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9.8: Watazamaji wa busara

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    Malengo ya kujifunza

    Mwishoni mwa sehemu hii, utaweza:

    • Kurahisisha maneno na\(a^{\frac{1}{n}}\)
    • Kurahisisha maneno na\(a^{\frac{m}{n}}\)
    • Matumizi Sheria ya Exponents kwa maneno tu na exponents busara
    Kuwa tayari

    Kabla ya kuanza, fanya jaribio hili la utayari.

    1. Ongeza:\(\frac{7}{15}+\frac{5}{12}\).
      Ikiwa umekosa tatizo hili, tathmini [kiungo].
    2. Kurahisisha:\((4x^{2}y^{5})^3\).
      Ikiwa umekosa tatizo hili, tathmini [kiungo].
    3. Kurahisisha:\(5^{−3}\).
      Ikiwa umekosa tatizo hili, tathmini [kiungo].

    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\((a^m)^n=a^{m·n}\) wakati m na n ni namba nzima. Hebu kudhani sisi sasa si mdogo kwa idadi nzima.

    Tuseme tunataka kupata idadi p vile kwamba\((8^p)^3=8\). Tutatumia Mali ya Nguvu ya Maonyesho ili kupata thamani ya p.

    \[\begin{array}{cc} {}&{(8^p)^3=8}\\ {\text{Multiply the exponents on the left.}}&{8^{3p}=8}\\ {\text{Write the exponent 1 on the right.}}&{8^{3p}=8^1}\\ {\text{The exponents must be equal.}}&{3p=1}\\ {\text{Solve for p.}}&{p=\frac{1}{3}}\\ \nonumber \end{array}\]

    Lakini tunajua pia\((\sqrt[3]{8})^3=8\). Basi ni lazima kwamba\(8^{\frac{1}{3}}=\sqrt[3]{8}\)

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

    Ufafanuzi: MTAZAMO MTA\(a^{\frac{1}{n}}\)

    Kama\(\sqrt[n]{a}\) ni idadi halisi na\(n \ge 2\),\(a^{\frac{1}{n}}=\sqrt[n]{a}\).

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

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

    1. \(x^{\frac{1}{2}}\)
    Denominator ya exponent ni 2, hivyo index ya radical ni 2. Hatuonyeshi index wakati ni 2. \(\sqrt{x}\)
    2. \(y^{\frac{1}{3}}\)
    Denominator ya exponent ni 3, hivyo index ni 3. \(\sqrt[3]{y}\)
    3. \(z^\frac{1}{4}}\)
    Denominator ya exponent ni 4, hivyo index ni 4. \(\sqrt[4]{z}\)
    Mfano\(\PageIndex{2}\)

    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. \(\sqrt[4]{r}\)
    Mfano\(\PageIndex{3}\)

    Andika kama kujieleza kwa kiasi kikubwa:

    1. \(b^{\frac{1}{2}}\)
    2. \(z^{\frac{1}{3}}\)
    3. \(p^{\frac{1}{4}}\).
    Jibu
    1. \(\sqrt{b}\)
    2. \(\sqrt[3]{z}\)
    3. \(\sqrt[4]{p}\)
    Mfano\(\PageIndex{4}\)

    Andika kwa kielelezo cha busara:

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

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

    1. \(\sqrt{x}\)
    Hakuna index ni umeonyesha, hivyo ni 2. denominator ya exponent itakuwa 2. \(x^{\frac{1}{2}}\)
    2. \(\sqrt[3]{y}\)
    Ripoti ni 3, hivyo denominator ya exponent ni 3. \(y^{\frac{1}{3}}\)
    3. \(\sqrt[4]{z}\)
    Ripoti ni 4, hivyo denominator ya exponent ni 4. \(z^{\frac{1}{4}}\)
    Mfano\(\PageIndex{5}\)

    Andika kwa kielelezo cha busara:

    1. \(\sqrt{s}\)
    2. \(\sqrt[3]{x}\)
    3. \(\sqrt[4]{b}\).
    Jibu
    1. \(s^{\frac{1}{2}}\)
    2. \(x^{\frac{1}{3}}\)
    3. \ (b^ {\ frac {1} {4}}}\
    Mfano\(\PageIndex{6}\)

    Andika kwa kielelezo cha busara:

    1. \(\sqrt{v}\)
    2. \(\sqrt[3]{p}\)
    3. \(\sqrt[4]{p}\).
    Jibu
    1. \(v^{\frac{1}{2}}\)
    2. \(p^{\frac{1}{3}}\)
    3. \(p^{\frac{1}{4}}\)
    Mfano\(\PageIndex{7}\)

    Andika kwa kielelezo cha busara:

    1. \(\sqrt{5y}\)
    2. \(\sqrt[3]{4x}\)
    3. \(3\sqrt[4]{5z}\).
    Jibu
    1. \(\sqrt{5y}\)
    Hakuna index ni umeonyesha, hivyo ni 2. denominator ya exponent itakuwa 2. \((5y)^{\frac{1}{2}}\)
    2. \(\sqrt[3]{4x}\)
    Ripoti ni 3, hivyo denominator ya exponent ni 3. \((4x)^{\frac{1}{3}}\)
    3. \(3\sqrt[4]{5z}\)
    Ripoti ni 4, hivyo denominator ya exponent ni 4. \(3(5z)^{\frac{1}{4}}\)
    Mfano\(\PageIndex{8}\)

    Andika kwa kielelezo cha busara:

    1. \(\sqrt{10m}\)
    2. \(\sqrt[5]{3n}\)
    3. \(3\sqrt[4]{6y}\).
    Jibu
    1. \((10^m)^{\frac{1}{2}}\)
    2. \((3n)^{\frac{1}{5}}\)
    3. \((486y)^{\frac{1}{4}}\)
    Mfano\(\PageIndex{9}\)

    Andika kwa kielelezo cha busara:

    1. \(\sqrt[7]{3k}\)
    2. \(\sqrt[4]{5j}\)
    3. \(\sqrt[3]{82a}\).
    Jibu
    1. \((3k)^{\frac{1}{7}}\)
    2. \((5j)^{\frac{1}{4}}\)
    3. \((1024a)^{\frac{1}{3}}\)

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

    Mfano\(\PageIndex{10}\)

    Kurahisisha:

    1. \(25^{\frac{1}{2}}\)
    2. \(64^{\frac{1}{3}}\)
    3. \(256^{\frac{1}{4}}\).
    Jibu
    1. \(25^{\frac{1}{2}}\)
    Andika upya kama mizizi ya mraba. \(\sqrt{25}\)
    Kurahisisha. 5
    2. \(64^{\frac{1}{3}}\)
    Andika upya kama mizizi ya mchemraba. \(\sqrt[3]{64}\)
    Kutambua 64 ni mchemraba kamilifu. \(\sqrt[3]{4^3}\)
    Kurahisisha. 4
    3. \(256^{\frac{1}{4}}\)
    Andika upya kama mizizi ya nne. \(\sqrt[4]{256}\)
    Tambua 256 ni nguvu kamili ya nne. \(\sqrt[4]{4^4}\)
    Kurahisisha. 4
    Mfano\(\PageIndex{11}\)

    Kurahisisha:

    1. \(36^{\frac{1}{2}}\)
    2. \(8^{\frac{1}{3}}\)
    3. \(16^{\frac{1}{4}}\).
    Jibu
    1. 6
    2. 2
    3. 2
    Mfano\(\PageIndex{12}\)

    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\(\PageIndex{13}\)

    Kurahisisha:

    1. \((−64)^{\frac{1}{3}}\)
    2. \(−64^{\frac{1}{3}}\)
    3. \((64)^{−\frac{1}{3}}\).
    Jibu
    1. \((−64)^{\frac{1}{3}}\)
    Andika upya kama mizizi ya mchemraba. \(\sqrt[3]{−64}\)
    Andika upya -64 kama mchemraba kamilifu. \(\sqrt[3]{(−4)^3}\)
    Kurahisisha. —4
    2. \(−64^{\frac{1}{3}}\)
    exponent inatumika tu kwa 64. \(−(64^{\frac{1}{3}})\)
    Andika upya kama mizizi ya mchemraba. \(−\sqrt[3]{64}\)
    Andika upya 64 kama\(4^3\). \(−\sqrt[3]{4^3}\)
    Kurahisisha. —4
    3. \((64)^{−\frac{1}{3}}\)

    Rewrite kama sehemu na exponent chanya, kwa kutumia mali,\(a^{−n}=\frac{1}{a^n}\).

    Andika kama mizizi ya mchemraba.

    		 \(\frac{1}{\sqrt[3]{64}}\)
    Andika upya 64 kama\(4^3\). \(\frac{1}{\sqrt[3]{4^3}}\)
    Kurahisisha. \(\frac{1}{4}\)
    Mfano\(\PageIndex{14}\)

    Kurahisisha:

    1. \((−125)^{\frac{1}{3}}\)
    2. \(−125^{\frac{1}{3}}\)
    3. \((125)^{−\frac{1}{3}}\).
    Jibu
    1. -5
    2. -5
    3. \(\frac{1}{5}\)
    Mfano\(\PageIndex{15}\)

    Kurahisisha:

    1. \((−32)^{\frac{1}{5}}\)
    2. \(−32^{\frac{1}{5}}\)
    3. \((32)^{−\frac{1}{5}}\).
    Jibu
    1. -2
    2. -2
    3. \(\frac{1}{2}\)
    Mfano\(\PageIndex{16}\)

    Kurahisisha:

    1. \((−16)^{\frac{1}{4}}\)
    2. \(−16^{\frac{1}{4}}\)
    3. \((16)^{−\frac{1}{4}}\).
    Jibu
    1. \((−16)^{\frac{1}{4}}\)
    Andika upya kama mizizi ya nne. \(\sqrt[4]{−16}\)
    Hakuna namba halisi ambayo nguvu yake ya nne ni -16.  
    2. \(−16^{\frac{1}{4}}\)
    exponent inatumika tu kwa 16. \(−(16^{\frac{1}{4}})\)
    Andika upya kama mizizi ya nne. \(−\sqrt[4]{16}\)
    Andika upya 16 kama\(2^4\) \(−\sqrt[4]{2^4}\)
    Kurahisisha. -2
    3. \((16)^{−\frac{1}{4}}\)

    Rewrite kama sehemu na exponent chanya, kwa 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. \(\frac{1}{2}\)
    Mfano\(\PageIndex{17}\)

    Kurahisisha:

    1. \((−64)^{\frac{1}{2}}\)
    2. \(−64^{\frac{1}{2}}\)
    3. \((64)^{−\frac{1}{2}}\).
    Jibu
    1. —8
    2. —8
    3. \(\frac{1}{8}\)
    Mfano\(\PageIndex{18}\)

    Kurahisisha:

    1. \((−256)^{\frac{1}{4}}\)
    2. \(−256^{\frac{1}{4}}\)
    3. \((256)^{−\frac{1}{4}}\).
    Jibu
    1. —4
    2. —4
    3. \(\frac{1}{4}\)

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

    Hebu kazi na Power Mali kwa Exponents baadhi zaidi.

    Tuseme sisi\(a^{\frac{1}{n}}\) kuongeza kwa nguvu m.

    \[\begin{array}{ll} {}&{(a^{\frac{1}{n}})^m}\\ {\text{Multiply the exponents.}}&{a^{\frac{1}{n}·m}}\\ {\text{Simplify.}}&{a^{\frac{m}{n}}}\\ {\text{So} a^{\frac{m}{n}}=(\sqrt[n]{a})^m \text{also.}}&{}\\ \nonumber \end{array}\]

    Sasa tuseme tunachukua\(a^m\)\(\frac{1}{n}\) nguvu.

    \[\begin{array}{ll} {}&{(a^m)^{\frac{1}{n}}}\\ {\text{Multiply the exponents.}}&{a^{m·\frac{1}{n}}}\\ {\text{Simplify.}}&{a^{\frac{m}{n}}}\\ {\text{So} a^{\frac{m}{n}}=\sqrt[n]{a^m} \text{also.}}&{}\\ \nonumber \end{array}\]

    Ni aina gani tunayotumia ili kurahisisha kujieleza? Kwa kawaida tunachukua mizizi kwanza-kwa njia hiyo tunaweka idadi katika radicna ndogo.

    Ufafanuzi: MTAZAMO MTA\(a^{\frac{m}{n}}\)

    Kwa integers yoyote nzuri m na n,

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

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

    Mfano\(\PageIndex{19}\)

    Andika kwa kielelezo cha busara:

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

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

    1. Takwimu hii inasema, “Nambari ya exponent ni exponent ya y, 3.” Halafu inaonyesha mizizi ya mraba ya y cubed. Takwimu hiyo inasema, “Denominator ya exponent ni index ya radical, 2.” Kisha inaonyesha y kwa nguvu 3/2.
    2. Takwimu hii inasema, “Nambari ya exponent ni exponent ya x, 2.” Halafu inaonyesha mizizi ya cubed ya x squared. Takwimu hiyo inasoma, “Denominator ya exponent ni index ya radical, 3.” Kisha inaonyesha y kwa nguvu 2/3.
    3. Takwimu hii inasoma, “Nambari ya exponent ni exponent ya z, 3.” Kisha inaonyesha mizizi ya nne ya mchemraba. Takwimu hiyo inasoma, “Denominator ya exponent ni index ya radical, 4.” Kisha inaonyesha z kwa 3/4 nguvu.
    Mfano\(\PageIndex{20}\)

    Andika kwa kielelezo cha busara:

    1. \(\sqrt{x^5}\)
    2. \(\sqrt[4]{z^3}\)
    3. \(\sqrt[5]{y^2}\).
    Jibu
    1. \(x^{\frac{5}{2}}\)
    2. \(z^{\frac{3}{4}}\)
    3. \(y^{\frac{2}{5}}\)
    Mfano\(\PageIndex{21}\)

    Andika kwa kielelezo cha busara:

    1. \(\sqrt[5]{a^2}\)
    2. \(\sqrt[3]{b^7}\)
    3. \(\sqrt[4]{m^5}\).
    Jibu
    1. \(a^{\frac{2}{5}}\)
    2. \(b^{\frac{7}{3}}\)
    3. \(m^{\frac{5}{4}}\)
    Mfano\(\PageIndex{22}\)

    Kurahisisha:

    1. \(9^{\frac{3}{2}}\)
    2. \(125^{\frac{2}{3}}\)
    3. \(81^{\frac{3}{4}}\).
    Jibu

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

    1. \(9^{\frac{3}{2}}\)
    Nguvu ya radical ni nambari ya exponent, 3. Kwa kuwa denominator ya exponent ni 2, hii ni mizizi ya mraba. \((\sqrt{9})^3\)
    Kurahisisha. \(3^3\)
      27
    2. \(125^{\frac{2}{3}}\)
    Nguvu ya radical ni nambari ya exponent, 2. Kwa kuwa denominator ya exponent ni 3, hii ni mizizi ya mraba. \((\sqrt[3]{125})^2\)
    Kurahisisha. \(5^2\)
      25
    3. \(81^{\frac{3}{4}}\)
    Nguvu ya radical ni nambari ya exponent, 2. Kwa kuwa denominator ya exponent ni 3, hii ni mizizi ya mraba. \((\sqrt[4]{81})^3\)
    Kurahisisha. \(3^3\)
      27
    Mfano\(\PageIndex{23}\)

    Kurahisisha:

    1. \(4^{\frac{3}{2}}\)
    2. \(27^{\frac{2}{3}}\)
    3. \(625^{\frac{3}{4}}\).
    Jibu
    1. 8
    2. 9
    3. 125
    Mfano\(\PageIndex{24}\)

    Kurahisisha:

    1. \(8^{\frac{5}{3}}\)
    2. \(81^{\frac{3}{2}}\)
    3. \(16^{\frac{3}{4}}\).
    Jibu
    1. 32
    2. 729
    3. 8

    Kumbuka hilo\(b^{−p}=\frac{1}{b^p}\). Ishara mbaya katika maonyesho haibadili ishara ya kujieleza.

    Mfano\(\PageIndex{25}\)

    Kurahisisha:

    1. \(16^{−\frac{3}{2}}\)
    2. \(32^{−\frac{2}{5}}\)
    3. \(4^{−\frac{5}{2}}\)
    Jibu

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

    1. \(16^{−\frac{3}{2}}\)
    Andika upya kutumia\(b^{−p}=\frac{1}{b^p}\). \(\frac{1}{16^{\frac{3}{2}}}\)
    Badilisha kwa fomu kali. Nguvu ya radical ni nambari ya exponent, 3. Ripoti ni denominator ya exponent, 2. \(\frac{1}{(\sqrt{16})^3}\)
    Kurahisisha. \(\frac{1}{4^3}\)
      \(\frac{1}{64}\)
    2. \(32^{−\frac{2}{5}}\)
    Andika upya kutumia\(b^{−p}=\frac{1}{b^p}\). \(\frac{1}{32^{\frac{2}{5}}}\)
    Badilisha kwa fomu kali. \(\frac{1}{(\sqrt[5]{32})^2}\)
    Andika upya radicand kama nguvu. \(\frac{1}{(\sqrt[5]{2^5})^2}\)
    Kurahisisha. \(\frac{1}{2^2}\)
      \(\frac{1}{4}\)
    3. \(4^{−\frac{5}{2}}\)
    Andika upya kutumia\(b^{−p}=\frac{1}{b^p}\). \(\frac{1}{4^{\frac{5}{2}}}\)
    Badilisha kwa fomu kali. \(\frac{1}{(\sqrt{4})^5}\)
    Kurahisisha. \(\frac{1}{2^5}\)
      \(\frac{1}{32}\)
    Mfano\(\PageIndex{26}\)

    Kurahisisha:

    1. \(8^{−\frac{5}{3}}\)8
    2. \(81^{−\frac{3}{2}}\)
    3. \(16^{−\frac{3}{4}}\).
    Jibu
    1. \(\frac{1}{32}\)
    2. \(\frac{1}{729}\)
    3. \(\frac{1}{8}\)
    Mfano\(\PageIndex{27}\)

    Kurahisisha:

    1. \(4^{−\frac{3}{2}}\)
    2. \(27^{−\frac{2}{3}}\)
    3. \(625^{−\frac{3}{4}}\).
    Jibu
    1. \(\frac{1}{8}\)
    2. \(\frac{1}{9}\)
    3. \(\frac{1}{125}\)
    Mfano\(\PageIndex{28}\)

    Kurahisisha:

    1. \(−25^{\frac{3}{2}}\)
    2. \(−25^{−\frac{3}{2}}\)
    3. \((−25)^{\frac{3}{2}}\).
    Jibu
    1. \(−25^{\frac{3}{2}}\)
    Andika upya kwa fomu kali. \(−(\sqrt{25})^3\)
    Kurahisisha radical \(−5^3\)
    Kurahisisha. -125
    2. \(−25^{−\frac{3}{2}}\)
    Andika upya kutumia\(b^{−p}=\frac{1}{b^p}\). \(−(\frac{1}{25^{\frac{3}{2}}})\)
    Andika upya kwa fomu kali. \(−(\frac{1}{(\sqrt{25})^3})\)
    Kurahisisha radical. \(−(\frac{1}{5^3})\)
    Kurahisisha. \(−\frac{1}{125}\)
    3. \((−25)^{\frac{3}{2}}\).
    Andika upya kwa fomu kali. \((\sqrt{−25})^3\)
    Hakuna namba halisi ambayo mizizi ya mraba ni -25. Si idadi halisi.
    Mfano\(\PageIndex{29}\)

    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
    Mfano\(\PageIndex{30}\)

    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 Sheria za Watazamaji ili kurahisisha Maneno na Maonyesho ya Mantiki

    sheria hiyo ya exponents kwamba sisi tayari kutumika kutumika kwa exponents busara, pia. Sisi orodha Mali Exponent hapa kuwa nao kwa ajili ya kumbukumbu kama sisi kurahisisha maneno.

    MUHTASARI WA MALI EXPONENT

    Ikiwa, b ni namba halisi na m, n ni namba za busara, basi

    \[\begin{array}{ll} {\textbf{Product Property}}&{a^m·a^n=a^{m+n}}\\ {\textbf{Power Property}}&{(a^m)^n=a^{m·n}}\\ {\textbf{Product to a Power}}&{(ab)^m=a^{m}b^{m}}\\ {\textbf{Quotient Property}}&{\frac{a^m}{a^n}=a^{m−n} , a \ne 0, m>n}\\ {}&{\frac{a^m}{a^n}=\frac{1}{a^{n−m}}, a \ne 0, n>m}\\ {\textbf{Zero Exponent Definition}}&{a^0=1, a \ne 0}\\ {\textbf{Quotient to a Power Property}}&{(\frac{a}{b})^m=\frac{a^m}{b^m}, b \ne 0}\\ \nonumber \end{array}\]

    Wakati sisi kuzidisha msingi huo, sisi kuongeza exponents.

    Mfano\(\PageIndex{31}\)

    Kurahisisha:

    1. \(2^{\frac{1}{2}}·2^{\frac{5}{2}}\)
    2. \(x^{\frac{2}{3}}·x^{\frac{4}{3}}\)
    3. \(z^{\frac{3}{4}}·z^{\frac{5}{4}}\).
    Jibu
    1. \(2^{\frac{1}{2}}·2^{\frac{5}{2}}\)
    Msingi ni sawa, kwa hiyo tunaongeza wafuasi. \(2^{\frac{1}{2}+\frac{5}{2}}\)
    Ongeza sehemu ndogo. \(2^{\frac{6}{2}}\)
    Kurahisisha exponent. \(2^3\)
    Kurahisisha. 8
    2. \(x^{\frac{2}{3}}·x^{\frac{4}{3}}\)
    Msingi ni sawa, kwa hiyo tunaongeza wafuasi. \(x^{\frac{2}{3}+\frac{4}{3}}\)
    Ongeza sehemu ndogo. \(x^{\frac{6}{3}}\)
    Kurahisisha. \(x^2\)
    3. \(z^{\frac{3}{4}}·z^{\frac{5}{4}}\)
    Msingi ni sawa, kwa hiyo tunaongeza wafuasi. \(z^{\frac{3}{4}+\frac{5}{4}}\)
    Ongeza sehemu ndogo. \(z^{\frac{8}{4}}\)
    Kurahisisha. \(z^2\)
    Mfano\(\PageIndex{32}\)

    Kurahisisha:

    1. \(3^{\frac{2}{3}}·3^{\frac{4}{3}}\)
    2. \(y^{\frac{1}{3}}·y^{\frac{8}{3}}\)
    3. \(m^{\frac{1}{4}}·m^{\frac{3}{4}}\).
    Jibu
    1. 9
    2. \(y^3\)
    3. m
    Mfano\(\PageIndex{33}\)

    Kurahisisha:

    1. \(5^{\frac{3}{5}}·5^{\frac{7}{5}}\)
    2. \(z^{\frac{1}{8}}·z^{\frac{7}{8}}\)
    3. \(n^{\frac{2}{7}}·n^{\frac{5}{7}}\).
    Jibu
    1. 25
    2. z
    3. n

    Tutatumia Mali ya Nguvu katika mfano unaofuata.

    Mfano\(\PageIndex{34}\)

    Kurahisisha:

    1. \((x^4)^{\frac{1}{2}}\)
    2. \((y^6)^{\frac{1}{3}}\)
    3. \((z^9)^{\frac{2}{3}}\).
    Jibu
    1. \((x^4)^{\frac{1}{2}}\)
    Ili kuongeza nguvu kwa nguvu, tunazidisha vielelezo. \(x^{4·\frac{1}{2}}\)
    Kurahisisha. \(x^2\)
    2. \((y^6)^{\frac{1}{3}}\)
    Ili kuongeza nguvu kwa nguvu, tunazidisha vielelezo. \(y^{6·\frac{1}{3}}\)
    Kurahisisha. \(y^2\)
    3. \((z^9)^{\frac{2}{3}}\)
    Ili kuongeza nguvu kwa nguvu, tunazidisha vielelezo. \(z^{9·\frac{2}{3}}\)
    Kurahisisha. \(z^6\)
    Mfano\(\PageIndex{35}\)

    Kurahisisha:

    1. \((p^{10})^{\frac{1}{5}}\)
    2. \((q^8)^{\frac{3}{4}}\)
    3. \((x^6)^{\frac{4}{3}}\)
    Jibu
    1. \(p^\)
    2. \(q^6\)
    3. \(x^8\)
    Mfano\(\PageIndex{36}\)

    Kurahisisha:

    1. \((r^6)^{\frac{5}{3}}\)
    2. \((s^{12})^{\frac{3}{4}}\)
    3. \((m^9)^{\frac{2}{9}}\)
    Jibu
    1. \(r^{10}\)
    2. \(s^9\)
    3. \(m^2\)

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

    Mfano\(\PageIndex{37}\)

    Kurahisisha:

    1. \(\frac{x^{\frac{4}{3}}}{x^{\frac{1}{3}}}\)
    2. \(\frac{y^{\frac{3}{4}}}{y^{\frac{1}{4}}}\)
    3. \(\frac{z^{\frac{2}{3}}}{z^{\frac{5}{3}}}\).
    Jibu
    1. \(\frac{x^{\frac{4}{3}}}{x^{\frac{1}{3}}}\)
    Ili kugawanywa na msingi huo, tunaondoa wafuasi. \(x^{\frac{4}{3}−\frac{1}{3}}\)
    Kurahisisha. x
    2. \(\frac{y^{\frac{3}{4}}}{y^{\frac{1}{4}}}\)
    Ili kugawanywa na msingi huo, tunaondoa wafuasi. \(y^{\frac{3}{4}−\frac{1}{4}}\)
    Kurahisisha. \(y^{\frac{1}{2}}\)
    3. \(\frac{z^{\frac{2}{3}}}{z^{\frac{5}{3}}}\)
    Ili kugawanywa na msingi huo, tunaondoa wafuasi. \(z^{\frac{2}{3}−\frac{5}{3}}\)
    Andika upya bila exponent hasi. \(\frac{1}{z}\)
    Mfano\(\PageIndex{38}\)

    Kurahisisha:

    1. \(\frac{u^{\frac{5}{4}}}{u^{\frac{1}{4}}}\)
    2. \(\frac{v^{\frac{3}{5}}}{v^{\frac{2}{5}}}\)
    3. \(\frac{x^{\frac{2}{3}}}{x^{\frac{5}{3}}}\).
    Jibu
    1. u
    2. \(v^{\frac{1}{5}}\)
    3. \(\frac{1}{x}\)
    Mfano\(\PageIndex{39}\)

    Kurahisisha:

    1. \(\frac{c^{\frac{12}{5}}}{c^{\frac{2}{5}}}\)
    2. \(\frac{m^{\frac{5}{4}}}{m^{\frac{9}{4}}}\)
    3. \(\frac{d^{\frac{1}{5}}}{d^{\frac{6}{5}}}\).
    Jibu
    1. \(c^2\)
    2. \(\frac{1}{m}\)
    3. \(\frac{1}{d}\)

    Wakati mwingine tunahitaji kutumia mali zaidi ya moja. Katika mifano miwili ijayo, tutatumia Bidhaa zote kwa Mali ya Nguvu na kisha Mali ya Nguvu.

    Mfano\(\PageIndex{40}\)

    Kurahisisha:

    1. \((27u^{\frac{1}{2}})^{\frac{2}{3}}\)
    2. \((8v^{\frac{1}{4}})^{\frac{2}{3}}\).
    Jibu
    1. \((27u^{\frac{1}{2}})^{\frac{2}{3}}\)
    Kwanza tunatumia Bidhaa kwa Mali ya Nguvu. \((27)^{\frac{2}{3}}(u^{\frac{1}{2}})^{\frac{2}{3}}\)
    Andika upya 27 kama nguvu ya 3. \((3^3)^{\frac{2}{3}}(u^{\frac{1}{2}})^{\frac{2}{3}}\)
    Ili kuongeza nguvu kwa nguvu, tunazidisha vielelezo. \((3^2)(u^{\frac{1}{3}})\)
    Kurahisisha. \(9u^{\frac{1}{3}}\)
    2. \((8v^{\frac{1}{4}})^{\frac{2}{3}}\).
    Kwanza tunatumia Bidhaa kwa Mali ya Nguvu. \((8)^{\frac{2}{3}}(v^{\frac{1}{4}})^{\frac{2}{3}}\)
    Andika upya 8 kama nguvu ya 2. \((2^3)^{\frac{2}{3}}(v^{\frac{1}{4}})^{\frac{2}{3}}\)
    Ili kuongeza nguvu kwa nguvu, tunazidisha vielelezo. \((2^2)(v^{\frac{1}{6}})\)
    Kurahisisha. \(4v^{\frac{1}{6}}\)
    Mfano\(\PageIndex{41}\)

    Kurahisisha:

    1. \(32x^{\frac{1}{3}})^{\frac{3}{5}}\)
    2. \((64y^{\frac{2}{3}})^{\frac{1}{3}}\).
    Jibu
    1. \(8x^{\frac{1}{5}}\)
    2. \(4y^{\frac{2}{9}}\)
    Mfano\(\PageIndex{42}\)

    Kurahisisha:

    1. \((16m^{\frac{1}{3}})^{\frac{3}{2}}\)
    2. \((81n^{\frac{2}{5}})^{\frac{3}{2}}\).
    Jibu
    1. \(64m^{\frac{1}{2}}\)
    2. \(729n^{\frac{3}{5}}\)
    Mfano\(\PageIndex{43}\)

    Kurahisisha:

    1. \((m^{3}n^{9})^{\frac{1}{3}}\)
    2. \((p^{4}q^{8})^{\frac{1}{4}}\).
    Jibu
    1. \((m^{3}n^{9})^{\frac{1}{3}}\)
    Kwanza tunatumia Bidhaa kwa Mali ya Nguvu. \((m^{3})^{\frac{1}{3}}(n^{9})^{\frac{1}{3}}\)
    Ili kuongeza nguvu kwa nguvu, tunazidisha vielelezo. \(mn^3\)
    2. \((p^{4}q^{8})^{\frac{1}{4}}\)
    Kwanza tunatumia Bidhaa kwa Mali ya Nguvu. \((p^{4})^{\frac{1}{4}}(q^{8})^{\frac{1}{4}}\)
    Ili kuongeza nguvu kwa nguvu, tunazidisha vielelezo. \(pq^2\)

    Tutatumia Bidhaa zote na Quotient Mali katika mfano unaofuata.

    Zoezi\(\PageIndex{44}\)

    Kurahisisha:

    1. \(\frac{x^{\frac{3}{4}}·x^{−\frac{1}{4}}}{x^{−\frac{6}{4}}}\)
    2. \(\frac{y^{\frac{4}{3}}·y}{y^{−\frac{2}{3}}}\).
    Jibu
    1. \(\frac{x^{\frac{3}{4}}·x^{−\frac{1}{4}}}{x^{−\frac{6}{4}}}\)
    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. \(x^2\)
    2. \(\frac{y^{\frac{4}{3}}·y}{y^{−\frac{2}{3}}}\)
    Tumia Mali ya Bidhaa katika nambari, ongeza vielelezo. \(\frac{y^{\frac{7}{3}}}{y^{−\frac{2}{3}}}\)
    Tumia Mali ya Quotient, Ondoa wafuasi. \(y^{\frac{9}{3}}\)
    Kurahisisha. \(y^3\)
    Mfano\(\PageIndex{45}\)

    Kurahisisha:

    1. \(\frac{m^{\frac{2}{3}}·m^{−\frac{1}{3}}}{m^{−\frac{5}{3}}}\)
    2. \(\frac{n^{\frac{1}{6}}·n}{n^{−\frac{11}{6}}}\).
    Jibu
    1. \(m^2\)
    2. \(n^3\)
    Mfano\(\PageIndex{46}\)

    Kurahisisha:

    1. \(\frac{u^{\frac{4}{5}}·u^{−\frac{2}{5}}}{u^{−\frac{13}{5}}}\)
    2. \(\frac{v^{\frac{1}{2}}·v}{v^{−\frac{7}{2}}}\).
    Jibu
    1. \(u^3\)
    2. \(v^5\)

    Dhana muhimu

    • Muhtasari wa Mali Exponent
    • Ikiwa, b ni namba halisi na m, n ni namba za busara, basi
      • Bidhaa Mali\(a^m·a^n=a^{m+n}\)
      • Power Mali\((a^m)^n=a^{m·n}\)
      • Bidhaa kwa Nguvu\((ab)^m=a^{m}b^{m}\)
      • Mali ya Quotient:

        \(\frac{a^m}{a^n}=a^{m−n} , a \ne 0, m>n\)

        \(\frac{a^m}{a^n}=\frac{1}{a^{n−m}}, a \ne 0, n>m\)

      • Ufafanuzi wa sifuri\(a^0=1, a \ne 0\)
      • Quotient kwa Mali Nguvu\((\frac{a}{b})^m=\frac{a^m}{b^m}, b \ne 0\)

    faharasa

    watetezi wa busara
    • Kama\(\sqrt[n]{a}\) ni idadi halisi na\(n \ge 2\),\(a^{\frac{1}{n}}=\sqrt[n]{a}\)
    • Kwa integers yoyote nzuri m na n,\(a^{\frac{m}{n}}=(\sqrt[n]{a})^m\) na\(a^{\frac{m}{n}}=\sqrt[n]{a^m}\)