9.2: Punguza Mizizi ya Mraba

Malengo ya kujifunza

Mwishoni mwa sehemu hii, utaweza:

• Tumia Mali ya Bidhaa ili kurahisisha mizizi ya mraba
• Tumia Mali ya Quotient ili kurahisisha mizizi ya mraba

KUWA TAYARI

Kabla ya kupata kuanza kuchukua jaribio hili utayari.

1. Kurahisisha:$\frac{80}{176}$.
   Ikiwa umekosa tatizo hili, tathmini [kiungo].
2. Kurahisisha:$\frac{n^9}{n^3}$.
   Ikiwa umekosa tatizo hili, tathmini [kiungo].
3. Kurahisisha:$\frac{q^4}{q^{12}}$.
   Ikiwa umekosa tatizo hili, tathmini [kiungo].

Katika sehemu ya mwisho, sisi inakadiriwa mizizi mraba ya idadi kati ya namba mbili mfululizo mzima. Tunaweza kusema kwamba$\sqrt{50}$ ni kati ya 7 na 8. Hii ni haki rahisi kufanya wakati idadi ni ndogo ya kutosha kwamba tunaweza kutumia [kiungo].

Lakini vipi ikiwa tunataka kukadiria$\sqrt{500}$? Kama sisi kurahisisha mizizi mraba kwanza, tutaweza kukadiria kwa urahisi. Kuna sababu nyingine, pia, ili kurahisisha mizizi ya mraba kama utaona baadaye katika sura hii.

Mzizi wa mraba unachukuliwa kuwa rahisi ikiwa radicana yake haina sababu kamili za mraba.

Ufafanuzi: SQUARE ROOT RAHISI

$\sqrt{a}$inachukuliwa kilichorahisishwa kama hana sababu kamili za mraba.

Hivyo$\sqrt{31}$ ni rahisi. Lakini$\sqrt{32}$ si rahisi, kwa sababu 16 ni sababu kamili ya mraba ya 32.

Tumia mali ya Bidhaa ili kurahisisha Mizizi ya Mraba

Mali ambayo tutatumia ili kurahisisha maneno na mizizi ya mraba ni sawa na mali ya wasimamizi. Tunajua kwamba$(ab)^m=a^{m}b^{m}$. Mali sambamba ya mizizi ya mraba inasema hivyo$\sqrt{ab}=\sqrt{a}·\sqrt{b}$.

Ufafanuzi: PRODUCT PROPERTY YA MIZIZI

Ikiwa a, b ni namba zisizo hasi halisi, basi$\sqrt{ab}=\sqrt{a}·\sqrt{b}$.

Tunatumia Bidhaa Mali ya Mizizi ya Mraba ili kuondoa mambo yote ya mraba kamili kutoka kwa radical. Tutaonyesha jinsi ya kufanya hivyo katika Mfano.

Jinsi ya Kutumia Mali ya Bidhaa ili kurahisisha Mizizi ya Mraba

Mfano$\PageIndex{1}$

Kurahisisha:$\sqrt{50}$.

Jibu

Mfano$\PageIndex{2}$

Kurahisisha:$\sqrt{48}$.

Jibu

$4\sqrt{3}$

Mfano$\PageIndex{3}$

Kurahisisha:$\sqrt{45}$.

Jibu

$3\sqrt{5}$

Taarifa katika mfano uliopita kuwa fomu rahisi ya$\sqrt{50}$ is $5\sqrt{2}$, which is the product of an integer and a square root. We always write the integer in front of the square root.

ufafanuzi: KURAHISISHA MIZIZI SQUARE KUTUMIA PRODUCT PROPERTY

1. Pata sababu kubwa zaidi ya mraba ya radicand. Andika upya radicna kama bidhaa kwa kutumia sababu kamilifu ya mraba.
2. Tumia utawala wa bidhaa ili uandike upya radical kama bidhaa ya radicals mbili.
3. Kurahisisha mizizi ya mraba wa mraba kamilifu.

Mfano$\PageIndex{4}$

Kurahisisha:$\sqrt{500}$.

Jibu

$$\begin{array}{ll} {}&{\sqrt{500}}\\ {\text{Rewrite the radicand as a product using the largest perfect square factor}}&{\sqrt{100·5}}\\ {\text{Rewrite the radical as the product of two radicals}}&{\sqrt{100}·\sqrt{5}}\\ {\text{Simplify}}&{10\sqrt{5}}\\ \end{array}$$

Mfano$\PageIndex{5}$

Kurahisisha:$\sqrt{288}$.

Jibu

$12\sqrt{2}$

Mfano$\PageIndex{6}$

Kurahisisha:$\sqrt{432}$.

Jibu

$12\sqrt{3}$

Tunaweza kutumia fomu kilichorahisishwa$10\sqrt{5}$ ili kukadiria$\sqrt{500}$. Tunajua$\sqrt{5}$ ni kati ya 2 na 3, na$\sqrt{500}$ ni$10\sqrt{5}$. Hivyo$\sqrt{500}$ ni kati ya 20 na 30.

Mfano unaofuata ni sawa na mifano ya awali, lakini kwa vigezo.

Mfano$\PageIndex{7}$

Kurahisisha:$\sqrt{x^3}$.

Jibu

$$\begin{array}{ll} {}&{\sqrt{x^3}}\\ {\text{Rewrite the radicand as a product using the largest perfect square factor}}&{\sqrt{x^2·x}}\\ {\text{Rewrite the radical as the product of two radicals}}&{\sqrt{x^2}·\sqrt{x}}\\ {\text{Simplify}}&{x\sqrt{x}}\\ \end{array}$$

Mfano$\PageIndex{8}$

Kurahisisha:$\sqrt{b^5}$.

Jibu

$b^2\sqrt{b}$

Mfano$\PageIndex{9}$

Kurahisisha:$\sqrt{p^9}$.

Jibu

$p^4\sqrt{p}$

Tunafuata utaratibu huo wakati kuna mgawo katika radical, pia.

Mfano$\PageIndex{10}$

Kurahisisha:$\sqrt{25y^5}$.

Jibu

$$\begin{array}{ll} {}&{\sqrt{25y^5}}\\ {\text{Rewrite the radicand as a product using the largest perfect square factor.}}&{\sqrt{25y^4·y}}\\ {\text{Rewrite the radical as the product of two radicals.}}&{\sqrt{25y^4}·\sqrt{y}}\\ {\text{Simplify.}}&{5y^2\sqrt{y}}\\ \end{array}$$

Mfano$\PageIndex{11}$

Kurahisisha:$\sqrt{16x^7}$.

Jibu

$4x^3\sqrt{x}$

Mfano$\PageIndex{12}$

Kurahisisha:$\sqrt{49v^9}$.

Jibu

$7v^4\sqrt{v}$

Katika mfano unaofuata wote mara kwa mara na kutofautiana wana mambo kamili ya mraba.

Mfano$\PageIndex{13}$

Kurahisisha:$\sqrt{72n^7}$.

Jibu

$$\begin{array}{ll} {}&{\sqrt{72n^7}}\\ {\text{Rewrite the radicand as a product using the largest perfect square factor.}}&{\sqrt{36n^{6}·2n}}\\ {\text{Rewrite the radical as the product of two radicals.}}&{\sqrt{36n^{6}}·\sqrt{2n}}\\ {\text{Simplify.}}&{6n^3\sqrt{2n}}\\ \end{array}$$

Mfano$\PageIndex{14}$

Kurahisisha:$\sqrt{32y^5}$.

Jibu

$4y^2\sqrt{2y}$

Mfano$\PageIndex{15}$

Kurahisisha:$\sqrt{75a^9}$.

Jibu

$5a^4\sqrt{3a}$

Mfano$\PageIndex{16}$

Kurahisisha:$\sqrt{63u^{3}v^{5}}$.

Jibu

$$\begin{array}{ll} {}&{\sqrt{63u^{3}v^{5}}}\\ {\text{Rewrite the radicand as a product using the largest perfect square factor.}}&{\sqrt{9u^{2}v^{4}·7uv}}\\ {\text{Rewrite the radical as the product of two radicals.}}&{\sqrt{9u^{2}v^{4}}·\sqrt{7uv}}\\ {\text{Simplify.}}&{3uv^{2}\sqrt{7uv}}\\ \end{array}$$

Mfano$\PageIndex{17}$

Kurahisisha:$\sqrt{98a^{7}b^{5}}$.

Jibu

$7a^{3}b^{2}\sqrt{2ab}$

Mfano$\PageIndex{18}$

Kurahisisha:$\sqrt{180m^{9}n^{11}}$.

Jibu

$6m^{4}n^{5}\sqrt{5mn}$

Tumeona jinsi ya kutumia Order of Operations kurahisisha baadhi ya maneno na radicals. Ili kurahisisha$\sqrt{25}+\sqrt{144}$ we must simplify each square root separately first, then add to get the sum of 17.

Maneno$\sqrt{17}+\sqrt{7}$ hayawezi kurahisishwa-kuanza tunatarajia kurahisisha kila mizizi ya mraba, lakini wala 17 wala 7 ina sababu kamili ya mraba.

Katika mfano unaofuata, tuna jumla ya integer na mizizi ya mraba. Sisi kurahisisha mizizi mraba lakini hawezi kuongeza kujieleza kusababisha kwa integer.

Mfano$\PageIndex{19}$

Kurahisisha:$3+\sqrt{32}$.

Jibu

$$\begin{array}{ll} {}&{3+\sqrt{32}}\\ {\text{Rewrite the radicand as a product using the largest perfect square factor.}}&{3+\sqrt{16·2}}\\ {\text{Rewrite the radical as the product of two radicals.}}&{3+\sqrt{16}·\sqrt{2}}\\ {\text{Simplify.}}&{3+4\sqrt{2}}\\ \end{array}$$

Masharti hayapatikani na hivyo hatuwezi kuziongezea. Kujaribu kuongeza integer na radical ni kama kujaribu kuongeza integer na variable-wao si kama maneno!

Mfano$\PageIndex{20}$

Kurahisisha:$5+\sqrt{75}$.

Jibu

$5+5\sqrt{3}$

Mfano$\PageIndex{21}$

Kurahisisha:$2+\sqrt{98}$.

Jibu

$2+7\sqrt{2}$

Mfano unaofuata unajumuisha sehemu yenye radical katika nambari. Kumbuka kwamba ili kurahisisha sehemu unahitaji jambo la kawaida katika nambari na denominator.

Mfano$\PageIndex{22}$

Kurahisisha:$\frac{4−\sqrt{48}}{2}$.

Jibu

$$\begin{array}{ll} {}&{\frac{4−\sqrt{48}}{2}}\\ {\text{Rewrite the radicand as a product using thelargest perfect square factor.}}&{\frac{4−\sqrt{16·3}}{2}}\\ {\text{Rewrite the radical as the product of two radicals.}}&{\frac{4−\sqrt{16}·\sqrt{3}}{2}}\\ {\text{Simplify.}}&{\frac{4−4\sqrt{3}}{2}}\\ {\text{Factor the common factor from thenumerator.}}&{\frac{4(1−\sqrt{3})}{2}}\\ {\text{Remove the common factor, 2, from thenumerator and denominator.}}&{2(1−\sqrt{3})}\\ \end{array}$$

Mfano$\PageIndex{23}$

Kurahisisha:$\frac{10−\sqrt{75}}{5}$.

Jibu

$2−\sqrt{3}$

Mfano$\PageIndex{24}$

Kurahisisha:$\frac{6−\sqrt{45}}{3}$.

Jibu

$2−\sqrt{5}$

Tumia mali ya Quotient ili kurahisisha Mizizi ya Mraba

Wakati wowote unapaswa kurahisisha mizizi ya mraba, hatua ya kwanza unapaswa kuchukua ni kuamua kama radicand ni mraba kamilifu. Sehemu kamili ya mraba ni sehemu ambayo namba zote na denominator ni mraba kamilifu.

Mfano$\PageIndex{25}$

Kurahisisha:$\sqrt{\frac{9}{64}}$.

Jibu

$$\begin{array}{ll} {}&{\sqrt{\frac{9}{64}}}\\ {\text{Since} (\frac{3}{8})^2}&{\frac{3}{8}}\\ \end{array}$$

Mfano$\PageIndex{26}$

Kurahisisha:$\sqrt{\frac{25}{16}}$.

Jibu

$\frac{5}{4}$

Mfano$\PageIndex{27}$

Kurahisisha:$\sqrt{\frac{49}{81}}$.

Jibu

$\frac{7}{9}$

Ikiwa nambari na denominator zina mambo yoyote ya kawaida, uwaondoe. Unaweza kupata sehemu kamili ya mraba!

Mfano$\PageIndex{28}$

Kurahisisha:$\sqrt{\frac{45}{80}}$.

Jibu

$$\begin{array}{ll} {}&{\sqrt{\frac{45}{80}}}\\ {\text{Simplify inside the radical first. Rewrite showing the common factors of the numerator and denominator.}}&{\sqrt{\frac{5·9}{5·16}}}\\ {\text{Simplify the fraction by removing common factors.}}&{\sqrt{\frac{9}{16}}}\\ {\text{Simplify.} (\frac{3}{4})^2 =\frac{9}{16}}&{\frac{3}{4}}\\ \end{array}$$

Mfano$\PageIndex{29}$

Kurahisisha:$\sqrt{\frac{75}{48}}$.

Jibu

$\frac{5}{4}$

Mfano$\PageIndex{30}$

Kurahisisha:$\sqrt{\frac{98}{162}}$.

Jibu

$\frac{7}{9}$

Katika mfano wa mwisho, hatua yetu ya kwanza ilikuwa kurahisisha sehemu chini ya radical kwa kuondoa mambo ya kawaida. Katika mfano unaofuata tutatumia Mali ya Quotient ili kurahisisha chini ya radical. Sisi kugawanya besi kama kwa kutoa exponents yao,$\frac{a^m}{a^n} = a^{m-n}$,$a \ne 0$.

Mfano$\PageIndex{31}$

Kurahisisha:$\sqrt{\frac{m^6}{m^4}}$.

Jibu

$$\begin{array}{ll} {}&{\sqrt{\frac{m^6}{m^4}}}\\ {\text{Simplify the fraction inside the radical first}}&{}\\ {}&{\sqrt{m^2}}\\ {\text{Divide the like bases by subtracting the exponents.}}&{}\\ {\text{Simplify.}}&{m}\\ \end{array}$$

Mfano$\PageIndex{32}$

Kurahisisha:$\sqrt{\frac{a^8}{a^6}}$.

Jibu

a

Mfano$\PageIndex{33}$

Kurahisisha:$\sqrt{\frac{x^{14}}{x^{10}}}$.

Jibu

$x^2$

Mfano$\PageIndex{34}$

Kurahisisha:$\sqrt{\frac{48p^7}{3p^3}}$.

Jibu

$$\begin{array}{ll} {}&{\sqrt{\frac{48p^7}{3p^3}}}\\ {\text{Simplify the fraction inside the radical first.}}&{\sqrt{16p^4}}\\ {\text{Simplify.}}&{4p^2}\\ \end{array}$$

Mfano$\PageIndex{35}$

Kurahisisha:$\sqrt{\frac{75x^5}{3x}}$.

Jibu

$5x^2$

Mfano$\PageIndex{36}$

Kurahisisha:$\sqrt{\frac{72z^{12}}{2z^{10}}}$.

Jibu

6z

Kumbuka Quotient kwa Mali Nguvu? Ni alisema tunaweza kuongeza sehemu ya nguvu kwa kuongeza nambari na denominator kwa nguvu tofauti.

$(\frac{a}{b})^m=\frac{a^{m}}{b^{m}}$,$b \ne 0$

Tunaweza kutumia mali sawa ili kurahisisha mizizi ya mraba ya sehemu. Baada ya kuondoa mambo yote ya kawaida kutoka kwa nambari na denominator, ikiwa sehemu sio mraba kamili, tunapunguza namba na denominator tofauti.

Ufafanuzi: QUOTIENT PROPERTY YA SQUARE ROOTS

Ikiwa, b ni namba zisizo hasi halisi na$b \ne 0$, basi

$\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$

Mfano$\PageIndex{37}$

Kurahisisha:$\sqrt{\frac{21}{64}}$.

Jibu

$$\begin{array}{ll} {}&{\sqrt{\frac{21}{64}}}\\ {\text{We cannot simplify the fraction inside the radical. Rewrite using the quotient property.}}&{\frac{\sqrt{21}}{\sqrt{64}}}\\ {\text{Simplify the square root of 64. The numerator cannot be simplified.}}&{\frac{\sqrt{21}}{8}}\\ \end{array}$$

Mfano$\PageIndex{38}$

Kurahisisha:$\sqrt{\frac{19}{49}}$.

Jibu

$\frac{\sqrt{19}}{7}$

Mfano$\PageIndex{39}$

Kurahisisha:$\sqrt{\frac{28}{81}}$

Jibu

$\frac{2\sqrt{7}}{9}$

Jinsi ya kutumia Mali ya Quotient ili kurahisisha Mizizi ya Mraba

Mfano$\PageIndex{40}$

Kurahisisha:$\sqrt{\frac{27m^3}{196}}$.

Jibu

Mfano$\PageIndex{41}$

Kurahisisha:$\sqrt{\frac{24p^3}{49}}$

Jibu

$\frac{2p\sqrt{6p}}{7}$

Mfano$\PageIndex{42}$

Kurahisisha:$\sqrt{\frac{48x^5}{100}}$

Jibu

$\frac{2x^2\sqrt{3x}}{5}$

ufafanuzi: KURAHISISHA MIZIZI SQUARE KUTUMIA QUOTIENT PROPERTY.

1. Kurahisisha sehemu katika radicand, ikiwa inawezekana.
2. Matumizi ya Mali Quotient kuandika upya radical kama quotient ya radicals mbili.
3. Kurahisisha radicals katika nambari na denominator.

Mfano$\PageIndex{43}$

Kurahisisha:$\sqrt{\frac{45x^5}{y^4}}$.

Jibu

$$\begin{array}{ll} {}&{\sqrt{\frac{45x^5}{y^4}}}\\ {\text{We cannot simplify the fraction inside the radical. Rewrite using the quotient property.}}&{\frac{\sqrt{45x^5}}{\sqrt{y^4}}}\\ {\text{Simplify the radicals in the numerator and the denominator.}}&{\frac{\sqrt{9x^4}\sqrt{5x}}{y^2}}\\ {\text{Simplify.}}&{\frac{3x^2\sqrt{5x}}{y^2}}\\ \end{array}$$

Mfano$\PageIndex{44}$

Kurahisisha:$\sqrt{\frac{80m^3}{n^6}}$

Jibu

$\frac{4m\sqrt{5m}}{n^3}$

Mfano$\PageIndex{45}$

Kurahisisha:$\sqrt{\frac{54u^7}{v^8}}$.

Jibu

$\frac{3u^3\sqrt{6u}}{v^4}$

Hakikisha kurahisisha sehemu katika radicna kwanza, ikiwa inawezekana.

Mfano$\PageIndex{46}$

Kurahisisha:$\sqrt{\frac{81d^9}{25d^4}}$.

Jibu

$$\begin{array}{ll} {}&{\sqrt{\frac{81d^9}{25d^4}}}\\ {\text{Simplify the fraction in the radicand.}}&{\sqrt{\frac{81d^5}{25}}}\\ {\text{Rewrite using the quotient property.}}&{\frac{\sqrt{81d^5}}{\sqrt{25}}}\\ {\text{Simplify the radicals in the numerator and the denominator.}}&{\frac{\sqrt{81d^4}\sqrt{d}}{5}}\\ {\text{Simplify.}}&{\frac{9d^2\sqrt{d}}{5}}\\ \end{array}$$

Mfano$\PageIndex{47}$

Kurahisisha:$\sqrt{\frac{64x^7}{9x^3}}$.

Jibu

$\frac{8x^2}{3}$

Mfano$\PageIndex{48}$

Kurahisisha:$\sqrt{\frac{16a^9}{100a^5}}$.

Jibu

$\frac{2a^2}{5}$

Mfano$\PageIndex{49}$

Kurahisisha:$\sqrt{\frac{18p^5q^7}{32pq^2}}$.

Jibu

$$\begin{array}{ll} {}&{\sqrt{\frac{18p^5q^7}{32pq^2}}}\\ {\text{Simplify the fraction in the radicand.}}&{\sqrt{\frac{9p^4q^5}{16}}}\\ {\text{Rewrite using the quotient property.}}&{\frac{\sqrt{9p^4q^5}}{\sqrt{16}}}\\ {\text{Simplify the radicals in the numerator and the denominator.}}&{\frac{\sqrt{9p^4q^4}\sqrt{q}}{4}}\\ {\text{Simplify.}}&{\frac{3p^2q^2\sqrt{q}}{4}}\\ \end{array}$$

Mfano$\PageIndex{50}$

Kurahisisha:$\sqrt{\frac{50x^5y^3}{72x^4y}}$.

Jibu

$\frac{5y\sqrt{x}}{6}$

Mfano$\PageIndex{51}$

Kurahisisha:$\sqrt{\frac{48m^7n^2}{125m^5n^9}}$.

Jibu

$\frac{4m\sqrt{3}}{5n^3\sqrt{5n}}$

Dhana muhimu

• Kilichorahisishwa Square Root$\sqrt{a}$ inachukuliwa kilichorahisishwa kama hana sababu kamilifu za mraba.
• Bidhaa Mali ya Mizizi Square Kama, b ni zisizo hasi idadi halisi, basi

  $\sqrt{ab}=\sqrt{a}·\sqrt{b}$

• Kurahisisha Mizizi ya Mraba Kutumia Mali ya Bidhaa Ili kurahisisha mizizi ya mraba kwa kutumia Mali ya Bidhaa:
  1. Pata sababu kubwa zaidi ya mraba ya radicand. Andika upya radicna kama bidhaa kwa kutumia sababu kamili ya mraba.
  2. Tumia utawala wa bidhaa ili uandike upya radical kama bidhaa ya radicals mbili.
  3. Kurahisisha mizizi ya mraba wa mraba kamilifu.
• Mali ya Quotient ya Mizizi ya Mraba Kama, b ni namba zisizo hasi halisi na$b \ne 0$, basi

  $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$

• Kurahisisha Mizizi ya Mraba Kutumia Mali ya Quotient Ili kurahisisha mizizi ya mraba kwa kutumia Mali ya Quotient:
  1. Kurahisisha sehemu katika radicand, ikiwa inawezekana.
  2. Tumia Utawala wa Quotient kuandika upya radical kama quotient ya radicals mbili.
  3. Kurahisisha radicals katika nambari na denominator.