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6.2: Tumia Mali ya kuzidisha ya Watazamaji

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    Malengo ya kujifunza

    Mwishoni mwa sehemu hii, utaweza:

    • Kurahisisha maneno na watazamaji
    • Kurahisisha maneno kwa kutumia Mali ya Bidhaa kwa Watazamaji
    • Kurahisisha maneno kwa kutumia Mali ya Nguvu kwa Watazamaji
    • Kurahisisha maneno kwa kutumia Bidhaa kwa Mali ya Nguvu
    • Kurahisisha maneno kwa kutumia mali kadhaa
    • Kuzidisha monomials
    Kumbuka

    Kabla ya kuanza, fanya jaribio hili la utayari.

    1. kurahisisha:\(\frac{3}{4}\cdot \frac{3}{4}\)
      Kama amekosa tatizo hili, mapitio Zoezi 1.6.13.
    2. Kurahisisha:\((−2)(−2)(−2)\).
      Ikiwa umekosa tatizo hili, tathmini Zoezi 1.5.13.

    Kurahisisha Maneno na Watazamaji

    Kumbuka kwamba exponent inaonyesha kuzidisha mara kwa mara ya kiasi sawa. Kwa mfano,\(2^4\) ina maana ya bidhaa ya\(4\) mambo ya\(2\), hivyo\(2^4\) ina maana\(2·2·2·2\).

    Hebu tuchunguze msamiati wa maneno na watazamaji.

    NUKUU YA KIELELEZO

    Takwimu hii ina nguzo mbili. Katika safu ya kushoto ni kwa nguvu m. m ni lebo katika bluu kama exponent. a ni kinachoitwa katika nyekundu kama msingi. Katika safu ya haki ni maandishi “kwa m nguvu ina maana kuzidisha m sababu za a.” Chini ya hii ni kwa m nguvu sawa mara mara a, ikifuatiwa na ellipsis, na “m sababu” iliyoandikwa hapa chini katika bluu.

    Hii inasoma\(a\) kwa\(m^{th}\) nguvu.

    Katika kujieleza\(a^{m}\), exponent\(m\) inatuambia mara ngapi sisi kutumia msingi a kama sababu.

    Takwimu hii ina nguzo mbili. Safu ya kushoto ina cubed 4. Chini ya hii ni mara 4 mara 4, na “mambo 3" yaliyoandikwa hapa chini katika bluu. Safu ya haki ina hasi 9 hadi nguvu ya tano. Chini ya hii ni hasi mara 9 hasi mara 9 hasi mara 9 hasi mara 9 hasi 9, na “sababu 5" zilizoandikwa hapa chini katika bluu.

    Kabla ya kuanza kufanya kazi na maneno variable zenye exponents, hebu kurahisisha maneno machache kuwashirikisha idadi tu.

    Mfano\(\PageIndex{1}\)

    Kurahisisha:

    1. \(4^{3}\)
    2. \(7^{1}\)
    3. \(\left(\frac{5}{6}\right)^{2}\)
    4. \((0.63)^{2}\)
    Jibu
    1. \(\begin{array}{ll} & 4^{3}\\ {\text { Multiply three factors of } 4 .} & {4 \cdot 4 \cdot 4} \\ {\text { Simplify. }} & {64}\end{array}\)
    2. \(\begin{array}{ll} & 7^{1}\\ \text{Multiply one factor of 7.} & 7\end{array}\)
    3. \(\begin{array}{ll} &\left(\frac{5}{6}\right)^{2}\\ {\text { Multiply two factors. }} & {\left(\frac{5}{6}\right)\left(\frac{5}{6}\right)} \\ {\text { Simplify. }} & {\frac{25}{36}}\end{array}\)
    4. \(\begin{array}{ll} &(0.63)^{2}\\ {\text { Multiply two factors. }} & {(0.63)(0.63)} \\ {\text { Simplify. }} & {0.3969}\end{array}\)
    Mfano\(\PageIndex{2}\)

    Kurahisisha:

    1. \(6^{3}\)
    2. \(15^{1}\)
    3. \(\left(\frac{3}{7}\right)^{2}\)
    4. \((0.43)^{2}\)
    Jibu
    1. 216
    2. 15
    3. \(\frac{9}{49}\)
    4. 0.1849
    Mfano\(\PageIndex{3}\)

    Kurahisisha:

    1. \(2^{5}\)
    2. \(21^{1}\)
    3. \(\left(\frac{2}{5}\right)^{3}\)
    4. \((0.218)^{2}\)
    Jibu
    1. 32
    2. 21
    3. \(\frac{8}{125}\)
    4. 0.047524
    Mfano\(\PageIndex{4}\)

    Kurahisisha:

    1. \((-5)^{4}\)
    2. \(-5^{4}\)
    Jibu
    1. \(\begin{array}{ll} &(-5)^{4}\\{\text { Multiply four factors of }-5} & {(-5)(-5)(-5)} \\ {\text { Simplify. }} & {625}\end{array}\)
    2. \(\begin{array}{ll} &-5^{4}\\{\text { Multiply four factors of } 5 .} & {-(5 \cdot 5 \cdot 5 \cdot 5)} \\ {\text { Simplify. }} & {-625}\end{array}\)

    Angalia kufanana na tofauti katika Mfano\(\PageIndex{4}\) sehemu 1 na Mfano\(\PageIndex{4}\) sehemu 2! Kwa nini majibu ni tofauti? Kama sisi kufuata utaratibu wa shughuli katika sehemu ya 1 mabano kutuambia kuongeza\((−5)\) kwa 4 th nguvu. Katika sehemu ya 2 sisi kuongeza tu\(5\) kwa 4 th nguvu na kisha kuchukua kinyume.

    Mfano\(\PageIndex{5}\)

    Kurahisisha:

    1. \((-3)^{4}\)
    2. \(-3^{4}\)
    Jibu
    1. 81
    2. -81
    Mfano\(\PageIndex{6}\)

    Kurahisisha:

    1. \((-13)^{4}\)
    2. \(-13^{4}\)
    Jibu
    1. 169
    2. -169

    Kurahisisha Maneno Kutumia Mali ya Bidhaa kwa Watazamaji

    Umeona kwamba unapochanganya maneno kama hayo kwa kuongeza na kutoa, unahitaji kuwa na msingi sawa na kielelezo sawa. Lakini wakati wewe kuzidisha na kugawanya, exponents inaweza kuwa tofauti, na wakati mwingine besi inaweza kuwa tofauti, pia.

    Tutaweza hupata mali ya exponents kwa kuangalia kwa mwelekeo katika mifano kadhaa.

    Kwanza, tutaangalia mfano unaoongoza kwenye Mali ya Bidhaa.

      x mara za mraba x cubed.
    Hii ina maana gani?
    Ni mambo ngapi kabisa?    		 x mara x, kuzidisha kwa mara x x. x mara x ina sababu mbili. x mara x mara x ina mambo matatu. 2 pamoja 3 ni sababu tano.
    Hivyo, tuna x kwa nguvu ya tano.
    Kumbuka kwamba 5 ni jumla ya exponents, 2 na 3. x mara za mraba x cubed ni x kwa nguvu ya 2 pamoja na 3, au x kwa nguvu ya tano.

    Tunaandika:\[\begin{array}{c}{x^{2} \cdot x^{3}} \\ {x^{2+3}} \\ {x^{5}}\end{array}\]

    Msingi ulikaa sawa na sisi aliongeza exponents. Hii inasababisha Bidhaa Mali kwa Exponents.

    BIDHAA MALI KWA EXPONENTS

    Kama\(a\) ni idadi halisi,\(m\) na\(n\) ni kuhesabu idadi, basi

    \[a^{m} \cdot a^{n}=a^{m+n}\]

    Ili kuzidisha na besi kama, ongeza vielelezo.

    Mfano na namba husaidia kuthibitisha mali hii.

    \[\begin{array}{rll} {2^3\cdot2^2} &\stackrel{?}{=} & 2^{2+3}\\ {4\cdot 8} &\stackrel{?}{=} & 2^{5} \\ {32} &=& 32\checkmark\end{array}\]

    Mfano\(\PageIndex{7}\)

    Kurahisisha:\(y^{5} \cdot y^{6}\)

    Jibu
      y kwa nguvu ya tano mara y kwa nguvu ya sita.
    Tumia mali ya bidhaa,\(a^{m} \cdot a^{n}=a^{m+n}\). y kwa nguvu ya 5 pamoja na 6.
    Kurahisisha. y kwa nguvu ya kumi na moja.
    Mfano\(\PageIndex{8}\)

    Kurahisisha:\(b^{9} \cdot b^{8}\)

    Jibu

    \(b^{17}\)

    Mfano\(\PageIndex{9}\)

    Kurahisisha:\(x^{12} \cdot x^{4}\)

    Jibu

    \(x^{16}\)

    Mfano\(\PageIndex{10}\)

    Kurahisisha:

    1. \(2^{5} \cdot 2^{9}\)
    2. \(3\cdot 3^{4}\)
    Jibu

    a.

      2 kwa mara ya tano nguvu 2 kwa nguvu ya tisa.
    Tumia mali ya bidhaa,\(a^{m} \cdot a^{n}=a^{m+n}\). 2 kwa nguvu ya 5 pamoja na 9.
    Kurahisisha. 2 kwa nguvu ya 14.

    b.

      3 kwa mara ya tano nguvu 3 kwa nguvu ya nne.
    Tumia mali ya bidhaa,\(a^{m} \cdot a^{n}=a^{m+n}\). 3 kwa nguvu ya 5 pamoja na 4.
    Kurahisisha. 3 kwa nguvu ya tisa.
    Mfano\(\PageIndex{11}\)

    Kurahisisha:

    1. \(5\cdot 5^{5}\)
    2. \(4^{9} \cdot 4^{9}\)
    Jibu
    1. \(5^{6}\)
    2. \(4^{18}\)
    Mfano\(\PageIndex{12}\)

    Kurahisisha:

    1. \(7^{6} \cdot 7^{8}\)
    2. \(10 \cdot 10^{10}\)
    Jibu
    1. \(7^{14}\)
    2. \(10^{11}\)
    Mfano\(\PageIndex{13}\)

    Kurahisisha:

    1. \(a^{7} \cdot a\)
    2. \(x^{27} \cdot x^{13}\)
    Jibu

    a.

      a kwa mara saba nguvu a.
    Andika upya,\(a = a^1\) a kwa mara saba nguvu kwa nguvu ya kwanza.
    Tumia mali ya bidhaa,\(a^m\cdot a^n = a^{m+n}\). a kwa nguvu ya 7 pamoja na 1.
    Kurahisisha. a kwa nguvu ya nane.

    b.

      x kwa mara ishirini na saba nguvu x kwa nguvu kumi na tatu.
    Angalia, besi ni sawa, hivyo kuongeza exponents. x kwa nguvu ya 27 pamoja na 13.
    Kurahisisha. x kwa nguvu ya arobaini.
    Mfano\(\PageIndex{14}\)

    Kurahisisha:

    1. \(p^{5} \cdot p\)
    2. \(y^{14} \cdot y^{29}\)
    Jibu
    1. \(p^{6}\)
    2. \(y^{43}\)
    Mfano\(\PageIndex{15}\)

    Kurahisisha:

    1. \(z \cdot z^{7}\)
    2. \(b^{15} \cdot b^{34}\)
    Jibu
    1. \(z^{8}\)
    2. \(b^{49}\)

    Tunaweza kupanua Bidhaa Mali kwa Exponents kwa sababu zaidi ya mbili.

    Mfano\(\PageIndex{16}\)

    Kurahisisha:\(d^{4} \cdot d^{5} \cdot d^{2}\)

    Jibu
      d kwa mara ya nne ya nguvu d kwa mara ya nguvu ya tano d squared.
    Kuongeza exponents, tangu besi ni sawa. d kwa nguvu ya 4 pamoja na 5 pamoja na 2.
    Kurahisisha. d kwa nguvu ya kumi na moja.
    Mfano\(\PageIndex{17}\)

    Kurahisisha:\(x^{6} \cdot x^{4} \cdot x^{8}\)

    Jibu

    \(x^{18}\)

    Mfano\(\PageIndex{18}\)

    Kurahisisha:\(b^{5} \cdot b^{9} \cdot b^{5}\)

    Jibu

    \(b^{19}\)

    Kurahisisha Maneno Kutumia Mali ya Nguvu kwa Wasanii

    Sasa hebu tuangalie usemi wa kielelezo ambao una nguvu iliyoinuliwa kwa nguvu. Angalia kama unaweza kugundua mali ya jumla.

      x mraba, katika mabano, cubed.
    Hii ina maana gani?
    Ni mambo ngapi kabisa?    		 x squared cubed ni x mara squared x mara x squared, ambayo ni x mara x, kuongezeka kwa x mara x, kuongezeka kwa x mara x. x mara x ina sababu mbili. Mbili pamoja na mbili pamoja na mbili ni sababu sita.
    Hivyo tuna x kwa nguvu ya sita.
    Kumbuka kwamba 6 ni bidhaa ya exponents, 2 na 3. x squared cubed ni x kwa nguvu ya mara 2 3, au x kwa nguvu ya sita.

    Tunaandika:

    \[\begin{array}{c}{\left(x^{2}\right)^{3}} \\ {x^{2 \cdot 3}} \\ {x^{6}}\end{array}\]

    Sisi kuzidisha exponents. Hii inasababisha Power Mali kwa Exponents.

    NGUVU MALI KWA EXPONENTS

    Ikiwa\(a\) ni idadi halisi,\(m\) na\(n\) ni namba nzima, basi

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

    Ili kuongeza nguvu kwa nguvu, kuzidisha wafuasi.

    Mfano na namba husaidia kuthibitisha mali hii.

    \[\begin{array} {lll} \left(3^{2}\right)^{3} &\stackrel{?}{=}&3^{2 \cdot 3} \\(9)^{3} &\stackrel{?}{=} & 3^{6} \\ 729 &=&729\checkmark \end{array}\]

    Mfano\(\PageIndex{19}\)

    Kurahisisha:

    1. \(\left(y^{5}\right)^{9}\)
    2. \(\left(4^{4}\right)^{7}\)
    Jibu

    a.

      y kwa nguvu ya tano, katika mabano, kwa nguvu ya tisa.
    Tumia mali ya nguvu,\(\big(a^m\big)^n = a^{m\cdot n}\). y kwa nguvu ya mara 5 9.
    Kurahisisha. y kwa nguvu 45.

    b.

      4 kwa nguvu ya nne, katika mabano, kwa nguvu ya 7.
    Tumia mali ya nguvu. 4 kwa nguvu ya mara 4 7.
    Kurahisisha. 4 kwa nguvu ishirini na nane.
    Mfano\(\PageIndex{20}\)

    Kurahisisha:

    1. \( \left(b^{7}\right)^{5} \)
    2. \(\left(5^{4}\right)^{3}\)
    Jibu
    1. \( b^{35}\)
    2. \(5^{12}\)
    Mfano\(\PageIndex{21}\)

    Kurahisisha:

    1. \(\left(z^{6}\right)^{9}\)
    2. \(\left(3^{7}\right)^{7}\)
    Jibu
    1. \(z^{54}\)
    2. \(3^{49}\)

    Kurahisisha Maneno Kutumia Bidhaa kwa Mali ya Nguvu

    Sasa tutaangalia maneno yaliyo na bidhaa inayofufuliwa kwa nguvu. Je, unaweza kupata ruwaza hii?

    \(\begin{array}{ll}{\text { What does this mean? }} & {\text { (2x) }^{3}} \\ {\text { We group the like factors together. }} & {2 x \cdot 2 x \cdot 2 x} \\ {\text { How many factors of } 2 \text { and of } x ?} & {2 \cdot 2 \cdot x^{3}} \\ {\text { Notice that each factor was raised to the power and }(2 x)^{3} \text { is } 2^{3} \cdot x^{3}}\end{array}\)

    \(\begin{array}{ll}\text{We write:} & {(2 x)^{3}} \\ & {2^{3} \cdot x^{3}}\end{array}\)

    Mtazamo hutumika kwa kila sababu! Hii inasababisha Bidhaa kwa Power Mali kwa Exponents.

    BIDHAA KWA MALI NGUVU KWA EXPONENTS

    Kama\(a\) na\(b\) ni idadi halisi na\(m\) ni idadi nzima, basi

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

    Ili kuongeza bidhaa kwa nguvu, ongeza kila sababu kwa nguvu hiyo.

    Mfano na namba husaidia kuthibitisha mali hii:

    \ [kuanza {safu} {lll} (2\ cdot 3) ^ {2} &\ stackrel {?} {=} &2 ^ {2}\ cdot 3^ {2}\ 6 ^ {2} &\ stackrel {?} {=} &4\ cdot 9\\ 36 &=&36
    \ checkmark\ mwisho {array}\]

    Mfano\(\PageIndex{22}\)

    Kurahisisha:

    1. \((-9 d)^{2}\)
    2. \((3mn)^{3}\).
    Jibu

    a.

      Hasi 9 d mraba.
    Matumizi Nguvu ya Bidhaa Mali,\((ab)^m=a^m b^m\). hasi 9 squared d mraba.
    Kurahisisha. 81 d mraba.
    b.
      30 m katika cubed.
    Matumizi Nguvu ya Bidhaa Mali,\((ab)^m=a^m b^m\). 3 m cubed cubed katika cubed.
    Kurahisisha. 27 mm cubed katika cubed.
    Mfano\(\PageIndex{23}\)

    Kurahisisha:

    1. \((-12 y)^{2}\)
    2. \((2 w x)^{5}\)
    Jibu
    1. \(144y^{2}\)
    2. \(32w^{5} x^{5}\)
    Mfano\(\PageIndex{24}\)

    Kurahisisha:

    1. \((5 w x)^{3}\)
    2. \((-3 y)^{3}\)
    Jibu
    1. 125\(w^{3} x^{3}\)
    2. \(-27 y^{3}\)

    Kurahisisha Maneno kwa kutumia Mali kadhaa

    Sisi sasa kuwa na mali tatu kwa ajili ya kuzidisha maneno na exponents. Hebu muhtasari yao na kisha tutaweza kufanya baadhi ya mifano kwamba matumizi ya zaidi ya moja ya mali.

    MALI YA WAPIGANAJI

    Ikiwa\(a\) na\(b\) ni namba halisi,\(m\) na\(n\) ni namba nzima, basi

    \[\begin{array}{llll} \textbf{Product Property } & a^{m} \cdot a^{n}&=&a^{m+n} \\ \textbf {Power Property } &\left(a^{m}\right)^{n}&=&a^{m n} \\ \textbf {Product to a Power } &(a b)^{m}&=&a^{m} b^{m} \end{array}\]

    Mali zote exponent kushikilia kweli kwa idadi yoyote halisi\(m\) na\(n\). Hivi sasa, sisi tu kutumia idadi nzima exponents.

    Mfano\(\PageIndex{25}\)

    Kurahisisha:

    1. \(\left(y^{3}\right)^{6}\left(y^{5}\right)^{4}\)
    2. \(\left(-6 x^{4} y^{5}\right)^{2}\)
    Jibu
    1. \(\begin{array}{ll}& \left(y^{3}\right)^{6}\left(y^{5}\right)^{4}\\ {\text { Use the Power Property. }}& y^{18} \cdot y^{20} \\ {\text { Add the exponents. }} & y^{38} \end{array}\)
    2. \(\begin{array}{ll}& \left(-6 x^{4} y^{5}\right)^{2}\\ {\text { Use the Product to a Power Property. }} & {(-6)^{2}\left(x^{4}\right)^{2}\left(y^{5}\right)^{2}} \\ {\text { Use the Power Property. }} & {(-6)^{2}\left(x^{8}\right)\left(y^{10}\right)^{2}} \\ {\text { Simplify. }} & {36 x^{8} y^{10}}\end{array}\)
    Mfano\(\PageIndex{26}\)

    Kurahisisha:

    1. \(\left(a^{4}\right)^{5}\left(a^{7}\right)^{4}\)
    2. \(\left(-2 c^{4} d^{2}\right)^{3}\)
    Jibu
    1. \(a^{48}\)
    2. \(-8 c^{12} d^{6}\)
    Mfano\(\PageIndex{27}\)

    Kurahisisha:

    1. \(\left(-3 x^{6} y^{7}\right)^{4}\)
    2. \(\left(q^{4}\right)^{5}\left(q^{3}\right)^{3}\)
    Jibu
    1. 81\(x^{24} y^{28}\)
    2. \(q^{29}\)
    Mfano\(\PageIndex{28}\)

    Kurahisisha:

    1. \((5 m)^{2}\left(3 m^{3}\right)\)
    2. \(\left(3 x^{2} y\right)^{4}\left(2 x y^{2}\right)^{3}\)
    Jibu
    1. \(\begin{array}{ll}& (5 m)^{2}\left(3 m^{3}\right)\\{\text { Raise } 5 m \text { to the second power. }} & {5^{2} m^{2} \cdot 3 m^{3}} \\ {\text { Simplify. }} & {25 m^{2} \cdot 3 m^{3}} \\ {\text { Use the Commutative Property. }} & {25 \cdot 3 \cdot m^{2} \cdot m^{3}} \\ {\text { Multiply the constants and add the exponents. }} & {75 m^{5}}\end{array}\)
    2. \(\begin{array}{ll} & \left(3 x^{2} y\right)^{4}\left(2 x y^{2}\right)^{3} \\ \text{Use the Product to a Power Property.} & \left(3^{4} x^{8} y^{4}\right)\left(2^{3} x^{3} y^{6}\right)\\\text{Simplify.} & \left(81 x^{8} y^{4}\right)\left(8 x^{3} y^{6}\right)\\ \text{Use the Commutative Property.} &81\cdot 8 \cdot x^{8} \cdot x^{3} \cdot y^{4} \cdot y^{6} \\\text{Multiply the constants and add the exponents.} & 648x^{11} y^{10}\\ \end{array}\)
    Mfano\(\PageIndex{29}\)

    Kurahisisha:

    1. \((5 n)^{2}\left(3 n^{10}\right)\)
    2. \(\left(c^{4} d^{2}\right)^{5}\left(3 c d^{5}\right)^{4}\)
    Jibu
    1. 75\(n^{12}\)
    2. 81\(c^{24} d^{30}\)
    Mfano\(\PageIndex{30}\)

    Kurahisisha:

    1. \(\left(a^{3} b^{2}\right)^{6}\left(4 a b^{3}\right)^{4}\)
    2. \((2 x)^{3}\left(5 x^{7}\right)\)
    Jibu
    1. 256\(a^{22} b^{24}\)
    2. 40\(x^{10}\)

    Kuzidisha Monomials

    Kwa kuwa monomial ni kujieleza algebraic, tunaweza kutumia mali ya exponents kuzidisha monomials.

    Mfano\(\PageIndex{31}\)

    Kuzidisha:\(\left(3 x^{2}\right)\left(-4 x^{3}\right)\)

    Jibu

    \ (\ kuanza {safu} {ll} &\ kushoto (3 x ^ {2}\ kulia)\ kushoto (-4 x^ {3}\ kulia)\\ Nakala {Tumia Mali ya Kubadilisha upya masharti.} & 3\ cdot (-4)\ cdot x ^ {2}\ cdot x ^ {3}\
    \ maandishi {Kuzidisha.} & -12 x^ {5}\ mwisho {safu}\)

    Mfano\(\PageIndex{32}\)

    Kuzidisha:\(\left(5 y^{7}\right)\left(-7 y^{4}\right)\)

    Jibu

    \(-35 y^{11}\)

    Mfano\(\PageIndex{33}\)

    Kuzidisha:\(\left(-6 b^{4}\right)\left(-9 b^{5}\right)\)

    Jibu

    54\(b^{9}\)

    Mfano\(\PageIndex{34}\)

    Kuzidisha:\(\left(\frac{5}{6} x^{3} y\right)\left(12 x y^{2}\right)\)

    Jibu

    \(\begin{array}{ll} & \left(\frac{5}{6} x^{3} y\right)\left(12 x y^{2}\right)\\ \text{Use the Commutative Property to rearrange the terms.} & \frac{5}{6} \cdot 12 \cdot x^{3} \cdot x \cdot y \cdot y^{2}\\ \text{Multiply.} &10x^{4} y^{3}\end{array}\)

    Mfano\(\PageIndex{35}\)

    Kuzidisha:\(\left(\frac{2}{5} a^{4} b^{3}\right)\left(15 a b^{3}\right)\)

    Jibu

    6\(a^{5} b^{6}\)

    Mfano\(\PageIndex{36}\)

    Kuzidisha:\(\left(\frac{2}{3} r^{5} s\right)\left(12 r^{6} s^{7}\right)\)

    Jibu

    8\(r^{11} s^{8}\)

    Kumbuka

    Kupata rasilimali hizi online kwa maelekezo ya ziada na mazoezi kwa kutumia mali kuzidisha ya exponents:

    • Kuzidisha Mali ya Watazamaji

    Dhana muhimu

    • Nukuu ya kielelezo
      Takwimu hii ina nguzo mbili. Katika safu ya kushoto ni kwa nguvu m. m ni lebo katika bluu kama exponent. a ni kinachoitwa katika nyekundu kama msingi. Katika safu ya kulia ni maandishi “a kwa poda m inamaanisha kuzidisha mambo m ya a.” Chini ya hii ni kwa m nguvu sawa mara mara a, ikifuatiwa na ellipsis, na “m sababu” iliyoandikwa hapa chini katika bluu.
    • Mali ya Watazamaji
      • Ikiwa\(a\) na\(b\) ni namba halisi\(m\) na\(n\) ni namba nzima, basi

    \[\begin{array}{llll} \textbf{Product Property } & a^{m} \cdot a^{n}&=&a^{m+n} \\ \textbf {Power Property } &\left(a^{m}\right)^{n}&=&a^{m n} \\ \textbf {Product to a Power } &(a b)^{m}&=&a^{m} b^{m} \end{array}\]