10.7: Gawanya Monomials (Sehemu ya 2)

Kurahisisha Maneno kwa kutumia Mali kadhaa

Tutaweza sasa muhtasari mali yote ya exponents hivyo wote ni pamoja kwa kutaja kama sisi kurahisisha maneno kwa kutumia mali kadhaa. Kumbuka kwamba wao ni sasa defined kwa exponents idadi nzima.

Muhtasari wa Mali Exponent

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

Bidhaa Mali	$a^m • a^n = a^{m + n}$
Power Mali	$(a^m)^n = a^{m • n}$
Bidhaa kwa Mali ya Nguvu	$(ab)^m = a^mb^m$
Mali ya Quotient	$\dfrac{a^{m}}{a^{n}} = a^{m − n},\quad a ≠ 0,\, m > n$
	$\dfrac{a^{m}}{a^{n}} = \dfrac{1}{a^{n-m}}, \quad a ≠ 0, \,n > m$
Zero Exponent Mali	$a^0 = 1, \quad a ≠ 0$
Quotient kwa Mali Nguvu	$\left(\dfrac{a}{b}\right)^{m} = \dfrac{a^{m}}{b^{m}}, b ≠ 0$

Mfano$\PageIndex{8}$:

Kurahisisha:$\dfrac{(x^{2})^{3}}{x^{5}}$.

Suluhisho

Panua vielelezo katika nambari, kwa kutumia Mali ya Nguvu.	$\dfrac{x^{6}}{x^{5}} \label{10.4.46}$
Ondoa exponents.	$x \label{10.4.47}$

Zoezi$\PageIndex{15}$:

Kurahisisha:$\dfrac{(a^{4})^{5}}{a^{9}}$.

Jibu

$a^{11}$

Zoezi$\PageIndex{16}$:

Kurahisisha:$\dfrac{(b^{5})^{6}}{b^{11}}$.

Jibu

$b^{19}$

Mfano$\PageIndex{9}$:

Kurahisisha:$\dfrac{(m^{8})}{(m^{2})^{4}}$.

Suluhisho

Panua vielelezo katika nambari, kwa kutumia Mali ya Nguvu.	$\dfrac{m^{8}}{m^{8}} \label{10.4.48}$
Ondoa exponents.	$m^{0} \label{10.4.49}$
Zero nguvu mali	1

Zoezi$\PageIndex{17}$:

Kurahisisha:$\dfrac{(k^{11}}{(k^{3})^{3}}$.

Jibu

$k^2$

Zoezi$\PageIndex{18}$:

Kurahisisha:$\dfrac{(d^{23}}{(d^{4})^{6}}$.

Jibu

$\frac{1}{d}$

Mfano$\PageIndex{10}$:

Kurahisisha:$\left(\dfrac{x^{7}}{x^{3}}\right)^{2}$.

Suluhisho

Kumbuka mabano kuja kabla exponents, na besi ni sawa ili tuweze kurahisisha ndani ya mabano. Ondoa exponents.	$(x^{7-3})^{2} \label{10.4.50}$
Kurahisisha.	$(x^{4})^{2} \label{10.4.51}$
Kuzidisha exponents.	$x^{8} \label{10.4.52}$

Zoezi$\PageIndex{19}$:

Kurahisisha:$\left(\dfrac{f^{14}}{f^{8}}\right)^{2}$.

Jibu

$f^{12}$

Zoezi$\PageIndex{20}$:

Kurahisisha:$\left(\dfrac{b^{6}}{b^{11}}\right)^{2}$.

Jibu

$\frac{1}{b^{10}}$

Mfano$\PageIndex{11}$:

Kurahisisha:$\left(\dfrac{p^{2}}{q^{5}}\right)^{3}$.

Suluhisho

Hapa hatuwezi kurahisisha ndani ya mabano kwanza, kwani misingi si sawa.

Kuongeza nambari na denominator kwa nguvu ya tatu kwa kutumia Quotient kwa Mali Power,$\left(\dfrac{a}{b}\right)^{m} = \dfrac{a^{m}}{b^{m}}$	$\dfrac{(p^{2})^{3}}{(q^{5})^{3}} \label{10.4.53}$
Tumia Mali ya Nguvu, (^m) ^n = ^m • n.	$\dfrac{p^{6}}{q^{15}} \label{10.4.54}$

Zoezi$\PageIndex{21}$:

Kurahisisha:$\left(\dfrac{m^{3}}{n^{8}}\right)^{5}$.

Jibu

$\frac{m^{15}}{n^{40}}$

Zoezi$\PageIndex{22}$:

Kurahisisha:$\left(\dfrac{t^{10}}{u^{7}}\right)^{2}$.

Jibu

$\frac{t^{20}}{u^{14}}$

Mfano$\PageIndex{12}$:

Kurahisisha:$\left(\dfrac{2x^{3}}{3y}\right)^{4}$.

Suluhisho

Kuongeza nambari na denominator kwa nguvu ya nne kwa kutumia Quotient kwa Power Mali.	$\dfrac{(2x^{3})^{4}}{(3y)^{4}} \label{10.4.55}$
Kuongeza kila sababu kwa nguvu ya nne, kwa kutumia Nguvu ya Mali ya Nguvu.	$\dfrac{2^{4} (x^{3})^{4}}{3^{4} y^{4}} \label{10.4.56}$
Matumizi Power Mali na kurahisisha.	$\dfrac{16x^{12}}{81y^{4}} \label{10.4.57}$

Zoezi$\PageIndex{23}$:

Kurahisisha:$\left(\dfrac{5b}{9c^{3}}\right)^{2}$.

Jibu

$\frac{25b^2}{81c^6}$

Zoezi$\PageIndex{24}$:

Kurahisisha:$\left(\dfrac{4p^{4}}{7q^{5}}\right)^{3}$.

Jibu

$\frac{64p^{12}}{343q^{15}}$

Mfano$\PageIndex{13}$:

Kurahisisha:$\dfrac{(y^{2})^{3} (y^{2})^{4}}{(y^{5})^{4}}$.

Suluhisho

Tumia Mali ya Nguvu.	$\dfrac{(y^{6})(y^{8})}{y^{20}} \label{10.4.58}$
Ongeza vielelezo katika nambari, kwa kutumia Mali ya Bidhaa.	$\dfrac{y^{14}}{y^{20}} \label{10.4.59}$
Tumia Mali ya Quotient.	$\dfrac{1}{y^{6}} \label{10.4.60}$

Zoezi$\PageIndex{25}$

Kurahisisha:$\dfrac{(y^{4})^{4} (y^{3})^{5}}{(y^{7})^{6}}$.

Jibu

$\frac{1}{y^{11}}$

Zoezi$\PageIndex{26}$

Kurahisisha:$\dfrac{(3x^{4})^{2} (x^{3})^{4}}{(x^{5})^{3}}$.

Jibu

$9x^5$

Kugawanya Monomials

Sasa tumeona mali yote ya exponents. Tutatumia kugawanya monomials. Baadaye, utazitumia kugawanya polynomials.

Mfano$\PageIndex{14}$:

Pata quotient: 56x ⁵ ÷ 7x ².

Suluhisho

Andika upya kama sehemu.	$\dfrac{56x^{5}}{7x^{2}} \label{10.4.61}$
Tumia kuzidisha sehemu ili kutenganisha sehemu ya nambari kutoka sehemu ya kutofautiana.	$\dfrac{56}{7} \cdot \dfrac{x^{5}}{x^{2}} \label{10.4.62}$
Tumia Mali ya Quotient.	$8x^{3} \label{10.4.63}$

Zoezi$\PageIndex{27}$:

Pata quotient: 63x ⁸ ÷ 9x ⁴.

Jibu

$7x^4$

Zoezi$\PageIndex{28}$:

Pata quotient: 96y ¹¹ ÷ 6y ⁸.

Jibu

$16y^3$

Tunapogawanya monomials na variable zaidi ya moja, tunaandika sehemu moja kwa kila kutofautiana.

Mfano$\PageIndex{15}$:

Kupata quotient:$\dfrac{42x^{2} y^{3}}{−7xy^{5}}$.

Suluhisho

Tumia kuzidisha sehemu.	$\dfrac{42}{-7} \cdot \dfrac{x^{2}}{x} \cdot \dfrac{y^{3}}{y^{5}} \label{10.4.64}$
Kurahisisha na kutumia Quotient Mali.	$-6 \cdot x \cdot \dfrac{1}{y^{2}} \label{10.4.65}$
Kuzidisha.	$- \dfrac{6x}{y^{2}} \label{10.4.66}$

Zoezi$\PageIndex{29}$:

Kupata quotient:$\dfrac{-84x^{8} y^{3}}{7x^{10} y^{2}}$.

Jibu

$-\frac{12y}{x^2}$

Zoezi$\PageIndex{30}$:

Kupata quotient:$\dfrac{-72a^{4} b^{5}}{−8a^{9} b^{5}}$.

Jibu

$\frac{9}{a^5}$

Mfano$\PageIndex{16}$:

Kupata quotient:$\dfrac{24a^{5} b^{3}}{48ab^{4}}$.

Suluhisho

Tumia kuzidisha sehemu.	$\dfrac{24}{48} \cdot \dfrac{a^{5}}{a} \cdot \dfrac{b^{3}}{b^{4}} \label{10.4.67}$
Kurahisisha na kutumia Quotient Mali.	$\dfrac{1}{2} \cdot a^{4} \cdot \dfrac{1}{b} \label{10.4.68}$
Kuzidisha.	$\dfrac{a^{4}}{2b} \label{10.4.69}$

Zoezi$\PageIndex{31}$:

Kupata quotient:$\dfrac{16a^{7} b^{6}}{24ab^{8}}$.

Jibu

$\frac{2a^6}{3b^2}$

Zoezi$\PageIndex{32}$:

Kupata quotient:$\dfrac{27p^{4} q^{7}}{-45p^{12} q}$.

Jibu

$-\frac{3q^6}{5p^8}$

Mara baada ya kuwa ukoo na mchakato na umeifanya hatua kwa hatua mara kadhaa, unaweza kuwa na uwezo wa kurahisisha sehemu katika hatua moja.

Mfano$\PageIndex{17}$:

Kupata quotient:$\dfrac{14x^{7} y^{12}}{21x^{11} y^{6}}$.

Suluhisho

Kurahisisha na kutumia Quotient Mali.

$\dfrac{2y^{6}}{3x^{4}} \label{10.4.70}$

Kuwa makini sana kurahisisha$\dfrac{14}{21}$ kwa kugawa nje sababu ya kawaida, na kurahisisha vigezo kwa kutoa exponents yao.

Zoezi$\PageIndex{33}$:

Kupata quotient:$\dfrac{28x^{5} y^{14}}{49x^{9} y^{12}}$.

Jibu

$\frac{4y^2}{7x^4}$

Zoezi$\PageIndex{34}$:

Kupata quotient:$\dfrac{30m^{5} n^{11}}{48m^{10} n^{14}}$.

Jibu

$\frac{5}{8m^5 n^3}$

Katika mifano yote hadi sasa, hapakuwa na kazi ya kufanya katika nambari au denominator kabla ya kurahisisha sehemu. Katika mfano unaofuata, tutapata kwanza bidhaa za monomials mbili katika nambari kabla ya kurahisisha sehemu.

Mfano$\PageIndex{18}$:

Kupata quotient:$\dfrac{(3x^{3} y^{2})(10x^{2} y^{3})}{6x^{4} y^{5}}$.

Suluhisho

Kumbuka, bar ya sehemu ni ishara ya makundi. Sisi kurahisisha nambari ya kwanza.

Kurahisisha nambari.	$\dfrac{30x^{5} y^{5}}{6x^{4} y^{5}} \label{10.4.71}$
Kurahisisha, kwa kutumia Utawala wa Quotient.	$5x \label{10.4.72}$

Zoezi$\PageIndex{35}$:

Kupata quotient:$\dfrac{(3x^{4} y^{5})(8x^{2} y^{5})}{12x^{5} y^{8}}$.

Jibu

$2xy^2$

Zoezi$\PageIndex{36}$:

Kupata quotient:$\dfrac{(-6a^{6} b^{9})(-8a^{5} b^{8})}{-12a^{10} b^{12}}$.

Jibu

$-4ab^5$

PATA RASILIMALI ZA ZIADA

Kurahisisha Quotient

sifuri exponent

Utawala wa Quotient

Idara ya Polynomial

Idara ya Polynomial 2

Mazoezi hufanya kamili

Kurahisisha Maneno Kutumia Mali ya Quotient ya Watazamaji

Katika mazoezi yafuatayo, kurahisisha.

219. $\dfrac{4^{8}}{4^{2}}$
220. $\dfrac{3^{12}}{3^{4}}$
221. $\dfrac{x^{12}}{x^{3}}$
222. $\dfrac{u^{9}}{u^{3}}$
223. $\dfrac{r^{5}}{r}$
224. $\dfrac{y^{4}}{y}$
225. $\dfrac{y^{4}}{y^{20}}$
226. $\dfrac{x^{10}}{x^{30}}$
227. $\dfrac{10^{3}}{10^{15}}$
228. $\dfrac{r^{2}}{r^{8}}$
229. $\dfrac{a}{a^{9}}$
230. $\dfrac{2}{2^{5}}$

Kurahisisha Maneno na Zero Exponents

Katika mazoezi yafuatayo, kurahisisha.

231. 5 ⁰
232. 10 ⁰
233. a ⁰
234. x ⁰
235. -7 ⁰
236. -4 ⁰
237. (a) (10p) ⁰ (b) 10p ⁰
238. (a) (3a) ⁰ (b) 3a ⁰
239. (a) (-27x ⁵ y) ⁰ (b) -27x ⁵ y ⁰
240. (a) (-92y ⁸ z) ⁰ (b) -92y ⁸ z ⁰
241. (a) 15 ⁰ (b) 15 ¹
242. (a) -6 ⁰ (b) -6 ¹
243. 2 • x ⁰ + 5 • y ⁰
244. 8 • m ⁰ - 4 • n ⁰

Kurahisisha Maneno Kutumia Quotient kwa Mali ya Nguvu

Katika mazoezi yafuatayo, kurahisisha.

245. $\left(\dfrac{3}{2}\right)^{5}$
246. $\left(\dfrac{4}{5}\right)^{3}$
247. $\left(\dfrac{m}{6}\right)^{3}$
248. $\left(\dfrac{p}{2}\right)^{5}$
249. $\left(\dfrac{x}{y}\right)^{10}$
250. $\left(\dfrac{a}{b}\right)^{8}$
251. $\left(\dfrac{a}{3b}\right)^{2}$
252. $\left(\dfrac{2x}{y}\right)^{4}$

Kurahisisha Maneno kwa kutumia Mali kadhaa

Katika mazoezi yafuatayo, kurahisisha.

253. $\dfrac{(x^{2})^{4}}{x^{5}}$
254. $\dfrac{(y^{4})^{3}}{y^{7}}$
255. $\dfrac{(u^{3})^{4}}{u^{10}}$
256. $\dfrac{(y^{2})^{5}}{y^{6}}$
257. $\dfrac{y^{8}}{(y^{5})^{2}}$
258. $\dfrac{p^{11}}{(p^{5})^{3}}$
259. $\dfrac{r^{5}}{(r^{4} \cdot r}$
260. $\dfrac{a^{3} \cdot a^{4}}{(a^{7}}$
261. $\left(\dfrac{x^{2}}{x^{8}}\right)^{3}$
262. $\left(\dfrac{u}{u^{10}}\right)^{2}$
263. $\left(\dfrac{a^{4} \cdot a^{6}}{a^{3}}\right)^{2}$
264. $\left(\dfrac{x^{3 \cdot x^{8}}}{x^{4}}\right)^{3}$
265. $\dfrac{(y^{3})^{5}}{(y^{4})^{3}}$
266. $\dfrac{(z^{6})^{2}}{(z^{2})^{4}}$
267. $\dfrac{(x^{3})^{6}}{(x^{4})^{7}}$
268. $\dfrac{(x^{4})^{8}}{(x^{5})^{7}}$
269. $\left(\dfrac{2r^{3}}{5s}\right)^{4}$
270. $\left(\dfrac{3m^{2}}{4n}\right)^{3}$
271. $\left(\dfrac{3y^{2} \cdot y^{5}}{y^{15} \cdot y^{8}}\right)^{0}$
272. $\left(\dfrac{15z^{4} \cdot z^{9}}{0.3z^{2}}\right)^{0}$
273. $\dfrac{(r^{2})^{5} (r^{4})^{2}}{(r^{3})^{7}}$
274. $\dfrac{(p^{4})^{2} (p^{3})^{5}}{(p^{2})^{9}}$
275. $\dfrac{(3x^{4})^{3} (2x^{3})^{2}}{(6x^{5})^{2}}$
276. $\dfrac{(-2y^{3})^{4} (3y^{4})^{2}}{(-6y^{3})^{2}}$

Kugawanya Monomials

Katika mazoezi yafuatayo, ugawanye monomials.

277. 48b ⁸ ÷ 6b ²
278. 42a ¹⁴ ÷ 6a ²
279. 36x ³ ÷ (-2x ⁹)
280. 20u ⁸ ÷ (-4u ⁶)
281. $\dfrac{18x^{3}}{9x^{2}}$
282. $\dfrac{36y^{9}}{4y^{7}}$
283. $\dfrac{-35x^{7}}{-42x^{13}}$
284. $\dfrac{18x^{5}}{-27x^{9}}$
285. $\dfrac{18r^{5} s}{3r^{3} s^{9}}$
286. $\dfrac{24p^{7} q}{6p^{2} q^{5}}$
287. $\dfrac{8mn^{10}}{64mn^{4}}$
288. $\dfrac{10a^{4} b}{50a^{2} b^{6}}$
289. $\dfrac{-12x^{4} y^{9}}{15x^{6} y^{3}}$
290. $\dfrac{48x^{11} y^{9} z^{3}}{36x^{6} y^{8} z^{5}}$
291. $\dfrac{64x^{5} y^{9} z^{7}}{48x^{7} y^{12} z^{6}}$
292. $\dfrac{(10u^{2} v)(4u^{3} v^{6})}{5u^{9} v^{2}}$
293. $\dfrac{(6m^{2} n)(5m^{4} n^{3})}{3m^{10} n^{2}}$
294. $\dfrac{(6a^{4} b^{3})(4ab^{5})}{(12a^{8} b)(a^{3} b)}$
295. $\dfrac{(4u^{5} v^{4})(15u^{8} v)}{(12u^{3} v)(u^{6} v)}$

Mazoezi ya mchanganyiko

296. (a) 24a ⁵ + 2a ⁵ (b) 24a ⁵ ÷ 2a ⁵ (c) 24a ⁵ • 2a ⁵ (d) 24a ⁵ ÷ 2a ⁵
297. (a) 15n ¹⁰ + 3n ¹⁰ (b) 15n ¹⁰ - 3n ¹⁰ (c) 15n ¹⁰ • 3n ¹⁰ (d) 15n ¹⁰ ÷ 3n ¹⁰
298. (a) p ⁴ • p ⁶ (b) (p ⁴) ⁶
299. (a) q ⁵ • q ³ (b) (q ⁵) ³
300. (a)$\dfrac{ y^{3}}{y}$ (b)$\dfrac{y}{y^{3}}$
301. (a)$\dfrac{z^{6}}{z^{5}}$ (b)$\dfrac{z^{5}}{z^{6}}$
302. (8x ⁵) (9x) ÷ 6x ³
303. (4y ⁵) (12y ⁷) ÷ 8y ²
304. $\dfrac{27a^{7}}{3a^{3}} + \dfrac{54a^{9}}{9a^{5}}$
305. $\dfrac{32c^{11}}{4c^{5}} + \dfrac{42c^{9}}{6c^{3}}$
306. $\dfrac{32y^{5}}{8y^{2}} - \dfrac{60y^{10}}{5y^{7}}$
307. $\dfrac{48x^{6}}{6x^{4}} - \dfrac{35x^{9}}{7x^{7}}$
308. $\dfrac{63r^{6} s^{3}}{9r^{4} s^{2}} - \dfrac{72r^{2} s^{2}}{6s}$
309. $\dfrac{56y^{4} z^{5}}{7y^{3} z^{3}} - \dfrac{45y^{2} z^{2}}{5y}$

kila siku Math

310. Kumbukumbu Megabyte moja ni takriban 10 ⁶ bytes. Gigabyte moja ni takriban 10 ⁹ bytes. Ni megabytes ngapi katika gigabyte moja?
311. Kumbukumbu Megabyte moja ni takriban 10 ⁶ bytes. Terabyte moja ni takriban 10 ¹² bytes. Ni megabytes ngapi katika terabyte moja?

Mazoezi ya kuandika

312. Vic anadhani quotient$\dfrac{x^{20}}{x^{4}}$ simplifies kwa x ⁵. Ni nini kibaya na hoja zake?
313. Mai hurahisisha quotient$\dfrac{y^{3}}{y}$ kwa kuandika$\dfrac{y^{3}}{y}$ = 3. Ni nini kibaya na hoja yake?
314. Wakati Dimple aliporahisishwa -3 ⁰ na (1-3) ⁰ alipata jibu lile lile. Eleza jinsi ya kutumia Utaratibu wa Uendeshaji kwa usahihi hutoa majibu tofauti.
315. Roxie anadhani katika ¹⁰ simplifies kwa 0. Ungesema nini kumshawishi Roxie yeye ni sahihi?

Self Check

(a) Baada ya kukamilisha mazoezi, tumia orodha hii ili kutathmini ujuzi wako wa malengo ya sehemu hii.

(b) Kwa kiwango cha 1—10, ungewezaje kupima ujuzi wako wa sehemu hii kwa kuzingatia majibu yako kwenye orodha? Unawezaje kuboresha hii?

Wachangiaji na Majina

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