1. \(\sqrt{\dfrac{m^{6}}{m^{4}}}\)
2. \(\sqrt[3]{\dfrac{a^{8}}{a^{5}}}\)
3. \(\sqrt[4]{\dfrac{a^{10}}{a^{2}}}\)

Suluhisho:

a.

\(\sqrt{\dfrac{m^{6}}{m^{4}}}\)

Kurahisisha sehemu ndani ya radical kwanza. Gawanya misingi kama hiyo kwa kutoa exponents.

\(\sqrt{m^{2}}\)

Kurahisisha.

\(|m|\)

b.

\(\sqrt[3]{\dfrac{a^{8}}{a^{5}}}\)

Matumizi Quotient Mali ya exponents kurahisisha sehemu chini ya radical kwanza.

\(\sqrt[3]{a^{3}}\)

Kurahisisha.

\(a\)

c.

\(\sqrt[4]{\dfrac{a^{10}}{a^{2}}}\)

Matumizi Quotient Mali ya exponents kurahisisha sehemu chini ya radical kwanza.

\(\sqrt[4]{a^{8}}\)

Andika upya radicna kutumia mambo kamili ya nne ya nguvu.

\(\sqrt[4]{\left(a^{2}\right)^{4}}\)

Kurahisisha.

\(a^{2}\)

Jaribu\(\PageIndex{17}\)

Kurahisisha:

1. \(\sqrt{\dfrac{a^{8}}{a^{6}}}\)
2. \(\sqrt[4]{\dfrac{x^{7}}{x^{3}}}\)
3. \(\sqrt[4]{\dfrac{y^{17}}{y^{5}}}\)

Jibu

1. \(|a|\)
2. \(|x|\)
3. \(y^{3}\)

Jaribu\(\PageIndex{18}\)

Kurahisisha:

1. \(\sqrt{\dfrac{x^{14}}{x^{10}}}\)
2. \(\sqrt[3]{\dfrac{m^{13}}{m^{7}}}\)
3. \(\sqrt[5]{\dfrac{n^{12}}{n^{2}}}\)

Jibu

1. \(x^{2}\)
2. \(m^{2}\)
3. \(n^{2}\)

Kumbuka Quotient kwa Mali Nguvu? Ni alisema tunaweza kuongeza sehemu ya nguvu kwa kuongeza nambari na denominator kwa nguvu tofauti.

\(\left(\dfrac{a}{b}\right)^{m}=\dfrac{a^{m}}{b^{m}}, b \neq 0\)

Ufafanuzi\(\PageIndex{3}\)

Mali ya Quotient ya Maneno makubwa

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

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

Mfano\(\PageIndex{10}\) how to simplify the quotient of radical expressions

Kurahisisha:\(\sqrt{\dfrac{27 m^{3}}{196}}\)

Suluhisho:

Hatua ya 1: Kurahisisha sehemu katika radicna, ikiwa inawezekana.

\(\dfrac{27 m^{3}}{196}\)haiwezi kuwa rahisi.

\(\sqrt{\dfrac{27 m^{3}}{196}}\)

Hatua ya 2: Tumia Mali ya Quotient kuandika upya radical kama quotient ya radicals mbili.

Sisi kuandika upya\(\sqrt{\dfrac{27 m^{3}}{196}}\) kama quotient ya\(\sqrt{27 m^{3}}\) na\(\sqrt{196}\).

\(\dfrac{\sqrt{27 m^{3}}}{\sqrt{196}}\)

Hatua ya 3: Kurahisisha radicals katika nambari na denominator.

\(9m^{2}\)na\(196\) ni mraba kamilifu.

\(\dfrac{\sqrt{9 m^{2}} \cdot \sqrt{3 m}}{\sqrt{196}}\)

\(\dfrac{3 m \sqrt{3 m}}{14}\)

Jaribu\(\PageIndex{19}\)

Kurahisisha:\(\sqrt{\dfrac{24 p^{3}}{49}}\).

Jibu

\(\dfrac{2|p| \sqrt{6 p}}{7}\)

Jaribu\(\PageIndex{20}\)

Kurahisisha:\(\sqrt{\dfrac{48 x^{5}}{100}}\).

Jibu

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

Kurahisisha Mizizi ya Mraba Kutumia Mali ya Quotient

1. Kurahisisha sehemu katika radicand, ikiwa inawezekana.
2. Matumizi ya Mali Quotient kuandika upya radical kama quotient ya radicals mbili.
3. Kurahisisha radicals katika nambari na denominator.

Mfano\(\PageIndex{11}\)

Kurahisisha:

1. \(\sqrt{\dfrac{45 x^{5}}{y^{4}}}\)
2. \(\sqrt[3]{\dfrac{24 x^{7}}{y^{3}}}\)
3. \(\sqrt[4]{\dfrac{48 x^{10}}{y^{8}}}\)

Suluhisho:

a.

\(\sqrt{\dfrac{45 x^{5}}{y^{4}}}\)

Hatuwezi kurahisisha sehemu katika radicand. Andika upya kwa kutumia Mali ya Quotient.

\(\dfrac{\sqrt{45 x^{5}}}{\sqrt{y^{4}}}\)

Kurahisisha radicals katika nambari na denominator.

\(\dfrac{\sqrt{9 x^{4}} \cdot \sqrt{5 x}}{y^{2}}\)

Kurahisisha.

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

b.

\(\sqrt[3]{\dfrac{24 x^{7}}{y^{3}}}\)

Sehemu katika radicand haiwezi kuwa rahisi. Tumia Mali ya Quotient kuandika kama radicals mbili.

\(\dfrac{\sqrt[3]{24 x^{7}}}{\sqrt[3]{y^{3}}}\)

Andika upya kila radicna kama bidhaa kwa kutumia mambo kamili ya mchemraba.

\(\dfrac{\sqrt[3]{8 x^{6} \cdot 3 x}}{\sqrt[3]{y^{3}}}\)

Andika upya namba kama bidhaa ya radicals mbili.

\(\dfrac{\sqrt[3]{\left(2 x^{2}\right)^{3}} \cdot \sqrt[3]{3 x}}{\sqrt[3]{y^{3}}}\)

Kurahisisha.

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

c.

\(\sqrt[4]{\dfrac{48 x^{10}}{y^{8}}}\)

Sehemu katika radicand haiwezi kuwa rahisi.

\(\dfrac{\sqrt[4]{48 x^{10}}}{\sqrt[4]{y^{8}}}\)

Tumia Mali ya Quotient kuandika kama radicals mbili. Andika upya kila radicna kama bidhaa kwa kutumia mambo kamili ya nne ya nguvu.

\(\dfrac{\sqrt[4]{16 x^{8} \cdot 3 x^{2}}}{\sqrt[4]{y^{8}}}\)

Andika upya namba kama bidhaa ya radicals mbili.

\(\dfrac{\sqrt[4]{\left(2 x^{2}\right)^{4}} \cdot \sqrt[4]{3 x^{2}}}{\sqrt[4]{\left(y^{2}\right)^{4}}}\)

Kurahisisha.

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

Jaribu\(\PageIndex{21}\)

Kurahisisha:

1. \(\sqrt{\dfrac{80 m^{3}}{n^{6}}}\)
2. \(\sqrt[3]{\dfrac{108 c^{10}}{d^{6}}}\)
3. \(\sqrt[4]{\dfrac{80 x^{10}}{y^{4}}}\)

Jibu

1. \(\dfrac{4|m| \sqrt{5 m}}{\left|n^{3}\right|}\)
2. \(\dfrac{3 c^{3} \sqrt[3]{4 c}}{d^{2}}\)
3. \(\dfrac{2 x^{2} \sqrt[4]{5 x^{2}}}{|y|}\)

Jaribu\(\PageIndex{22}\)

Kurahisisha:

1. \(\sqrt{\dfrac{54 u^{7}}{v^{8}}}\)
2. \(\sqrt[3]{\dfrac{40 r^{3}}{s^{6}}}\)
3. \(\sqrt[4]{\dfrac{162 m^{14}}{n^{12}}}\)

Jibu

1. \(\dfrac{3 u^{3} \sqrt{6 u}}{v^{4}}\)
2. \(\dfrac{2 r \sqrt[3]{5}}{s^{2}}\)
3. \(\dfrac{3\left|m^{3}\right| \sqrt[4]{2 m^{2}}}{\left|n^{3}\right|}\)

Hakikisha kurahisisha sehemu katika radicna kwanza, ikiwa inawezekana.

Mfano\(\PageIndex{12}\)

Kurahisisha:

1. \(\sqrt{\dfrac{18 p^{5} q^{7}}{32 p q^{2}}}\)
2. \(\sqrt[3]{\dfrac{16 x^{5} y^{7}}{54 x^{2} y^{2}}}\)
3. \(\sqrt[4]{\dfrac{5 a^{8} b^{6}}{80 a^{3} b^{2}}}\)

Suluhisho:

a.

\(\sqrt{\dfrac{18 p^{5} q^{7}}{32 p q^{2}}}\)

Kurahisisha sehemu katika radicand, ikiwa inawezekana.

\(\sqrt{\dfrac{9 p^{4} q^{5}}{16}}\)

Andika upya kwa kutumia Mali ya Quotient.

\(\dfrac{\sqrt{9 p^{4} q^{5}}}{\sqrt{16}}\)

Kurahisisha radicals katika nambari na denominator.

\(\dfrac{\sqrt{9 p^{4} q^{4}} \cdot \sqrt{q}}{4}\)

Kurahisisha.

\(\dfrac{3 p^{2} q^{2} \sqrt{q}}{4}\)

b.

\(\sqrt[3]{\dfrac{16 x^{5} y^{7}}{54 x^{2} y^{2}}}\)

Kurahisisha sehemu katika radicand, ikiwa inawezekana.

\(\sqrt[3]{\dfrac{8 x^{3} y^{5}}{27}}\)

Andika upya kwa kutumia Mali ya Quotient.

\(\dfrac{\sqrt[3]{8 x^{3} y^{5}}}{\sqrt[3]{27}}\)

Kurahisisha radicals katika nambari na denominator.

\(\dfrac{\sqrt[3]{8 x^{3} y^{3}} \cdot \sqrt[3]{y^{2}}}{\sqrt[3]{27}}\)

Kurahisisha.

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

c.

\(\sqrt[4]{\dfrac{5 a^{8} b^{6}}{80 a^{3} b^{2}}}\)

Kurahisisha sehemu katika radicand, ikiwa inawezekana.

\(\sqrt[4]{\dfrac{a^{5} b^{4}}{16}}\)

Andika upya kwa kutumia Mali ya Quotient.

\(\dfrac{\sqrt[4]{a^{5} b^{4}}}{\sqrt[4]{16}}\)

Kurahisisha radicals katika nambari na denominator.

\(\dfrac{\sqrt[4]{a^{4} b^{4}} \cdot \sqrt[4]{a}}{\sqrt[4]{16}}\)

Kurahisisha.

\(\dfrac{|a b| \sqrt[4]{a}}{2}\)

Jaribu\(\PageIndex{23}\)

Kurahisisha:

1. \(\sqrt{\dfrac{50 x^{5} y^{3}}{72 x^{4} y}}\)
2. \(\sqrt[3]{\dfrac{16 x^{5} y^{7}}{54 x^{2} y^{2}}}\)
3. \(\sqrt[4]{\dfrac{5 a^{8} b^{6}}{80 a^{3} b^{2}}}\)

Jibu

1. \(\dfrac{5|y| \sqrt{x}}{6}\)
2. \(\dfrac{2 x y \sqrt[3]{y^{2}}}{3}\)
3. \(\dfrac{|a b| \sqrt[4]{a}}{2}\)

Jaribu\(\PageIndex{24}\)

Kurahisisha:

1. \(\sqrt{\dfrac{48 m^{7} n^{2}}{100 m^{5} n^{8}}}\)
2. \(\sqrt[3]{\dfrac{54 x^{7} y^{5}}{250 x^{2} y^{2}}}\)
3. \(\sqrt[4]{\dfrac{32 a^{9} b^{7}}{162 a^{3} b^{3}}}\)

Jibu

1. \(\dfrac{2|m| \sqrt{3}}{5\left|n^{3}\right|}\)
2. \(\dfrac{3 x y \sqrt[3]{x^{2}}}{5}\)
3. \(\dfrac{2|a b| \sqrt[4]{a^{2}}}{3}\)

Katika mfano unaofuata, hakuna kitu cha kurahisisha katika denominators. Tangu index juu ya radicals ni sawa, tunaweza kutumia Mali Quotient tena, kuchanganya yao katika radical moja. Sisi kisha kuangalia kuona kama tunaweza kurahisisha kujieleza.

Mfano\(\PageIndex{13}\)

Kurahisisha:

1. \(\dfrac{\sqrt{48 a^{7}}}{\sqrt{3 a}}\)
2. \(\dfrac{\sqrt[3]{-108}}{\sqrt[3]{2}}\)
3. \(\dfrac{\sqrt[4]{96 x^{7}}}{\sqrt[4]{3 x^{2}}}\)

Suluhisho:

a.

\(\dfrac{\sqrt{48 a^{7}}}{\sqrt{3 a}}\)

Denominator haiwezi kurahisishwa, hivyo tumia Mali ya Quotient kuandika kama radical moja.

\(\sqrt{\dfrac{48 a^{7}}{3 a}}\)

Kurahisisha sehemu chini ya radical.

\(\sqrt{16 a^{6}}\)

Kurahisisha.

\(4\left|a^{3}\right|\)

b.

\(\dfrac{\sqrt[3]{-108}}{\sqrt[3]{2}}\)

Denominator haiwezi kurahisishwa, hivyo tumia Mali ya Quotient kuandika kama radical moja.

\(\sqrt[3]{\dfrac{-108}{2}}\)

Kurahisisha sehemu chini ya radical.

\(\sqrt[3]{-54}\)

Andika upya radicand kama bidhaa kwa kutumia mambo kamili ya mchemraba.

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

Andika upya radical kama bidhaa ya radicals mbili.

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

Kurahisisha.

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

c.

\(\dfrac{\sqrt[4]{96 x^{7}}}{\sqrt[4]{3 x^{2}}}\)

Denominator haiwezi kurahisishwa, hivyo tumia Mali ya Quotient kuandika kama radical moja.

\(\sqrt[4]{\dfrac{96 x^{7}}{3 x^{2}}}\)

Kurahisisha sehemu chini ya radical.

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

Andika upya radicna kama bidhaa kwa kutumia mambo kamili ya nne ya nguvu.

\(\sqrt[4]{16 x^{4}} \cdot \sqrt[4]{2 x}\)

Andika upya radical kama bidhaa ya radicals mbili.

\(\sqrt[4]{(2 x)^{4}} \cdot \sqrt[4]{2 x}\)

Kurahisisha.

\(2|x| \sqrt[4]{2 x}\)

Jaribu\(\PageIndex{25}\)

Kurahisisha:

1. \(\dfrac{\sqrt{98 z^{5}}}{\sqrt{2 z}}\)
2. \(\dfrac{\sqrt[3]{-500}}{\sqrt[3]{2}}\)
3. \(\dfrac{\sqrt[4]{486 m^{11}}}{\sqrt[4]{3 m^{5}}}\)

Jibu

1. \(7z^{2}\)
2. \(-5 \sqrt[3]{2}\)
3. \(3|m| \sqrt[4]{2 m^{2}}\)

Jaribu\(\PageIndex{26}\)

Kurahisisha:

1. \(\dfrac{\sqrt{128 m^{9}}}{\sqrt{2 m}}\)
2. \(\dfrac{\sqrt[3]{-192}}{\sqrt[3]{3}}\)
3. \(\dfrac{\sqrt[4]{324 n^{7}}}{\sqrt[4]{2 n^{3}}}\)

Jibu

1. \(8m^{4}\)
2. \(-4\)
3. \(3|n| \sqrt[4]{2}\)

Fikia rasilimali hizi za mtandaoni kwa maelekezo ya ziada na mazoezi na kurahisisha maneno makubwa.

• Kurahisisha Mizizi ya Mraba na Mizizi ya mchemraba
• Express Radical katika Simplified Fomu-Mraba na Cube mizizi na Vigezo na Exponents
• Kurahisisha mizizi mchemraba

Dhana muhimu

• Rahisi radical kujieleza
  • Kwa idadi halisi\(a, m\) na\(n≥2\)
    \(\sqrt[n]{a}\) ni kuchukuliwa kilichorahisishwa kama\(a\) hana sababu ya\(m^{n}\)
• Bidhaa Mali ya\(n^{th}\) 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\(\sqrt[n]{a} \cdot \sqrt[n]{b}=\sqrt[n]{a b}\)
• Jinsi ya kurahisisha kujieleza radical kutumia Mali Bidhaa
  1. Pata sababu kubwa zaidi katika radicana hiyo ni nguvu kamili ya index.
     Andika upya radicna kama bidhaa ya mambo mawili, kwa kutumia jambo hilo.
  2. Tumia utawala wa bidhaa ili uandike upya radical kama bidhaa ya radicals mbili.
  3. Kurahisisha mizizi ya nguvu kamili.
• Mali ya Quotient ya Maneno ya Radical
  • 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\(\dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\dfrac{a}{b}}\)
• Jinsi ya kurahisisha kujieleza radical kwa kutumia Mali Quotient.
  1. Kurahisisha sehemu katika radicand, ikiwa inawezekana.
  2. Matumizi ya Mali Quotient kuandika upya radical kama quotient ya radicals mbili.
  3. Kurahisisha radicals katika nambari na denominator.