10.6: Gawanya Monomials (Sehemu ya 1)

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Malengo ya kujifunza

Kurahisisha maneno kwa kutumia Mali Quotient ya Exponents
Kurahisisha maneno na watazamaji sifuri
Kurahisisha maneno kwa kutumia Quotient kwa Mali Nguvu
Kurahisisha maneno kwa kutumia mali kadhaa
Gawanya monomials

kuwa tayari!

Kabla ya kuanza, fanya jaribio hili la utayari.

Kurahisisha:$\dfrac{8}{24}$. Ikiwa umekosa tatizo, kagua Mfano 4.3.1.
Kurahisisha: (2m ³) ⁵. Ikiwa umekosa tatizo, tathmini Mfano 10.3.13.
Kurahisisha:$\dfrac{12x}{12y}$. Ikiwa umekosa tatizo, kagua Mfano 4.3.5.

Kurahisisha Maneno Kutumia Mali ya Quotient ya Watazamaji

Mapema katika sura hii, sisi maendeleo ya mali ya exponents kwa kuzidisha. Sisi muhtasari mali hizi hapa.

Muhtasari wa Mali Exponent kwa Kuzidisha

Ikiwa, b ni namba halisi na m, n ni namba nzima, basi

Bidhaa Mali	^m • ⁿ = ^{m + n}
Power Mali	(^m) ⁿ = ^{m • n}
Bidhaa kwa Nguvu	(ab) ^m = ^m b ^m

Sasa tutaangalia mali exponent kwa mgawanyiko. Kumbukumbu ya haraka ya kumbukumbu inaweza kusaidia kabla ya kuanza. Katika sehemu ndogo ulijifunza kwamba sehemu ndogo zinaweza kuwa rahisi kwa kugawa mambo ya kawaida kutoka kwa nambari na denominator kwa kutumia Mali sawa ya Fractions. Mali hii pia itatusaidia kufanya kazi na sehemu za algebraic-ambazo pia ni quotients.

Ufafanuzi: Sawa Fractions Mali

Ikiwa a, b, c ni namba nzima ambapo b ∙ 0, c ∙ 0, basi

\[\dfrac{a}{b} = \dfrac{a \cdot c}{b \cdot c}\quad and\quad \dfrac{a \cdot c}{b \cdot c} = \dfrac{a}{b}\]

Kama hapo awali, tutajaribu kugundua mali kwa kuangalia mifano fulani.

Fikiria	$$\ dfrac {x^ {5}} {x^ {2}} $$	na	$$\ dfrac {x^ {2}} {x^ {3}} $$
Wanamaanisha nini?	$$\ dfrac {x\ cdot x\ cdot x\ cdot x} {x\ cdot x} $$		$$\ drac {x\ dot x} {x\ dot x\ dot x} $$
Tumia Mali sawa ya FRACTIONS	$$\ dfrac {\ kufuta {x}\ cdot\ kufuta {x}\ cdot x\ cdot x} {\ kufuta {x}\ cdot\ cdot\ kufuta {x}\ cdot 1} $$		$$\ dfrac {\ kufuta {x}\ cdot\ kufuta {x}\ cdot 1} {\ kufuta {x}\ cdot\ cdot\ kufuta {x}\ cdot x} $$
Kurahisisha.	$x^ {3} $$		$$\ dfrac {1} {x} $$

Kumbuka kwamba katika kila kesi besi zilikuwa sawa na sisi subtracted exponents.

Wakati exponent kubwa ilikuwa katika nambari, tuliachwa na mambo katika nambari na 1 katika denominator, ambayo sisi kilichorahisishwa.
Wakati kielelezo kikubwa kilikuwa katika denominator, tuliachwa na mambo katika denominator, na 1 katika nambari, ambayo haikuweza kuwa rahisi.

Tunaandika:

\[\begin{split} \dfrac{x^{5}}{x^{2}} \qquad &\quad \dfrac{x^{2}}{x^{3}} \\ x^{5-2} \qquad &\; \dfrac{1}{x^{3-2}} \\ x^{3} \qquad \quad &\quad \dfrac{1}{x} \end{split}\]

Ufafanuzi: Mali ya Quotient ya Watazamaji

Ikiwa ni namba halisi, a - 0, na m, n ni namba nzima, basi

\[\dfrac{a^{m}}{a^{n}} = a^{m-n}, \; m>n \quad and \quad \dfrac{a^{m}}{a^{n}} = \dfrac{1}{a^{n-m}},\; n>m\]

Mifano michache yenye namba inaweza kusaidia kuthibitisha mali hii.

\[\begin{split} \dfrac{3^{4}}{3^{2}} &\stackrel{?}{=} 3^{4-2} \qquad \; \dfrac{5^{2}}{5^{3}} \stackrel{?}{=} \dfrac{1}{5^{3-2}} \\ \dfrac{81}{9} &\stackrel{?}{=} 3^{2} \qquad \; \; \dfrac{25}{125} \stackrel{?}{=} \dfrac{1}{5^{1}} \\ 9 &= 9\; \checkmark \qquad \; \; \; \dfrac{1}{5} = \dfrac{1}{5}\; \checkmark \end{split}\]

Wakati sisi kazi na idadi na exponent ni chini ya au sawa na 3, sisi kuomba exponent. Wakati exponent ni kubwa kuliko 3, sisi kuondoka jibu katika fomu kielelezo.

Mfano$\PageIndex{1}$:

Kurahisisha: (a)$\dfrac{x^{10}}{x^{8}}$ (b)$\dfrac{2^{9}}{2^{2}}$

Suluhisho

Ili kurahisisha kujieleza kwa quotient, tunahitaji kwanza kulinganisha vielelezo katika nambari na denominator.

(a)

Tangu 10> 8, kuna mambo zaidi ya x katika nambari.	$$\ dfrac {x^ {10}} {x^ {8}} $
Tumia mali ya quotient na m> n,$\dfrac{a^{m}}{a^{n}} = a^{m − n}$.	$x^ {\ textcolor {nyekundu} {10-8}} $$
Kurahisisha.	$x^ {2} $$

(b)

Tangu 9> 2, kuna mambo zaidi ya 2 katika nambari.	$$\ dfrac {2^ {9}} {2^ {2}} $
Tumia mali ya quotient na m> n,$\dfrac{a^{m}}{a^{n}} = a^{m − n}$.	$2^ {\ textcolor {nyekundu} {9-2}} $$
Kurahisisha.	$2^ {7} $$

Kumbuka kwamba wakati exponent kubwa ni katika nambari, sisi ni wa kushoto na mambo katika nambari.

Zoezi$\PageIndex{1}$:

Kurahisisha: (a)$\dfrac{x^{12}}{x^{9}}$ (b)$\dfrac{7^{14}}{7^{5}}$

Jibu: $x^3$

Jibu b: $7^9$

Zoezi$\PageIndex{2}$:

Kurahisisha: (a)$\dfrac{y^{23}}{y^{17}}$ (b)$\dfrac{8^{15}}{8^{7}}$

Jibu: $y^6$

Jibu b: $8^8$

Mfano$\PageIndex{2}$:

Kurahisisha: (a)$\dfrac{b^{10}}{b^{15}}$ (b)$\dfrac{3^{3}}{3^{5}}$

Suluhisho

Ili kurahisisha kujieleza kwa quotient, tunahitaji kwanza kulinganisha vielelezo katika nambari na denominator.

(a)

Tangu 15> 10, kuna mambo zaidi ya b katika denominator.	$$\ dfrac {b^ {10}} {b^ {15}} $
Tumia mali ya quotient na n> m,$\dfrac{a^{m}}{a^{n}} = \dfrac{1}{a^{n − m}}$.	$$\ dfrac {\ textcolor {nyekundu} {1}} {b^ {\ textcolor {nyekundu} {15-10}}} $$
Kurahisisha.	$$\ dfrac {1} {b^ {5}} $

(b)

Tangu 5> 3, kuna mambo zaidi ya 3 katika denominator.	$$\ dfrac {3^ {3}} {3^ {5}} $$
Tumia mali ya quotient na n> m,$\dfrac{a^{m}}{a^{n}} = \dfrac{1}{a^{n − m}}$.	$$\ dfrac {\ textcolor {nyekundu} {1}} {3^ {\ textcolor {nyekundu} {5-3}}} $$
Kurahisisha.	$$\ dfrac {1} {3^ {2}} $
Kuomba exponent.	$$\ dfrac {1} {9} $$

Kumbuka kwamba wakati exponent kubwa ni katika denominator, sisi ni wa kushoto na mambo katika denominator na 1 katika nambari.

Zoezi$\PageIndex{3}$:

Kurahisisha: (a)$\dfrac{x^{8}}{x^{15}}$ (b)$\dfrac{12^{11}}{12^{21}}$

Jibu: $\frac{1}{x^7}$

Jibu b: $\frac{1}{12^10}$

Zoezi$\PageIndex{4}$:

Kurahisisha: (a)$\dfrac{m^{17}}{m^{26}}$ (b)$\dfrac{7^{8}}{7^{14}}$

Jibu: $\frac{1}{m^9}$

Jibu b: $\frac{1}{7^6}$

Mfano$\PageIndex{3}$:

Kurahisisha: (a)$\dfrac{a^{5}}{a^{9}}$ (b)$\dfrac{x^{11}}{x^{7}}$

Suluhisho

(a)

Tangu 9> 5, kuna zaidi ya a katika denominator na hivyo tutaishia na mambo katika denominator.	$$\ dfrac {a^ {5}} {a^ {9}} $$
Tumia mali ya quotient na n> m,$\dfrac{a^{m}}{a^{n}} = \dfrac{1}{a^{n − m}}$.	$$\ dfrac {\ textcolor {nyekundu} {1}} {a^ {\ textcolor {nyekundu} {9-5}}} $$
Kurahisisha.	$$\ dfrac {1} {a^ {4}} $

(b)

Angalia kuna mambo zaidi ya x katika nambari, tangu 11> 7. Kwa hiyo tutaishia na mambo katika nambari.	$$\ dfrac {x^ {11}} {x^ {97}} $
Tumia mali ya quotient na m> n,$\dfrac{a^{m}}{a^{n}} = a^{m − n}$.	$$a^ {\ textcolor {nyekundu} {11-7}} $$
Kurahisisha.	$x^ {4} $$

Zoezi$\PageIndex{5}$:

Kurahisisha: (a)$\dfrac{b^{19}}{b^{11}}$ (b)$\dfrac{z^{5}}{z^{11}}$

Jibu: $b^8$

Jibu b: $\frac{1}{z^6}$

Zoezi$\PageIndex{6}$:

Kurahisisha: (a)$\dfrac{p^{9}}{p^{17}}$ (b)$\dfrac{w^{13}}{w^{9}}$

Jibu: $\frac{1}{p^8}$

Jibu b: $w^4$

Kurahisisha Maneno na Zero Exponents

Kesi maalum ya Mali ya Quotient ni wakati maonyesho ya nambari na denominator ni sawa, kama vile kujieleza kama$\dfrac{a^{m}}{a^{m}}$. Kutoka kazi ya awali na sehemu ndogo, tunajua kwamba

\[\dfrac{2}{2} = 1 \qquad \dfrac{17}{17} = 1 \qquad \dfrac{-43}{-43} = 1\]

Kwa maneno, idadi iliyogawanywa na yenyewe ni 1. Hivyo$\dfrac{x}{x}$ = 1, kwa x yoyote (x ∙ 0), tangu idadi yoyote iliyogawanywa na yenyewe ni 1.

Mali ya Quotient ya Exponents inatuonyesha jinsi ya kurahisisha$\dfrac{a^{m}}{a^{n}}$ wakati m> n na wakati n <m kwa kutoa exponents. Nini kama m = n?

Sasa sisi kurahisisha$\dfrac{a^{m}}{a^{m}}$ kwa njia mbili kutuongoza kwa ufafanuzi wa exponent sifuri. Fikiria kwanza$\dfrac{8}{8}$, ambayo tunajua ni 1.

	$$\ dfrac {8} {8} = $$1
Andika 8 kama 2 ³.	$$\ dfrac {2^ {3}} {2^ {3}} = $1
Ondoa watetezi.	$2^ {3-3} = $1 $
Kurahisisha.	$2^ {0} = $1 $

$CNX_BMath_Figure_10_04_019_img.jpg$

Tunaona$\dfrac{a^{m}}{a^{n}}$ simplifies kwa ⁰ na 1. Hivyo ⁰ = 1.

ufafanuzi: Zero Exponent

Ikiwa ni nambari isiyo ya sifuri, basi ⁰ = 1. Nambari yoyote isiyo ya zero iliyoinuliwa kwa nguvu ya sifuri ni 1.

Katika maandishi haya, sisi kudhani variable yoyote kwamba sisi kuongeza kwa nguvu sifuri si sifuri.

Mfano$\PageIndex{4}$:

Kurahisisha: (a) 12 ⁰ (b) y ⁰

Suluhisho

Ufafanuzi anasema nambari yoyote isiyo ya sifuri iliyoinuliwa kwa nguvu ya sifuri ni 1.

(a) 12 ⁰

Tumia ufafanuzi wa exponent sifuri.

(b) na ⁰

Tumia ufafanuzi wa exponent sifuri.

Zoezi$\PageIndex{7}$:

Kurahisisha: (a) 17 ⁰ (b) m ⁰

Jibu: 1

Jibu b: 1

Zoezi$\PageIndex{8}$:

Kurahisisha: (a) k ⁰ (b) 29 ⁰

Jibu: 1

Jibu b: 1

Sasa kwa kuwa tuna defined exponent sifuri, tunaweza kupanua Mali yote ya Exponents ni pamoja na idadi nzima exponents.

Nini kuhusu kuinua maneno kwa nguvu ya sifuri? Hebu tuangalie (2x) ⁰. Tunaweza kutumia bidhaa kwa utawala wa nguvu ili kuandika tena maneno haya.

	(2x⁾
Tumia Bidhaa kwa Utawala wa Nguvu.	2 ⁰ x ⁰
Tumia Zero Exponent Mali.	1 • 1
Kurahisisha.	1

Hii inatuambia kwamba maneno yoyote yasiyo ya sifuri yaliyofufuliwa kwa nguvu ya sifuri ni moja.

Mfano$\PageIndex{5}$:

Kurahisisha: (7z) ⁰.

Suluhisho

Tumia ufafanuzi wa exponent sifuri.

Zoezi$\PageIndex{9}$:

Kurahisisha: (-4y) ⁰.

Jibu: 1

Zoezi$\PageIndex{10}$:

Kurahisisha:$\left(\dfrac{2}{3} x\right)^{0}$.

Jibu: 1

Mfano$\PageIndex{6}$:

Kurahisisha: (a) (-3x ² y) ⁰ (b) -3x ² y ⁰

Suluhisho

(a) (-3x ² y ⁰)

Bidhaa hiyo inafufuliwa kwa nguvu ya sifuri.	(-3x ² y ⁰)
Tumia ufafanuzi wa exponent sifuri.	1

(b) -3x ² y ⁰

Kumbuka kwamba tu y variable ni kuwa alimfufua kwa nguvu sifuri.	-3x ² y ⁰
Tumia ufafanuzi wa exponent sifuri.	-3x ² • 1
Kurahisisha.	-3x ²

Zoezi$\PageIndex{11}$:

Kurahisisha: (a) (7x ² y) ⁰ (b) 7x ² y ⁰

Jibu: 1

Jibu b: $7x^2$

Zoezi$\PageIndex{12}$:

Kurahisisha: (a) -23x ² y ⁰ (b) (-23x ² y) ⁰

Jibu: $-23x^2$

Jibu b: 1

Kurahisisha Maneno Kutumia Quotient kwa Mali ya Nguvu

Sasa tutaangalia mfano ambao utatuongoza kwenye Quotient kwa Mali ya Nguvu.

	$$\ kushoto (\ dfrac {x} {y}\ haki) ^ {3} $$
Hii ina maana	$$\ dfrac {x} {y}\ cdot\ dfrac {x} {y}\ cdot\ dfrac {x} {y} $$
Panua sehemu ndogo.	$$\ dfrac {x\ cdot x\ cdot x} {y\ dot y\ cdot y} $$
Andika na watazamaji.	$$\ dfrac {x^ {3}} {y^ {3}} $

Kumbuka kwamba exponent inatumika kwa nambari zote mbili na denominator. Tunaona kwamba$\left(\dfrac{x}{y}\right)^{3}$ ni$\dfrac{x^{3}}{y^{3}}$. Tunaandika:

\[\left(\dfrac{x}{y}\right)^{3} = \dfrac{x^{3}}{y^{3}}\]

Hii inasababisha Quotient kwa Power Mali kwa Exponents.

Ufafanuzi: Quotient kwa Mali Nguvu ya Watazamaji

Ikiwa a na b ni namba halisi, b ∙ 0, na m ni namba ya kuhesabu, basi

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

Ili kuongeza sehemu kwa nguvu, ongeza nambari na denominator kwa nguvu hiyo.

Mfano na namba inaweza kukusaidia kuelewa mali hii:

\[\begin{split} \left(\dfrac{2}{3}\right)^{3} &\stackrel{?}{=} \dfrac{2^{3}}{3^{3}} \\ \dfrac{2}{3} \cdot \dfrac{2}{3} \cdot \dfrac{2}{3} &\stackrel{?}{=} \dfrac{8}{27} \\ \dfrac{8}{27} &= \dfrac{8}{27}\; \checkmark \end{split}\]

Mfano$\PageIndex{7}$:

Kurahisisha: (a)$\left(\dfrac{5}{8}\right)^{2}$ (b)$\left(\dfrac{x}{3}\right)^{4}$ (c)$\left(\dfrac{y}{m}\right)^{3}$

Suluhisho

(a)$\left(\dfrac{5}{8}\right)^{2}$

Matumizi Quotient kwa Power Mali,$\left(\dfrac{a}{b}\right)^{m} = \dfrac{a^{m}}{b^{m}}$.	$$\ dfrac {5^ {\ textcolor {nyekundu} {2}}} {8^ {\ textcolor {nyekundu} {2}}} $$
Kurahisisha.	$$\ dfrac {25} {64} $$

(b)$\left(\dfrac{x}{3}\right)^{4}$

Matumizi Quotient kwa Power Mali,$\left(\dfrac{a}{b}\right)^{m} = \dfrac{a^{m}}{b^{m}}$.	$$\ dfrac {x^ {\ textcolor {nyekundu} {4}}} {3^ {\ textcolor {nyekundu} {4}}} $$
Kurahisisha.	$$\ dfrac {x^ {4}} {81} $$

(c)$\left(\dfrac{y}{m}\right)^{3}$

Kuongeza nambari na denominator kwa nguvu ya tatu.

$$\ dfrac {y^ {\ textcolor {nyekundu} {3}}} {m^ {\ textcolor {nyekundu} {3}}} $$

Zoezi$\PageIndex{13}$:

Kurahisisha: (a)$\left(\dfrac{7}{9}\right)^{2}$ (b)$\left(\dfrac{y}{8}\right)^{3}$ (c)$\left(\dfrac{p}{q}\right)^{6}$

Jibu: $\dfrac{49}{81}$

Jibu b: $\dfrac{y^3}{512}$

Jibu c: $\dfrac{p^6}{q^6}$

Zoezi$\PageIndex{14}$:

Kurahisisha: (a)$\left(\dfrac{1}{8}\right)^{2}$ (b)$\left(\dfrac{-5}{m}\right)^{3}$ (c)$\left(\dfrac{r}{s}\right)^{4}$

Jibu: $\dfrac{1}{64}$

Jibu b: $-\dfrac{125}{m^3}$

Jibu c: $\dfrac{r^4}{s^4}$

Wachangiaji na Majina

Template:ContribOpenStaxPreAlgebra

Muhtasari wa Mali Exponent kwa Kuzidisha

Ufafanuzi: Sawa Fractions Mali

Ufafanuzi: Mali ya Quotient ya Watazamaji

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ufafanuzi: Zero Exponent

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Ufafanuzi: Quotient kwa Mali Nguvu ya Watazamaji

Mfano\(\PageIndex{7}\):

Zoezi\(\PageIndex{13}\):

Zoezi\(\PageIndex{14}\):

Tangu 10> 8, kuna mambo zaidi ya x katika nambari.	$$\ dfrac {x^ {10}} {x^ {8}} $
Tumia mali ya quotient na m> n,\(\dfrac{a^{m}}{a^{n}} = a^{m − n}\).	$x^ {\ textcolor {nyekundu} {10-8}} $$
Kurahisisha.	$x^ {2} $$

Tangu 15> 10, kuna mambo zaidi ya b katika denominator.	$$\ dfrac {b^ {10}} {b^ {15}} $
Tumia mali ya quotient na n> m,\(\dfrac{a^{m}}{a^{n}} = \dfrac{1}{a^{n − m}}\).	$$\ dfrac {\ textcolor {nyekundu} {1}} {b^ {\ textcolor {nyekundu} {15-10}}} $$
Kurahisisha.	$$\ dfrac {1} {b^ {5}} $

Matumizi Quotient kwa Power Mali,\(\left(\dfrac{a}{b}\right)^{m} = \dfrac{a^{m}}{b^{m}}\).	$$\ dfrac {5^ {\ textcolor {nyekundu} {2}}} {8^ {\ textcolor {nyekundu} {2}}} $$
Kurahisisha.	$$\ dfrac {25} {64} $$