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Matemáticas 2 · Tema 4

Álgebra con fracciones

Mini apunte

Multiplicación (xa)(xb)=xa+b(x^a)(x^b) = x^{a+b} — los exponentes se suman
División xaxb=xa−b\dfrac{x^a}{x^b} = x^{a-b} — producto cruzado o ley de la herradura; exponentes se restan
Potencias (xa)b=xa⋅b(x^a)^b = x^{a \cdot b} — los exponentes se multiplican
Raíces xan=xa÷n\sqrt[n]{x^a} = x^{a \div n} — el exponente se divide entre el índice
Jerarquía Potencias y raíces → Multiplicación y división → Suma y resta

Ejemplos

Multiplicación (5x8b36a4b2) ⁣(−4x2b3a5c)=−20x10b418a9b2c=−10x10b29a9c\left(\frac{5x^8b^3}{6a^4b^2}\right)\!\left(\frac{-4x^2b}{3a^5c}\right) = \frac{-20x^{10}b^4}{18a^9b^2c} = \mathbf{\dfrac{-10x^{10}b^2}{9a^9c}}
División 8x7y3aa46x2−4a3y=−32x7y4a46a4x2=−16x5y43\dfrac{\dfrac{8x^7y^3a}{a^4}}{\dfrac{6x^2}{-4a^3y}} = \frac{-32x^7y^4a^4}{6a^4x^2} = \mathbf{\dfrac{-16x^5y^4}{3}}
Potenciación (4x3y2b68x4y3b2) ⁣2=16x6y4b1264x8y6b4=b84x2y2\left(\frac{4x^3y^2b^6}{8x^4y^3b^2}\right)^{\!2} = \frac{16x^6y^4b^{12}}{64x^8y^6b^4} = \mathbf{\dfrac{b^8}{4x^2y^2}}
Radicación 25x6y4b236x8=5x3y2b6x4=5y2b6x\sqrt{\frac{25x^6y^4b^2}{36x^8}} = \frac{5x^3y^2b}{6x^4} = \mathbf{\dfrac{5y^2b}{6x}}
Jerarquía — potencia primero 8x3−4(5x2) ⁣3=8x3−4(125x6)=8x3−500x6=8x7−15003x6\begin{aligned} \frac{8x}{3} - 4\left(\frac{5}{x^2}\right)^{\!3} &= \frac{8x}{3} - 4\left(\frac{125}{x^6}\right) \\[14pt] &= \frac{8x}{3} - \frac{500}{x^6} \\[14pt] &= \mathbf{\dfrac{8x^7 - 1500}{3x^6}} \end{aligned}
Binomio al cuadrado (3x22+8x53) ⁣2=(3x22) ⁣2+2 ⁣(3x22) ⁣(8x53)+(8x53) ⁣2=9x44+48x76+64x109=9x44+8x7+64x109\begin{aligned} \left(\frac{3x^2}{2} + \frac{8x^5}{3}\right)^{\!2} &= \left(\frac{3x^2}{2}\right)^{\!2} + 2\!\left(\frac{3x^2}{2}\right)\!\left(\frac{8x^5}{3}\right) + \left(\frac{8x^5}{3}\right)^{\!2} \\[14pt] &= \frac{9x^4}{4} + \frac{48x^7}{6} + \frac{64x^{10}}{9} \\[14pt] &= \mathbf{\dfrac{9x^4}{4} + 8x^7 + \dfrac{64x^{10}}{9}} \end{aligned}
Simplificación — factorizar y cancelar x2−2x−3x2+5x+4=(x−3)(x+1)(x+4)(x+1)=x−3x+4\frac{x^2-2x-3}{x^2+5x+4} = \frac{(x-3)(x+1)}{(x+4)(x+1)} = \mathbf{\dfrac{x-3}{x+4}}

Ejercicios

Bloque 1
(5x4b3a3b) ⁣(−6x2ab4)\left(\dfrac{5x^4b}{3a^3b}\right)\!\left(\dfrac{-6x^2a}{b^4}\right)
3x5y2ab34xb2−3ab\dfrac{\dfrac{3x^5y^2a}{b^3}}{\dfrac{4xb^2}{-3ab}}
(5x2y3b4−2y5a) ⁣3\left(\dfrac{5x^2y^3b^4}{-2y^5a}\right)^{\!3}
36x12b44y2a6\sqrt{\dfrac{36x^{12}b^4}{4y^2a^6}}
7x2−5(2x3) ⁣2\dfrac{7x}{2} - 5\left(\dfrac{2}{x^3}\right)^{\!2}
(4x23+5x4) ⁣2\left(\dfrac{4x^2}{3} + \dfrac{5x}{4}\right)^{\!2}
x2+6x+5x2+3x−10\dfrac{x^2+6x+5}{x^2+3x-10}
Bloque 2
(−6x7y2x5) ⁣(−5x8y3bax)\left(\dfrac{-6x^7y}{2x^5}\right)\!\left(\dfrac{-5x^8y}{3bax}\right)
−2x4y3ba25x3b2a2b\dfrac{\dfrac{-2x^4y^3b}{a^2}}{\dfrac{5x^3b}{2a^2b}}
(6x3y4−2y8a3x) ⁣2\left(\dfrac{6x^3y^4}{-2y^8a^3x}\right)^{\!2}
−8x9a33125y63\dfrac{\sqrt[3]{-8x^9a^3}}{\sqrt[3]{125y^6}}
2x3−4(5x4) ⁣2\dfrac{2x}{3} - 4\left(\dfrac{5}{x^4}\right)^{\!2}
(6x32−4x26) ⁣2\left(\dfrac{6x^3}{2} - \dfrac{4x^2}{6}\right)^{\!2}
x2+2x−3x2−3x+2\dfrac{x^2+2x-3}{x^2-3x+2}
Bloque 3
(−2x5b35xb) ⁣(−2a3x4b3)\left(\dfrac{-2x^5b^3}{5xb}\right)\!\left(\dfrac{-2a^3x}{4b^3}\right)
−4x3yab25xb6a3\dfrac{\dfrac{-4x^3ya}{b^2}}{\dfrac{5xb}{6a^3}}
(4x3y2a6x7a) ⁣2\left(\dfrac{4x^3y^2a}{6x^7a}\right)^{\!2}
25x6b84y6x4\sqrt{\dfrac{25x^6b^8}{4y^6x^4}}
2x3−6(3x4) ⁣2\dfrac{2x}{3} - 6\left(\dfrac{3}{x^4}\right)^{\!2}
(5x32+4x3) ⁣2\left(\dfrac{5x^3}{2} + \dfrac{4x}{3}\right)^{\!2}
x2−6x+8x2+x−20\dfrac{x^2-6x+8}{x^2+x-20}
Bloque 4
(5x4a32a6) ⁣(−3x8ya2)\left(\dfrac{5x^4a^3}{2a^6}\right)\!\left(\dfrac{-3x^8}{ya^2}\right)
−4x5y3a2a5x710x9y4\dfrac{\dfrac{-4x^5y^3a}{2a}}{\dfrac{5x^7}{10x^9y^4}}
(−4x6y35x7b3) ⁣3\left(\dfrac{-4x^6y^3}{5x^7b^3}\right)^{\!3}
64x12a631000a33\dfrac{\sqrt[3]{64x^{12}a^6}}{\sqrt[3]{1000a^3}}
3x5−4(2x5) ⁣2\dfrac{3x}{5} - 4\left(\dfrac{2}{x^5}\right)^{\!2}
(3x82−5x43) ⁣2\left(\dfrac{3x^8}{2} - \dfrac{5x^4}{3}\right)^{\!2}
x2−6x+8x2+4x−12\dfrac{x^2-6x+8}{x^2+4x-12}

Resoluciones

▼
Bloque 1
−30x6ba3a3b5=−10x6a2b4\frac{-30x^6ba}{3a^3b^5} = \mathbf{\dfrac{-10x^6}{a^2b^4}}
−9x5y2a2b4b5x=−9x4y2a24b4\frac{-9x^5y^2a^2b}{4b^5x} = \mathbf{\dfrac{-9x^4y^2a^2}{4b^4}}
125x6y9b12−8y15a3=−125x6b128y6a3\frac{125x^6y^9b^{12}}{-8y^{15}a^3} = \mathbf{\dfrac{-125x^6b^{12}}{8y^6a^3}}
6x6b22ya3=3x6b2ya3\frac{6x^6b^2}{2ya^3} = \mathbf{\dfrac{3x^6b^2}{ya^3}}
7x2−20x6=7x7−402x6\begin{aligned} \frac{7x}{2} - \frac{20}{x^6} &= \mathbf{\dfrac{7x^7-40}{2x^6}} \end{aligned}
(4x23) ⁣2+2 ⁣(4x23) ⁣(5x4)+(5x4) ⁣2=16x49+10x33+25x216\begin{aligned} \left(\tfrac{4x^2}{3}\right)^{\!2} + 2\!\left(\tfrac{4x^2}{3}\right)\!\left(\tfrac{5x}{4}\right) + \left(\tfrac{5x}{4}\right)^{\!2} \\[14pt] = \mathbf{\dfrac{16x^4}{9} + \dfrac{10x^3}{3} + \dfrac{25x^2}{16}} \end{aligned}
(x+5)(x+1)(x+5)(x−2)=x+1x−2\frac{(x+5)(x+1)}{(x+5)(x-2)} = \mathbf{\dfrac{x+1}{x-2}}
Bloque 2
30x15y26x6ba=5x9y2ba\frac{30x^{15}y^2}{6x^6ba} = \mathbf{\dfrac{5x^9y^2}{ba}}
−4x4y3b2a25a2x3b=−4xy3b5\frac{-4x^4y^3b^2a^2}{5a^2x^3b} = \mathbf{\dfrac{-4xy^3b}{5}}
36x6y84y16a6x2=9x4y8a6\frac{36x^6y^8}{4y^{16}a^6x^2} = \mathbf{\dfrac{9x^4}{y^8a^6}}
−8x9a33125y63=−2x3a5y2\dfrac{\sqrt[3]{-8x^9a^3}}{\sqrt[3]{125y^6}} = \mathbf{\dfrac{-2x^3a}{5y^2}}
2x3−100x8=2x9−3003x8\begin{aligned} \frac{2x}{3} - \frac{100}{x^8} &= \mathbf{\dfrac{2x^9-300}{3x^8}} \end{aligned}
(6x32) ⁣2+2 ⁣(6x32) ⁣(−4x26)+(−4x26) ⁣2=9x6−4x5+4x49\begin{aligned} \left(\tfrac{6x^3}{2}\right)^{\!2} + 2\!\left(\tfrac{6x^3}{2}\right)\!\left(\tfrac{-4x^2}{6}\right) + \left(\tfrac{-4x^2}{6}\right)^{\!2} \\[14pt] = \mathbf{9x^6 - 4x^5 + \dfrac{4x^4}{9}} \end{aligned}
(x−1)(x+3)(x−2)(x−1)=x+3x−2\frac{(x-1)(x+3)}{(x-2)(x-1)} = \mathbf{\dfrac{x+3}{x-2}}
Bloque 3
4x6a3b320xb4=x5a35b\frac{4x^6a^3b^3}{20xb^4} = \mathbf{\dfrac{x^5a^3}{5b}}
−24a4x3y5b3x=−24a4x2y5b3\frac{-24a^4x^3y}{5b^3x} = \mathbf{\dfrac{-24a^4x^2y}{5b^3}}
16x6y4a236x14a2=4y49x8\frac{16x^6y^4a^2}{36x^{14}a^2} = \mathbf{\dfrac{4y^4}{9x^8}}
5xb42y3\mathbf{\dfrac{5xb^4}{2y^3}}
2x3−54x8=2x9−1623x8\begin{aligned} \frac{2x}{3} - \frac{54}{x^8} &= \mathbf{\dfrac{2x^9-162}{3x^8}} \end{aligned}
25x64+20x43+16x29\mathbf{\dfrac{25x^6}{4} + \dfrac{20x^4}{3} + \dfrac{16x^2}{9}}
(x−4)(x−2)(x−4)(x+5)=x−2x+5\frac{(x-4)(x-2)}{(x-4)(x+5)} = \mathbf{\dfrac{x-2}{x+5}}
Bloque 4
−15x12a32a8y=−15x122a5y\frac{-15x^{12}a^3}{2a^8y} = \mathbf{\dfrac{-15x^{12}}{2a^5y}}
−40x14y7a10ax7=−4x7y7\frac{-40x^{14}y^7a}{10ax^7} = \mathbf{-4x^7y^7}
−64x18y9125x21b9=−64y9125x3b9\frac{-64x^{18}y^9}{125x^{21}b^9} = \mathbf{\dfrac{-64y^9}{125x^3b^9}}
64x12a631000a33=4x4a210a=2x4a5\dfrac{\sqrt[3]{64x^{12}a^6}}{\sqrt[3]{1000a^3}} = \dfrac{4x^4a^2}{10a} = \mathbf{\dfrac{2x^4a}{5}}
3x5−16x10=3x11−805x10\begin{aligned} \frac{3x}{5} - \frac{16}{x^{10}} &= \mathbf{\dfrac{3x^{11}-80}{5x^{10}}} \end{aligned}
9x164−5x12+25x89\mathbf{\dfrac{9x^{16}}{4} - 5x^{12} + \dfrac{25x^8}{9}}
(x−4)(x−2)(x+6)(x−2)=x−4x+6\frac{(x-4)(x-2)}{(x+6)(x-2)} = \mathbf{\dfrac{x-4}{x+6}}

Ejercicios extra

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  • (7a86a3c) ⁣(−9x4ac7)\left(\dfrac{7a^8}{6a^3c}\right)\!\left(\dfrac{-9x^4a}{c^7}\right)
  • (5a812a2c) ⁣(−6c3ac5)\left(\dfrac{5a^8}{12a^2c}\right)\!\left(\dfrac{-6c^3a}{c^5}\right)
  • (4x35a4b) ⁣(−6x2bb6)\left(\dfrac{4x^3}{5a^4b}\right)\!\left(\dfrac{-6x^2b}{b^6}\right)
  • (5x26a2b) ⁣(−4x3ab5)\left(\dfrac{5x^2}{6a^2b}\right)\!\left(\dfrac{-4x^3a}{b^5}\right)
  • 6x8y3c58xc4−7xc\dfrac{\dfrac{6x^8y^3}{c^5}}{\dfrac{8xc^4}{-7xc}}
  • −x4y33c47xc3−5yc\dfrac{\dfrac{-x^4y^3}{3c^4}}{\dfrac{7xc^3}{-5yc}}
  • 5x6y2b57xb3−6y3\dfrac{\dfrac{5x^6y^2}{b^5}}{\dfrac{7xb^3}{-6y^3}}
  • 7x7y2b43xb3−5xy\dfrac{\dfrac{7x^7y^2}{b^4}}{\dfrac{3xb^3}{-5xy}}
  • (2x5y3c6−5y9a) ⁣3\left(\dfrac{2x^5y^3c^6}{-5y^9a}\right)^{\!3}
  • (6x3y4c5−5y8c) ⁣3\left(\dfrac{6x^3y^4c^5}{-5y^8c}\right)^{\!3}
  • (3x3y5b6−2y7a) ⁣3\left(\dfrac{3x^3y^5b^6}{-2y^7a}\right)^{\!3}
  • (4x4y2b5−3y6a) ⁣3\left(\dfrac{4x^4y^2b^5}{-3y^6a}\right)^{\!3}
  • 81x14c636c6a8\sqrt{\dfrac{81x^{14}c^6}{36c^6a^8}}
  • −10x12c6364y3a93\dfrac{\sqrt[3]{-10x^{12}c^6}}{\sqrt[3]{64y^3a^9}}
  • 36x14b825y4a6\sqrt{\dfrac{36x^{14}b^8}{25y^4a^6}}
  • −48x15a9327y6a33\dfrac{\sqrt[3]{-48x^{15}a^9}}{\sqrt[3]{27y^6a^3}}
  • −10x6+2(4x7) ⁣3-\dfrac{10x}{6} + 2\left(\dfrac{4}{x^7}\right)^{\!3}
  • −7x4−5(2x4) ⁣2-\dfrac{7x}{4} - 5\left(\dfrac{2}{x^4}\right)^{\!2}
  • 11x5−3(5x5) ⁣3\dfrac{11x}{5} - 3\left(\dfrac{5}{x^5}\right)^{\!3}
  • 9x6−4(6x6) ⁣2\dfrac{9x}{6} - 4\left(\dfrac{6}{x^6}\right)^{\!2}
  • (5x27+3x4) ⁣2\left(\dfrac{5x^2}{7} + \dfrac{3x}{4}\right)^{\!2}
  • (4x35−7x6) ⁣2\left(\dfrac{4x^3}{5} - \dfrac{7x}{6}\right)^{\!2}
  • (7x28+5x6) ⁣2\left(\dfrac{7x^2}{8} + \dfrac{5x}{6}\right)^{\!2}
  • (5x27−4x3) ⁣2\left(\dfrac{5x^2}{7} - \dfrac{4x}{3}\right)^{\!2}
  • x2−8x+15x2−x−20\dfrac{x^2-8x+15}{x^2-x-20}
  • x2−4x+5x2−3x−10\dfrac{x^2-4x+5}{x^2-3x-10}
  • x2+6x+9x2−x−12\dfrac{x^2+6x+9}{x^2-x-12}
  • x2−7x+10x2+6x−16\dfrac{x^2-7x+10}{x^2+6x-16}
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