9.3: Ongeza na Ondoa Mizizi ya Mraba
- Page ID
- 177388
Mwishoni mwa sehemu hii, utaweza:
- Ongeza na uondoe kama mizizi ya mraba
- Ongeza na uondoe mizizi ya mraba ambayo inahitaji kurahisisha
Tunajua kwamba tunapaswa kufuata utaratibu wa shughuli ili kurahisisha maneno na mizizi ya mraba. Radical ni ishara ya makundi, kwa hiyo tunafanya kazi ndani ya radical kwanza. Sisi kurahisisha\(\sqrt{2+7}\) kwa njia hii:
\[\begin{array}{ll} {}&{\sqrt{2+7}}\\ {\text{Add inside the radical.}}&{\sqrt{9}}\\ {\text{Simplify.}}&{3}\\ \end{array}\]
Kwa hiyo ikiwa tunapaswa kuongeza\(\sqrt{2}+\sqrt{7}\), hatupaswi kuchanganya nao katika radical moja.
\(\sqrt{2}+\sqrt{7} \ne \sqrt{2+7}\)
Kujaribu kuongeza mizizi ya mraba na radicands tofauti ni kama kujaribu kuongeza tofauti na maneno.
\[\begin{array}{llll} {\text{But, just like we can}}&{x+x}&{\text{we can add}}&{\sqrt{3}+\sqrt{3}}\\ {}&{x+x=2x}&{}&{\sqrt{3}+\sqrt{3}=2\sqrt{3}}\\ \end{array}\]
Kuongeza mizizi ya mraba na radicand sawa ni kama kuongeza kama maneno. Tunaita mizizi ya mraba na radicana sawana kama mizizi ya mraba kutukumbusha wanafanya kazi sawa na maneno kama hayo.
Mizizi ya mraba yenye radicand sawa huitwa kama mizizi ya mraba.
Tunaongeza na kuondoa kama mizizi ya mraba kwa njia ile ile tunayoongeza na kuondoa kama maneno. Tunajua kwamba 3x +8x ni 11x. Vile vile tunaongeza\(3\sqrt{x}+8\sqrt{x}\) and the result is \(11\sqrt{x}\).
Ongeza na Ondoa Kama Mizizi ya Mraba
Fikiria juu ya kuongeza maneno kama na vigezo kama unavyofanya mifano michache ijayo. Wakati una kama radicands, wewe tu kuongeza au Ondoa coefficients. Wakati radicands si kama, huwezi kuchanganya maneno.
Kurahisisha:\(2\sqrt{2}−7\sqrt{2}\).
- Jibu
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\[\begin{array}{ll} {}&{2\sqrt{2}−7\sqrt{2}}\\ {\text{Since the radicals are like, we subtract the coefficients.}}&{−5\sqrt{2}}\\ \end{array}\]
Kurahisisha:\(8\sqrt{2}−9\sqrt{2}\).
- Jibu
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\(−\sqrt{2}\)
Kurahisisha:\(5\sqrt{3}−9\sqrt{3}\).
- Jibu
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\(−4\sqrt{3}\)
Kurahisisha:\(3\sqrt{y}+4\sqrt{y}\).
- Jibu
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\[\begin{array}{ll} {}&{3\sqrt{y}+4\sqrt{y}}\\ {\text{Since the radicals are like, we add the coefficients.}}&{7\sqrt{y}}\\ \end{array}\]
Kurahisisha:\(2\sqrt{x}+7\sqrt{x}\).
- Jibu
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\(9\sqrt{x}\)
Kurahisisha:\(5\sqrt{u}+3\sqrt{u}\).
- Jibu
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\(8\sqrt{u}\)
Kurahisisha:\(4\sqrt{x}−2\sqrt{y}\)
- Jibu
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\[\begin{array}{ll} {}&{4\sqrt{x}−2\sqrt{y}}\\ {\text{Since the radicals are not like, we cannot subtract them. We leave the expression as is.}}&{4\sqrt{x}−2\sqrt{y}}\\ \end{array}\]
Kurahisisha:\(7\sqrt{p}−6\sqrt{q}\).
- Jibu
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\(7\sqrt{p}−6\sqrt{q}\)
Kurahisisha:\(6\sqrt{a}−3\sqrt{b}\).
- Jibu
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\(6\sqrt{a}−3\sqrt{b}\)
Kurahisisha:\(5\sqrt{13}+4\sqrt{13}+2\sqrt{13}\).
- Jibu
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\[\begin{array}{ll} {}&{5\sqrt{13}+4\sqrt{13}+2\sqrt{13}}\\ {\text{Since the radicals are like, we add the coefficients.}}&{11\sqrt{13}}\\ \end{array}\]
Kurahisisha:\(4\sqrt{11}+2\sqrt{11}+3\sqrt{11}\).
- Jibu
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\(9\sqrt{11}\)
Kurahisisha:\(6\sqrt{10}+2\sqrt{10}+3\sqrt{10}\).
- Jibu
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\(11\sqrt{10}\)
Kurahisisha:\(2\sqrt{6}−6\sqrt{6}+3\sqrt{3}\).
- Jibu
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\[\begin{array}{ll} {}&{2\sqrt{6}−6\sqrt{6}+3\sqrt{3}}\\ {\text{Since the first two radicals are like, we subtract their coefficients.}}&{−4\sqrt{6}+3\sqrt{3}}\\ \end{array}\]
Kurahisisha:\(5\sqrt{5}−4\sqrt{5}+2\sqrt{6}\).
- Jibu
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\(\sqrt{5}+2\sqrt{6}\)
Kurahisisha:\(3\sqrt{7}−8\sqrt{7}+2\sqrt{5}\).
- Jibu
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\(−5\sqrt{7}+2\sqrt{5}\)
Kurahisisha:\(2\sqrt{5n}−6\sqrt{5n}+4\sqrt{5n}\).
- Jibu
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\[\begin{array}{ll} {}&{2\sqrt{5n}−6\sqrt{5n}+4\sqrt{5n}}\\ {\text{Since the radicals are like, we combine them.}}&{−0\sqrt{5n}}\\ {\text{Simplify.}}&{0}\\ \end{array}\]
Kurahisisha:\(\sqrt{7x}−7\sqrt{7x}+4\sqrt{7x}\).
- Jibu
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\(−2\sqrt{7x}\)
Kurahisisha:\(4\sqrt{3y}−7\sqrt{3y}+2\sqrt{3y}\).
- Jibu
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\(−3\sqrt{y}\)
Wakati radicals vyenye variable zaidi ya moja, kwa muda mrefu kama vigezo vyote na exponents yao ni sawa, radicals ni kama.
Kurahisisha:\(\sqrt{3xy}+5\sqrt{3xy}−4\sqrt{3xy}\).
- Jibu
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\[\begin{array}{ll} {}&{\sqrt{3xy}+5\sqrt{3xy}−4\sqrt{3xy}}\\ {\text{Since the radicals are like, we combine them.}}&{2\sqrt{3xy}}\\ \end{array}\]
Kurahisisha:\(\sqrt{5xy}+4\sqrt{5xy}−7\sqrt{5xy}\).
- Jibu
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\(−2\sqrt{5xy}\)
Kurahisisha:\(3\sqrt{7mn}+\sqrt{7mn}−4\sqrt{7mn}\).
- Jibu
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0
Ongeza na Ondoa Mizizi ya Mraba ambayo inahitaji kurahisisha
Kumbuka kwamba sisi daima kurahisisha mizizi ya mraba kwa kuondoa sababu kubwa ya mraba kamilifu. Wakati mwingine tunapoongeza au kuondoa mizizi ya mraba ambayo haionekani kuwa kama radicals, tunapata kama radicals baada ya kurahisisha mizizi ya mraba.
Kurahisisha:\(\sqrt{20}+3\sqrt{5}\).
- Jibu
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\[\begin{array}{ll} {}&{\sqrt{20}+3\sqrt{5}}\\ {\text{Simplify the radicals, when possible.}}&{\sqrt{4}·\sqrt{5}+3\sqrt{5}}\\ {}&{2\sqrt{5}+3\sqrt{5}}\\ {\text{Combine the like radicals.}}&{5\sqrt{5}}\\ \end{array}\]
Kurahisisha:\(\sqrt{18}+6\sqrt{2}\).
- Jibu
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\(9\sqrt{2}\)
Kurahisisha:\(\sqrt{27}+4\sqrt{3}\).
- Jibu
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\(7\sqrt{3}\)
Kurahisisha:\(\sqrt{48}−\sqrt{75}\)
- Jibu
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\[\begin{array}{ll} {}&{\sqrt{48}−\sqrt{75}}\\ {\text{Simplify the radicals.}}&{\sqrt{16}·\sqrt{3}−\sqrt{25}·\sqrt{3}}\\ {}&{4\sqrt{3}−5\sqrt{3}}\\ {\text{Combine the like radicals.}}&{−\sqrt{3}}\\ \end{array}\]
Kurahisisha:\(\sqrt{32}−\sqrt{18}\).
- Jibu
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\(\sqrt{2}\)
Kurahisisha:\(\sqrt{20}−\sqrt{45}\).
- Jibu
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\(−\sqrt{5}\)
Kama sisi kutumia Associative Mali ya Kuzidisha ili kurahisisha 5 (3x) na kupata 15x, tunaweza kurahisisha\(5(3\sqrt{x})\) and get \(15\sqrt{x}\). We will use the Associative Property to do this in the next example.
Kurahisisha:\(5\sqrt{18}−2\sqrt{8}\).
- Jibu
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\[\begin{array}{ll} {}&{5\sqrt{18}−2\sqrt{8}}\\ {\text{Simplify the radicals.}}&{5·\sqrt{9}·\sqrt{2}−2·\sqrt{4}·\sqrt{2}}\\ {}&{5·3·\sqrt{2}−2·2·\sqrt{2}}\\ {}&{15\sqrt{2}−4\sqrt{2}}\\ {\text{Combine the like radicals.}}&{11\sqrt{2}}\\ \end{array}\]
Kurahisisha:\(4\sqrt{27}−3\sqrt{12}\).
- Jibu
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\(6\sqrt{3}\)
Kurahisisha:\(3\sqrt{20}−7\sqrt{45}\).
- Jibu
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\(−15\sqrt{5}\)
Kurahisisha:\(\frac{3}{4}\sqrt{192}−\frac{5}{6}\sqrt{108}\).
- Jibu
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\[\begin{array}{ll} {}&{\frac{3}{4}\sqrt{192}−\frac{5}{6}\sqrt{108}}\\ {\text{Simplify the radicals.}}&{\frac{3}{4}\sqrt{64}·\sqrt{3}−\frac{5}{6}\sqrt{36}·\sqrt{3}}\\ {}&{\frac{3}{4}·8·\sqrt{3}−\frac{5}{6}·6·\sqrt{3}}\\ {}&{6\sqrt{3}−5\sqrt{3}}\\ {\text{Combine the like radicals.}}&{\sqrt{3}}\\ \end{array}\]
Kurahisisha:\(\frac{2}{3}\sqrt{108}−\frac{5}{7}\sqrt{147}\).
- Jibu
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\(−\sqrt{3}\)
Kurahisisha:\(\frac{3}{5}\sqrt{200}−\frac{3}{4}\sqrt{128}\).
- Jibu
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0
Kurahisisha:\(\frac{2}{3}\sqrt{48}−\frac{3}{4}\sqrt{12}\).
- Jibu
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\[\begin{array}{ll} {}&{\frac{2}{3}\sqrt{48}−\frac{3}{4}\sqrt{12}}\\ {\text{Simplify the radicals.}}&{\frac{2}{3}\sqrt{16}·\sqrt{3}−\frac{3}{4}\sqrt{4}·\sqrt{3}}\\ {}&{\frac{2}{3}·4·\sqrt{3}−\frac{3}{4}·2·\sqrt{3}}\\ {}&{\frac{8}{3}\sqrt{3}−\frac{3}{2}\sqrt{3}}\\ {\text{Find a common denominator to subtract the coefficients of the like radicals.}}&{\frac{16}{6}\sqrt{3}−\frac{9}{6}\sqrt{3}}\\ {\text{Simplify.}}&{\frac{7}{6}\sqrt{3}} \end{array}\]
Kurahisisha:\(\frac{2}{5}\sqrt{32}−\frac{1}{3}\sqrt{8}\)
- Jibu
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\(\frac{14}{15}\sqrt{2}\)
Kurahisisha:\(\frac{1}{3}\sqrt{80}−\frac{1}{4}\sqrt{125}\)
- Jibu
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\(\frac{1}{12}[\sqrt{5}\)
Katika mfano unaofuata, tutaondoa mambo ya mara kwa mara na ya kutofautiana kutoka mizizi ya mraba.
Kurahisisha:\(\sqrt{18n^5}−\sqrt{32n^5}\)
- Jibu
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\[\begin{array}{ll} {}&{\sqrt{18n^5}−\sqrt{32n^5}}\\ {\text{Simplify the radicals.}}&{\sqrt{9n^4}·\sqrt{2n}−\sqrt{16n^4}·\sqrt{2n}}\\ {}&{3n^2\sqrt{2n}−4n^2\sqrt{2n}}\\ {\text{Combine the like radicals.}}&{−n^2\sqrt{2n}}\\ \end{array}\]
Kurahisisha:\(\sqrt{32m^7}−\sqrt{50m^7}\).
- Jibu
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\(−m^3\sqrt{2m}\)
Kurahisisha:\(\sqrt{27p^3}−\sqrt{48p^3}\)
- Jibu
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\(−p^3\sqrt{p}\)
Kurahisisha:\(9\sqrt{50m^2}−6\sqrt{48m^2}\).
- Jibu
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\[\begin{array}{ll} {}&{9\sqrt{50m^{2}}−6\sqrt{48m^{2}}}\\ {\text{Simplify the radicals.}}&{9\sqrt{25m^{2}}·\sqrt{2}−6·\sqrt{16m^{2}}·\sqrt{3}}\\ {}&{9·5m·\sqrt{2}−6·4m·\sqrt{3}}\\ {}&{45m\sqrt{2}−24m\sqrt{3}}\\ \end{array}\]
Kurahisisha:\(5\sqrt{32x^2}−3\sqrt{48x^2}\).
- Jibu
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\(20x\sqrt{2}−12x\sqrt{3}\)
Kurahisisha:\(7\sqrt{48y^2}−4\sqrt{72y^2}\).
- Jibu
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\(28y\sqrt{3}−24y\sqrt{2}\)
Kurahisisha:\(2\sqrt{8x^2}−5x\sqrt{32}+5\sqrt{18x^2}\).
- Jibu
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\[\begin{array}{ll} {}&{2\sqrt{8x^2}−5x\sqrt{32}+5\sqrt{18x^2}}\\ {\text{Simplify the radicals.}}&{2\sqrt{4x^2}·\sqrt{2}−5x\sqrt{16}·\sqrt{2}+5\sqrt{9x^2}·\sqrt{2}}\\ {}&{2·2x·\sqrt{2}−5x·4·\sqrt{2}+5·3x·\sqrt{2}}\\ {}&{4x\sqrt{2}−20x\sqrt{2}+15x\sqrt{2}}\\ {\text{Combine the like radicals.}}&{−x\sqrt{2}}\\ \end{array}\]
Kurahisisha:\(3\sqrt{12x^2}−2x\sqrt{48}+4\sqrt{27x^2}\)
- Jibu
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\(10x\sqrt{3}\)
Kurahisisha:\(3\sqrt{18x^2}−6x\sqrt{32}+2\sqrt{50x^2}\).
- Jibu
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\(−5x\sqrt{2}\)
Fikia rasilimali hii ya mtandaoni kwa maelekezo ya ziada na mazoezi na kuongeza na kuondoa mizizi ya mraba.
- Kuongeza/Kutoa Mizizi ya Mraba
faharasa
- kama mizizi ya mraba
- Mizizi ya mraba yenye radicand sawa huitwa kama mizizi ya mraba.