# 5.3: Utawala wa Quotient wa Watazamaji

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##### Ufafanuzi: Utawala wa Quotient kwa Watazamaji

Kwa idadi yoyote halisi$$a$$ na idadi nzuri$$m$$ na$$n$$, wapi$$m > n$$.

Utawala wa Quotient Kwa Maonyesho ni yafuatayo.

$$\dfrac{a^m }{a^n} = a^{ m−n}$$

Kumbuka: Msingi lazima iwe sawa. Matokeo yatakuwa na msingi sawa.

## Wazo:

Kutoka sehemu ya mwisho,

$$x^3 = \textcolor{blue}{x \cdot x \cdot x} \qquad x^5 = \textcolor{red}{x \cdot x \cdot x \cdot x \cdot x}$$

Quotient yao

$$\dfrac{x^ 5 }{x^3} = \dfrac{\textcolor{red}{x \cdot x \cdot x \cdot x \cdot x }}{\textcolor{blue}{x \cdot x \cdot x }}= \dfrac{\textcolor{red}{\cancel{x \cdot x\cdot x \cdot x }\cdot x }}{\textcolor{blue}{\cancel{x \cdot x\cdot x }}}= \dfrac{\textcolor{red}{x \cdot x }}{1} = \textcolor{red}{x \cdot x}$$.

Hivyo,$$\dfrac{x^5 }{x^3 }= x^{5−3 }= x^2$$

##### Mfano Template:index

Kutumia utawala wa quotient wa exponents kurahisisha maneno.

1. $$\dfrac{k^3 }{k^2}$$
2. $$\dfrac{r^{32} }{r^{21}}$$
3. $$\dfrac{\sqrt{2}^ 7 }{\sqrt{2 }^4}$$
4. $$\dfrac{(−7)^9 }{(−7)^6}$$
5. $$\dfrac{(x \sqrt{5})^8 }{x\sqrt{ 5}}$$
6. $$\dfrac{(xy)^{18} }{(xy)^{17}}$$
###### Suluhisho
 Ufafanuzi Utawala wa Quotient Msingi $$\dfrac{k^3 }{k^2}$$ $$k^{3−2 }= k$$ $$k$$ $$\dfrac{r^{32} }{r^{21}}$$ $$r^{32−21 }= r^{11}$$ $$r$$ $$\dfrac{\sqrt{2}^ 7 }{\sqrt{2 }^4}$$ $$\sqrt{2 }^{7−4 }= \sqrt{2 }^3$$ $$\sqrt{2}$$ $$\dfrac{(−7)^9 }{(−7)^6}$$ $$(−7)^{9−6 }= (−7)^3$$ $$-7$$ $$\dfrac{(x \sqrt{5})^8 }{x\sqrt{ 5}}$$ $$(x \sqrt{5})^{8−1 }= (x \sqrt{5})^7$$ $$x\sqrt{5}$$ $$\dfrac{(xy)^{18} }{(xy)^{17}}$$ $$(xy)^{18−17 }= xy$$ $$xy$$

Kumbuka: Katika sehemu hii exponent ya nambari ilikuwa kubwa zaidi kuliko exponent ya denominator. Hiyo si mara zote kuwa kesi. Kesi ambapo exponent katika denominator ni kubwa kuliko exponent katika nambari itajadiliwa katika sehemu ya baadaye.

##### Zoezi Template:index

Tumia utawala wa quotient wa vielelezo ili kurahisisha usemi uliotolewa.

1. $$\dfrac{−y ^{13} }{−y^7}$$
2. $$\dfrac{(2x)^{25}}{ 2x}$$
3. $$\dfrac{\sqrt{7 }^{17 }}{\sqrt{7 }^{12}}$$
4. $$\dfrac{(−7)^9 }{(−7)^6}$$
5. $$\dfrac{(x + y) ^{78}}{ (x + y)^{43}}$$
6. $$\dfrac{\sqrt{xy }^{15 }}{\sqrt{xy }^{11}}$$