8.4: Kurahisisha Watazamaji wa busara

Malengo ya kujifunza

Mwishoni mwa sehemu hii, utaweza:

• Kurahisisha maneno na\(a^{\frac{1}{n}}\)
• Kurahisisha maneno na\(a^{\frac{m}{n}}\)
• Matumizi mali ya exponents kurahisisha maneno na exponents busara

Kabla ya kuanza, fanya jaribio hili la utayari.

1. Ongeza:\(\frac{7}{15}+\frac{5}{12}\).
   Kama amekosa tatizo hili, mapitio Mfano 1.28.
2. Kurahisisha:\((4x^{2}y^{5})^{3}\).
   Kama amekosa tatizo hili, mapitio Mfano 5.18.
3. Kurahisisha:\(5^{−3}\).
   Kama amekosa tatizo hili, mapitio Mfano 5.14.

Kurahisisha Maneno na\(a^{\frac{1}{n}}\)

Watazamaji wa busara ni njia nyingine ya kuandika maneno na radicals. Wakati sisi kutumia exponents busara, tunaweza kutumia mali ya exponents kurahisisha maneno.

Power Mali kwa Exponents anasema kwamba\(\left(a^{m}\right)^{n}=a^{m \cdot n}\) wakati\(m\) na\(n\) ni idadi nzima. Hebu kudhani sisi sasa si mdogo kwa idadi nzima.

Tuseme tunataka kupata idadi\(p\) kama hiyo\(\left(8^{p}\right)^{3}=8\). Tutatumia Power Mali ya Exponents kupata thamani ya\(p\).

\(\left(8^{p}\right)^{3}=8\)

Multiple exponents upande wa kushoto.

\(8^{3p}=8\)

Andika kielelezo\(1\) upande wa kulia.

\(8^{3p}=8^{1}\)

Kwa kuwa besi ni sawa, watazamaji lazima wawe sawa.

\(3p=1\)

Kutatua kwa\(p\).

\(p=\frac{1}{3}\)

Hivyo\(\left(8^{\frac{1}{3}}\right)^{3}=8\). Lakini tunajua pia\((\sqrt[3]{8})^{3}=8\). Kisha ni lazima iwe hivyo\(8^{\frac{1}{3}}=\sqrt[3]{8}\).

Hii mantiki hiyo inaweza kutumika kwa ajili ya yoyote chanya integer exponent\(n\) kuonyesha kwamba\(a^{\frac{1}{n}}=\sqrt[n]{a}\).

Ufafanuzi\(\PageIndex{1}\): Rational Exponent \(a^{\frac{1}{n}}\)

Kama\(\sqrt[n]{a}\) ni idadi halisi na\(n \geq 2\), basi

\(a^{\frac{1}{n}}=\sqrt[n]{a}\)

Denominator ya exponent busara ni index ya radical.

Kutakuwa na nyakati ambapo kufanya kazi na maneno itakuwa rahisi ikiwa unatumia vielelezo vya busara na nyakati ambapo itakuwa rahisi ikiwa unatumia radicals. Katika mifano michache ya kwanza, utasikia mazoezi ya kubadilisha maneno kati ya maelezo haya mawili.

Mfano\(\PageIndex{1}\)

Andika kama kujieleza kwa kiasi kikubwa:

1. \(x^{\frac{1}{2}}\)
2. \(y^{\frac{1}{3}}\)
3. \(z^{\frac{1}{4}}\)

Suluhisho:

Tunataka kuandika kila kujieleza kwa fomu\(\sqrt[n]{a}\).

a.

\(x^{\frac{1}{2}}\)

Denominator ya exponent busara ni\(2\), hivyo index ya radical ni\(2\). Hatuonyeshi index wakati ni\(2\).

\(\sqrt{x}\)

b.

\(y^{\frac{1}{3}}\)

Denominator ya exponent ni\(3\), hivyo index ni\(3\).

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

c.

\(z^{\frac{1}{4}}\)

denominator ya exponent ni\\(4\), hivyo index ni\(4\).

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

Zoezi\(\PageIndex{1}\)

Andika kama kujieleza kwa kiasi kikubwa:

1. \(t^{\frac{1}{2}}\)
2. \(m^{\frac{1}{3}}\)
3. \(r^{\frac{1}{4}}\)

Jibu

1. \(\sqrt{t}\)
2. \(\sqrt[3]{m}\)
3. \(\sqrt[4]{r}\)

Zoezi\(\PageIndex{2}\)

Andika kama kujieleza kwa kiasi kikubwa:

1. \(b^{\frac{1}{6}}\)
2. \(z^{\frac{1}{5}}\)
3. \(p^{\frac{1}{4}}\)

Jibu

1. \(\sqrt[6]{b}\)
2. \(\sqrt[5]{z}\)
3. \(\sqrt[4]{p}\)

Katika mfano unaofuata, tutaandika kila radical kwa kutumia exponent busara. Ni muhimu kutumia mabano karibu na kujieleza nzima katika radicana tangu kujieleza nzima hufufuliwa kwa nguvu ya busara.

Mfano\(\PageIndex{2}\)

Andika kwa kielelezo cha busara:

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

Suluhisho:

Tunataka kuandika kila radical katika fomu\(a^{\frac{1}{n}}\)

a.

\(\sqrt{5y}\)

Hakuna index inavyoonyeshwa, hivyo ni\(2\).

Denominator ya exponent itakuwa\(2\).

Weka mabano karibu na kujieleza nzima\(5y\).

\((5 y)^{\frac{1}{2}}\)

b.

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

Ripoti ni\(3\), hivyo denominator ya exponent ni\(3\). Jumuisha mabano\((4x)\).

\((4 x)^{\frac{1}{3}}\)

c.

\(3 \sqrt[4]{5 z}\)

Ripoti ni\(4\), hivyo denominator ya exponent ni\(4\). Weka mabano tu karibu\(5z\) tangu 3 si chini ya ishara radical.

\(3(5 z)^{\frac{1}{4}}\)

Zoezi\(\PageIndex{3}\)

Andika kwa kielelezo cha busara:

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

Jibu

1. \((10 m)^{\frac{1}{2}}\)
2. \((3 n)^{\frac{1}{5}}\)
3. \(3(6 y)^{\frac{1}{4}}\)

Zoezi\(\PageIndex{4}\)

Andika kwa kielelezo cha busara:

1. \(\sqrt[7]{3 k}\)
2. \(\sqrt[4]{5 j}\)
3. \(8 \sqrt[3]{2 a}\)

Jibu

1. \((3 k)^{\frac{1}{7}}\)
2. \((5 j)^{\frac{1}{4}}\)
3. \(8(2 a)^{\frac{1}{3}}\)

Katika mfano unaofuata, unaweza kupata rahisi kurahisisha maneno ikiwa utaandika tena kama radicals kwanza.

Mfano\(\PageIndex{3}\)

Kurahisisha:

1. \(25^{\frac{1}{2}}\)
2. \(64^{\frac{1}{3}}\)
3. \(256^{\frac{1}{4}}\)

Suluhisho:

a.

\(25^{\frac{1}{2}}\)

Andika upya kama mizizi ya mraba.

\(\sqrt{25}\)

Kurahisisha.

\(5\)

b.

\(64^{\frac{1}{3}}\)

Andika upya kama mizizi ya mchemraba.

\(\sqrt[3]{64}\)

Kutambua\(64\) ni mchemraba kamili.

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

Kurahisisha.

\(4\)

c.

\(256^{\frac{1}{4}}\)

Andika upya kama mizizi ya nne.

\(\sqrt[4]{256}\)

Kutambua\(256\) ni nguvu kamili ya nne.

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

Kurahisisha.

\(4\)

Zoezi\(\PageIndex{5}\)

Kurahisisha:

1. \(36^{\frac{1}{2}}\)
2. \(8^{\frac{1}{3}}\)
3. \(16^{\frac{1}{4}}\)

Jibu

1. \(6\)
2. \(2\)
3. \(2\)

Zoezi\(\PageIndex{6}\)

Kurahisisha:

1. \(100^{\frac{1}{2}}\)
2. \(27^{\frac{1}{3}}\)
3. \(81^{\frac{1}{4}}\)

Jibu

1. \(10\)
2. \(3\)
3. \(3\)

Kuwa makini na kuwekwa kwa ishara hasi katika mfano unaofuata. Tutahitaji kutumia mali\(a^{-n}=\frac{1}{a^{n}}\) katika kesi moja.

Mfano\(\PageIndex{4}\)

Kurahisisha:

1. \((-16)^{\frac{1}{4}}\)
2. \(-16^{\frac{1}{4}}\)
3. \((16)^{-\frac{1}{4}}\)

Suluhisho:

a.

\((-16)^{\frac{1}{4}}\)

Andika upya kama mizizi ya nne.

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

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

Kurahisisha.

Hakuna ufumbuzi halisi

b.

\(-16^{\frac{1}{4}}\)

exponent inatumika tu kwa\(16\). Andika upya kama mizizi ya nne.

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

Andika upya\(16\) kama\(2^{4}\)

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

Kurahisisha.

\(-2\)

c.

\((16)^{-\frac{1}{4}}\)

Andika upya kutumia mali\(a^{-n}=\frac{1}{a^{n}}\).

\(\frac{1}{(16)^{\frac{1}{4}}}\)

Andika upya kama mizizi ya nne.

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

Andika upya\(16\) kama\(2^{4}\).

\(\frac{1}{\sqrt[4]{2^{4}}}\)

Kurahisisha.

\(\frac{1}{2}\)

Zoezi\(\PageIndex{7}\)

Kurahisisha:

1. \((-64)^{-\frac{1}{2}}\)
2. \(-64^{\frac{1}{2}}\)
3. \((64)^{-\frac{1}{2}}\)

Jibu

1. Hakuna ufumbuzi halisi
2. \(-8\)
3. \(\frac{1}{8}\)

Zoezi\(\PageIndex{8}\)

Kurahisisha:

1. \((-256)^{\frac{1}{4}}\)
2. \(-256^{\frac{1}{4}}\)
3. \((256)^{-\frac{1}{4}}\)

Jibu

1. Hakuna ufumbuzi halisi
2. \(-4\)
3. \(\frac{1}{4}\)

Kurahisisha Maneno na\(a^{\frac{m}{n}}\)

Tunaweza kuangalia\(a^{\frac{m}{n}}\) kwa njia mbili. Kumbuka Power Mali inatuambia kuzidisha exponents na hivyo\(\left(a^{\frac{1}{n}}\right)^{m}\) na\(\left(a^{m}\right)^{\frac{1}{n}}\) wote sawa\(a^{\frac{m}{n}}\). Ikiwa tunaandika maneno haya kwa fomu kali, tunapata

\(a^{\frac{m}{n}}=\left(a^{\frac{1}{n}}\right)^{m}=(\sqrt[n]{a})^{m} \quad \text { and } \quad a^{\frac{m}{n}}=\left(a^{m}\right)^{^{\frac{1}{n}}}=\sqrt[n]{a^{m}}\)

Hii inatuongoza kwa ufafanuzi wafuatayo.

Ufafanuzi\(\PageIndex{2}\): Rational Exponent \(a^{\frac{m}{n}}\)

Kwa integers yoyote chanya\(m\) na\(n\),

\(a^{\frac{m}{n}}=(\sqrt[n]{a})^{m} \quad \text { and } \quad a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\)

Ni aina gani tunayotumia ili kurahisisha kujieleza? Kwa kawaida tunachukua mizizi kwanza-kwa njia hiyo tunaweka namba katika radicna ndogo, kabla ya kuinua kwa nguvu zilizoonyeshwa.

Mfano\(\PageIndex{5}\)

Andika kwa kielelezo cha busara:

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

Suluhisho:

Tunataka kutumia\(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\) kuandika kila radical katika fomu\(a^{\frac{m}{n}}\)

a.

Kielelezo 8.3.1

b.

Kielelezo 8.3.2

c.

Kielelezo 8.3.3

Zoezi\(\PageIndex{9}\)

Andika kwa kielelezo cha busara:

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

Jibu

1. \(x^{\frac{5}{2}}\)
2. \((3 y)^{\frac{3}{4}}\)
3. \(\left(\frac{2 m}{3 n}\right)^{\frac{5}{2}}\)

Zoezi\(\PageIndex{10}\)

Andika kwa kielelezo cha busara:

1. \(\sqrt[5]{a^{2}}\)
2. \((\sqrt[3]{5 a b})^{5}\)
3. \(\sqrt{\left(\frac{7 x y}{z}\right)^{3}}\)

Jibu

1. \(a^{\frac{2}{5}}\)
2. \((5 a b)^{\frac{5}{3}}\)
3. \(\left(\frac{7 x y}{z}\right)^{\frac{3}{2}}\)

Kumbuka hilo\(a^{-n}=\frac{1}{a^{n}}\). Ishara mbaya katika exponent haina mabadiliko ya ishara ya kujieleza.

Mfano\(\PageIndex{6}\)

Kurahisisha:

1. \(125^{\frac{2}{3}}\)
2. \(16^{-\frac{3}{2}}\)
3. \(32^{-\frac{2}{5}}\)

Suluhisho:

Sisi kuandika upya maneno kama radical kwanza kutumia defintion,\(a^{\frac{m}{n}}=(\sqrt[n]{a})^{m}\). Fomu hii inatuwezesha kuchukua mizizi kwanza na hivyo tunaweka namba katika radicna ndogo kuliko kama tulitumia fomu nyingine.

a.

\(125^{\frac{2}{3}}\)

Nguvu ya radical ni nambari ya exponent,\(2\). Ripoti ya radical ni denominator ya exponent,\(3\).

\((\sqrt[3]{125})^{2}\)

Kurahisisha.

\((5)^{2}\)

\(25\)

b Tutaandika upya kila kujieleza kwanza kwa kutumia\(a^{-n}=\frac{1}{a^{n}}\) na kisha kubadili fomu kali.

\(16^{-\frac{3}{2}}\)

Andika upya kwa kutumia\(a^{-n}=\frac{1}{a^{n}}\)

\(\frac{1}{16^{\frac{3}{2}}}\)

Badilisha kwa fomu kali. Nguvu ya radical ni nambari ya exponent,\(3\). Ripoti ni denominator ya exponent,\(2\).

\(\frac{1}{(\sqrt{16})^{3}}\)

Kurahisisha.

\(\frac{1}{4^{3}}\)

\(\frac{1}{64}\)

c.

\(32^{-\frac{2}{5}}\)

Andika upya kwa kutumia\(a^{-n}=\frac{1}{a^{n}}\)

\(\frac{1}{32^{\frac{2}{5}}}\)

Badilisha kwa fomu kali.

\(\frac{1}{(\sqrt[5]{32})^{2}}\)

Andika upya radicand kama nguvu.

\(\frac{1}{\left(\sqrt[5]{2^{5}}\right)^{2}}\)

Kurahisisha.

\(\frac{1}{2^{2}}\)

\(\frac{1}{4}\)

Zoezi\(\PageIndex{11}\)

Kurahisisha:

1. \(27^{\frac{2}{3}}\)
2. \(81^{-\frac{3}{2}}\)
3. \(16^{-\frac{3}{4}}\)

Jibu

1. \(9\)
2. \(\frac{1}{729}\)
3. \(\frac{1}{8}\)

Zoezi\(\PageIndex{12}\)

Kurahisisha:

1. \(4^{\frac{3}{2}}\)
2. \(27^{-\frac{2}{3}}\)
3. \(625^{-\frac{3}{4}}\)

Jibu

1. \(8\)
2. \(\frac{1}{9}\)
3. \(\frac{1}{125}\)

Mfano\(\PageIndex{7}\)

Kurahisisha:

1. \(-25^{\frac{3}{2}}\)
2. \(-25^{-\frac{3}{2}}\)
3. \((-25)^{\frac{3}{2}}\)

Suluhisho:

a.

\(-25^{\frac{3}{2}}\)

Andika upya kwa fomu kali.

\(-(\sqrt{25})^{3}\)

Kurahisisha radical.

\(-(5)^{3}\)

Kurahisisha.

\(-125\)

b.

\(-25^{-\frac{3}{2}}\)

Andika upya kwa kutumia\(a^{-n}=\frac{1}{a^{n}}\).

\(-\left(\frac{1}{25^{\frac{3}{2}}}\right)\)

Andika upya kwa fomu kali.

\(-\left(\frac{1}{(\sqrt{25})^{3}}\right)\)

Kurahisisha radical.

\(-\left(\frac{1}{(5)^{3}}\right)\)

Kurahisisha.

\(-\frac{1}{125}\)

c.

\((-25)^{\frac{3}{2}}\)

Andika upya kwa fomu kali.

\((\sqrt{-25})^{3}\)

Hakuna idadi halisi ambayo mizizi ya mraba ni\(-25\).

Si idadi halisi.

Zoezi\(\PageIndex{13}\)

Kurahisisha:

1. \(-16^{\frac{3}{2}}\)
2. \(-16^{-\frac{3}{2}}\)
3. \((-16)^{-\frac{3}{2}}\)

Jibu

1. \(-64\)
2. \(-\frac{1}{64}\)
3. Si idadi halisi

Zoezi\(\PageIndex{14}\)

Kurahisisha:

1. \(-81^{\frac{3}{2}}\)
2. \(-81^{-\frac{3}{2}}\)
3. \((-81)^{-\frac{3}{2}}\)

Jibu

1. \(-729\)
2. \(-\frac{1}{729}\)
3. Si idadi halisi

Tumia Mali ya Watazamaji ili kurahisisha Maneno na Maonyesho ya Mantiki

Mali sawa ya watazamaji ambao tayari tumetumia pia hutumika kwa watazamaji wa busara. Sisi orodha Mali ya Exponents hapa kuwa nao kwa ajili ya kumbukumbu kama sisi kurahisisha maneno.

Mali ya Watazamaji

Ikiwa\(a\) na\(b\) ni idadi halisi\(m\) na\(n\) ni namba za busara, basi

Bidhaa Mali

\(a^{m} \cdot a^{n}=a^{m+n}\)

Power Mali

\(\left(a^{m}\right)^{n}=a^{m \cdot n}\)

Bidhaa kwa Nguvu

\((a b)^{m}=a^{m} b^{m}\)

Mali ya Quotient

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

Ufafanuzi wa sifuri

\(a^{0}=1, a \neq 0\)

Quotient kwa Mali Nguvu

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

Hasi Exponent Mali

\(a^{-n}=\frac{1}{a^{n}}, a \neq 0\)

Tutatumia mali hizi katika mfano unaofuata.

Mfano\(\PageIndex{8}\)

Kurahisisha:

1. \(x^{\frac{1}{2}} \cdot x^{\frac{5}{6}}\)
2. \(\left(z^{9}\right)^{\frac{2}{3}}\)
3. \(\frac{x^{\frac{1}{3}}}{x^{\frac{5}{3}}}\)

Suluhisho

a. bidhaa Mali inatuambia kwamba wakati sisi nyingi msingi huo, sisi kuongeza exponents.

\(x^{\frac{1}{2}} \cdot x^{\frac{5}{6}}\)

Msingi ni sawa, kwa hiyo tunaongeza wafuasi.

\(x^{\frac{1}{2}+\frac{5}{6}}\)

Ongeza sehemu ndogo.

\(x^{\frac{8}{6}}\)

Kurahisisha exponent.

\(x^{\frac{4}{3}}\)

b. Power Mali inatuambia kwamba wakati sisi kuongeza nguvu kwa nguvu, sisi nyingi exponents.

\(\left(z^{9}\right)^{\frac{2}{3}}\)

Kuongeza nguvu kwa nguvu, sisi nyingi exponents.

\(z^{9 \cdot \frac{2}{3}}\)

Kurahisisha.

\(z^{6}\)

c. mali Quotient inatuambia kwamba wakati sisi kugawanya na msingi huo, sisi Ondoa exponents.

\(\frac{x^{\frac{1}{3}}}{x^{\frac{5}{3}}}\)

Ili kugawanywa na msingi huo, tunaondoa wafuasi.

\(\frac{1}{x^{\frac{5}{3}-\frac{1}{3}}}\)

Kurahisisha.

\(\frac{1}{x^{\frac{4}{3}}}\)

Zoezi\(\PageIndex{15}\)

Kurahisisha:

1. \(x^{\frac{1}{6}} \cdot x^{\frac{4}{3}}\)
2. \(\left(x^{6}\right)^{\frac{4}{3}}\)
3. \(\frac{x^{\frac{2}{3}}}{x^{\frac{5}{3}}}\)

Jibu

1. \(x^{\frac{3}{2}}\)
2. \(x^{8}\)
3. \(\frac{1}{x}\)

Zoezi\(\PageIndex{16}\)

Kurahisisha:

1. \(y^{\frac{3}{4}} \cdot y^{\frac{5}{8}}\)
2. \(\left(m^{9}\right)^{\frac{2}{9}}\)
3. \(\frac{d^{\frac{1}{5}}}{d^{\frac{6}{5}}}\)

Jibu

1. \(y^{\frac{11}{8}}\)
2. \(m^{2}\)
3. \(\frac{1}{d}\)

Wakati mwingine tunahitaji kutumia mali zaidi ya moja. Katika mfano unaofuata, tutatumia Bidhaa zote kwa Mali ya Nguvu na kisha Mali ya Nguvu.

Mfano\(\PageIndex{9}\)

Kurahisisha:

1. \(\left(27 u^{\frac{1}{2}}\right)^{\frac{2}{3}}\)
2. \(\left(m^{\frac{2}{3}} n^{\frac{1}{2}}\right)^{\frac{3}{2}}\)

Suluhisho:

a.

\(\left(27 u^{\frac{1}{2}}\right)^{\frac{2}{3}}\)

Kwanza tunatumia Bidhaa kwa Mali ya Nguvu.

\((27)^{\frac{2}{3}}\left(u^{\frac{1}{2}}\right)^{\frac{2}{3}}\)

Andika upya\(27\) kama nguvu ya\(3\).

\(\left(3^{3}\right)^{\frac{2}{3}}\left(u^{\frac{1}{2}}\right)^{\frac{2}{3}}\)

Kuongeza nguvu kwa nguvu, sisi nyingi exponents.

\(\left(3^{2}\right)\left(u^{\frac{1}{3}}\right)\)

Kurahisisha.

\(9 u^{\frac{1}{3}}\)

b.

\(\left(m^{\frac{2}{3}} n^{\frac{1}{2}}\right)^{\frac{3}{2}}\)

Kwanza tunatumia Bidhaa kwa Mali ya Nguvu.

\(\left(m^{\frac{2}{3}}\right)^{\frac{3}{2}}\left(n^{\frac{1}{2}}\right)^{\frac{3}{2}}\)

Ili kuongeza nguvu kwa nguvu, tunazidisha vielelezo.

\(m n^{\frac{3}{4}}\)

Zoezi\(\PageIndex{17}\)

Kurahisisha:

1. \(\left(32 x^{\frac{1}{3}}\right)^{\frac{3}{5}}\)
2. \(\left(x^{\frac{3}{4}} y^{\frac{1}{2}}\right)^{\frac{2}{3}}\)

Jibu

1. \(8 x^{\frac{1}{5}}\)
2. \(x^{\frac{1}{2}} y^{\frac{1}{3}}\)

Zoezi\(\PageIndex{18}\)

Kurahisisha:

1. \(\left(81 n^{\frac{2}{5}}\right)^{\frac{3}{2}}\)
2. \(\left(a^{\frac{3}{2}} b^{\frac{1}{2}}\right)^{\frac{4}{3}}\)

Jibu

1. \(729 n^{\frac{3}{5}}\)
2. \(a^{2} b^{\frac{2}{3}}\)

Tutatumia Mali ya Bidhaa na Mali ya Quotient katika mfano unaofuata.

Mfano\(\PageIndex{10}\)

Kurahisisha:

1. \(\frac{x^{\frac{3}{4}} \cdot x^{-\frac{1}{4}}}{x^{-\frac{6}{4}}}\)
2. \(\left(\frac{16 x^{\frac{4}{3}} y^{-\frac{5}{6}}}{x^{-\frac{2}{3}} y^{\frac{1}{6}}}\right)^{\frac{1}{2}}\)

Suluhisho:

a.

\(\frac{x^{\frac{3}{4}} \cdot x^{-\frac{1}{4}}}{x^{-\frac{6}{4}}}\)

Tumia Mali ya Bidhaa katika nambari, ongeza vielelezo.

\(\frac{x^{\frac{2}{4}}}{x^{-\frac{6}{4}}}\)

Tumia Mali ya Quotient, Ondoa wafuasi.

\(x^{\frac{8}{4}}\)

Kurahisisha.

\(x^{2}\)

b.

\(\left(\frac{16 x^{\frac{4}{3}} y^{-\frac{5}{6}}}{x^{-\frac{2}{3}} y^{\frac{1}{6}}}\right)^{\frac{1}{2}}\)

Tumia Mali ya Quotient, Ondoa wafuasi.

\(\left(\frac{16 x^{\frac{6}{3}}}{y^{\frac{6}{6}}}\right)^{\frac{1}{2}}\)

Kurahisisha.

\(\left(\frac{16 x^{2}}{y}\right)^{\frac{1}{2}}\)

Matumizi ya Bidhaa kwa Power Mali, kuzidisha exponents.

\(\frac{4 x}{y^{\frac{1}{2}}}\)

Zoezi\(\PageIndex{19}\)

Kurahisisha:

1. \(\frac{m^{\frac{2}{3}} \cdot m^{-\frac{1}{3}}}{m^{-\frac{5}{3}}}\)
2. \(\left(\frac{25 m^{\frac{1}{6}} n^{\frac{11}{6}}}{m^{\frac{2}{3}} n^{-\frac{1}{6}}}\right)^{\frac{1}{2}}\)

Jibu

1. \(m^{2}\)
2. \(\frac{5 n}{m^{\frac{1}{4}}}\)

Zoezi\(\PageIndex{20}\)

Kurahisisha:

1. \(\frac{u^{\frac{4}{5}} \cdot u^{-\frac{2}{5}}}{u^{-\frac{13}{5}}}\)
2. \(\left(\frac{27 x^{\frac{4}{5}} y^{\frac{1}{6}}}{x^{\frac{1}{5}} y^{-\frac{5}{6}}}\right)^{\frac{1}{3}}\)

Jibu

1. \(u^{3}\)
2. \(3 x^{\frac{1}{5}} y^{\frac{1}{3}}\)

Kupata rasilimali hizi online kwa maelekezo ya ziada na mazoezi na kurahisisha exponents busara.

• Mapitio ya busara
• Kutumia Sheria za Watazamaji juu ya Radicals: Mali ya Maonyesho ya Mantiki

Dhana muhimu

• Mtazamo wa busara\(a^{\frac{1}{n}}\)
  • Ikiwa\(\sqrt[n]{a}\) ni namba halisi na\(n≥2\), basi\(a^{\frac{1}{n}}=\sqrt[n]{a}\).
• Mtazamo wa busara\(a^{\frac{m}{n}}\)
  • Kwa integers yoyote chanya\(m\) na\(n\),
    \(a^{\frac{m}{n}}=(\sqrt[n]{a})^{m} \text { and } a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\)
• Mali ya Watazamaji
  • Ikiwa\(a, b\) ni idadi halisi na\(m, n\) ni namba za busara, basi
    • Bidhaa Mali\(a^{m} \cdot a^{n}=a^{m+n}\)
    • Power Mali\(\left(a^{m}\right)^{n}=a^{m \cdot n}\)
    • Bidhaa kwa Nguvu\((a b)^{m}=a^{m} b^{m}\)
    • Mali ya Quotient\(\frac{a^{m}}{a^{n}}=a^{m-n}, a \neq 0\)
    • Ufafanuzi wa sifuri\(a^{0}=1, a \neq 0\)
    • Quotient kwa Mali Nguvu\(\left(\frac{a}{b}\right)^{m}=\frac{a^{m}}{b^{m}}, b \neq 0\)
    • Hasi Exponent Mali\(a^{-n}=\frac{1}{a^{n}}, a \neq 0\)