Matemáticas 2 · Tema 4
Álgebra con fracciones
Mini apunte
| Multiplicación | $(x^a)(x^b) = x^{a+b}$ — los exponentes se suman |
| División | $\dfrac{x^a}{x^b} = x^{a-b}$ — producto cruzado o ley de la herradura; exponentes se restan |
| Potencias | $(x^a)^b = x^{a \cdot b}$ — los exponentes se multiplican |
| Raíces | $\sqrt[n]{x^a} = x^{a \div n}$ — el exponente se divide entre el índice |
| Jerarquía | Potencias y raíces → Multiplicación y división → Suma y resta |
Ejemplos
Multiplicación
$$\left(\frac{5x^8b^3}{6a^4b^2}\right)\!\left(\frac{-4x^2b}{3a^5c}\right) = \frac{-20x^{10}b^4}{18a^9b^2c} = \mathbf{\dfrac{-10x^{10}b^2}{9a^9c}}$$
División
$$\dfrac{\dfrac{8x^7y^3a}{a^4}}{\dfrac{6x^2}{-4a^3y}} = \frac{-32x^7y^4a^4}{6a^4x^2} = \mathbf{\dfrac{-16x^5y^4}{3}}$$
Potenciación
$$\left(\frac{4x^3y^2b^6}{8x^4y^3b^2}\right)^{\!2} = \frac{16x^6y^4b^{12}}{64x^8y^6b^4} = \mathbf{\dfrac{b^8}{4x^2y^2}}$$
Radicación
$$\sqrt{\frac{25x^6y^4b^2}{36x^8}} = \frac{5x^3y^2b}{6x^4} = \mathbf{\dfrac{5y^2b}{6x}}$$
Jerarquía — potencia primero
$$\begin{aligned}
\frac{8x}{3} - 4\left(\frac{5}{x^2}\right)^{\!3} &= \frac{8x}{3} - 4\left(\frac{125}{x^6}\right) \\[14pt]
&= \frac{8x}{3} - \frac{500}{x^6} \\[14pt]
&= \mathbf{\dfrac{8x^7 - 1500}{3x^6}}
\end{aligned}$$
Binomio al cuadrado
$$\begin{aligned}
\left(\frac{3x^2}{2} + \frac{8x^5}{3}\right)^{\!2} &= \left(\frac{3x^2}{2}\right)^{\!2} + 2\!\left(\frac{3x^2}{2}\right)\!\left(\frac{8x^5}{3}\right) + \left(\frac{8x^5}{3}\right)^{\!2} \\[14pt]
&= \frac{9x^4}{4} + \frac{48x^7}{6} + \frac{64x^{10}}{9} \\[14pt]
&= \mathbf{\dfrac{9x^4}{4} + 8x^7 + \dfrac{64x^{10}}{9}}
\end{aligned}$$
Simplificación — factorizar y cancelar
$$\frac{x^2-2x-3}{x^2+5x+4} = \frac{(x-3)(x+1)}{(x+4)(x+1)} = \mathbf{\dfrac{x-3}{x+4}}$$
Ejercicios
Bloque 1
$\left(\dfrac{5x^4b}{3a^3b}\right)\!\left(\dfrac{-6x^2a}{b^4}\right)$
$\dfrac{\dfrac{3x^5y^2a}{b^3}}{\dfrac{4xb^2}{-3ab}}$
$\left(\dfrac{5x^2y^3b^4}{-2y^5a}\right)^{\!3}$
$\sqrt{\dfrac{36x^{12}b^4}{4y^2a^6}}$
$\dfrac{7x}{2} - 5\left(\dfrac{2}{x^3}\right)^{\!2}$
$\left(\dfrac{4x^2}{3} + \dfrac{5x}{4}\right)^{\!2}$
$\dfrac{x^2+6x+5}{x^2+3x-10}$
Bloque 2
$\left(\dfrac{-6x^7y}{2x^5}\right)\!\left(\dfrac{-5x^8y}{3bax}\right)$
$\dfrac{\dfrac{-2x^4y^3b}{a^2}}{\dfrac{5x^3b}{2a^2b}}$
$\left(\dfrac{6x^3y^4}{-2y^8a^3x}\right)^{\!2}$
$$\dfrac{\sqrt[3]{-8x^9a^3}}{\sqrt[3]{125y^6}}$$
$\dfrac{2x}{3} - 4\left(\dfrac{5}{x^4}\right)^{\!2}$
$\left(\dfrac{6x^3}{2} - \dfrac{4x^2}{6}\right)^{\!2}$
$\dfrac{x^2+2x-3}{x^2-3x+2}$
Bloque 3
$\left(\dfrac{-2x^5b^3}{5xb}\right)\!\left(\dfrac{-2a^3x}{4b^3}\right)$
$\dfrac{\dfrac{-4x^3ya}{b^2}}{\dfrac{5xb}{6a^3}}$
$\left(\dfrac{4x^3y^2a}{6x^7a}\right)^{\!2}$
$\sqrt{\dfrac{25x^6b^8}{4y^6x^4}}$
$\dfrac{2x}{3} - 6\left(\dfrac{3}{x^4}\right)^{\!2}$
$\left(\dfrac{5x^3}{2} + \dfrac{4x}{3}\right)^{\!2}$
$\dfrac{x^2-6x+8}{x^2+x-20}$
Bloque 4
$\left(\dfrac{5x^4a^3}{2a^6}\right)\!\left(\dfrac{-3x^8}{ya^2}\right)$
$\dfrac{\dfrac{-4x^5y^3a}{2a}}{\dfrac{5x^7}{10x^9y^4}}$
$\left(\dfrac{-4x^6y^3}{5x^7b^3}\right)^{\!3}$
$$\dfrac{\sqrt[3]{64x^{12}a^6}}{\sqrt[3]{1000a^3}}$$
$\dfrac{3x}{5} - 4\left(\dfrac{2}{x^5}\right)^{\!2}$
$\left(\dfrac{3x^8}{2} - \dfrac{5x^4}{3}\right)^{\!2}$
$\dfrac{x^2-6x+8}{x^2+4x-12}$
Resoluciones
▼Bloque 1
$$\frac{-30x^6ba}{3a^3b^5} = \mathbf{\dfrac{-10x^6}{a^2b^4}}$$
$$\frac{-9x^5y^2a^2b}{4b^5x} = \mathbf{\dfrac{-9x^4y^2a^2}{4b^4}}$$
$$\frac{125x^6y^9b^{12}}{-8y^{15}a^3} = \mathbf{\dfrac{-125x^6b^{12}}{8y^6a^3}}$$
$$\frac{6x^6b^2}{2ya^3} = \mathbf{\dfrac{3x^6b^2}{ya^3}}$$
$$\begin{aligned}
\frac{7x}{2} - \frac{20}{x^6} &= \mathbf{\dfrac{7x^7-40}{2x^6}}
\end{aligned}$$
$$\begin{aligned}
\left(\tfrac{4x^2}{3}\right)^{\!2} + 2\!\left(\tfrac{4x^2}{3}\right)\!\left(\tfrac{5x}{4}\right) + \left(\tfrac{5x}{4}\right)^{\!2} \\[14pt]
= \mathbf{\dfrac{16x^4}{9} + \dfrac{10x^3}{3} + \dfrac{25x^2}{16}}
\end{aligned}$$
$$\frac{(x+5)(x+1)}{(x+5)(x-2)} = \mathbf{\dfrac{x+1}{x-2}}$$
Bloque 2
$$\frac{30x^{15}y^2}{6x^6ba} = \mathbf{\dfrac{5x^9y^2}{ba}}$$
$$\frac{-4x^4y^3b^2a^2}{5a^2x^3b} = \mathbf{\dfrac{-4xy^3b}{5}}$$
$$\frac{36x^6y^8}{4y^{16}a^6x^2} = \mathbf{\dfrac{9x^4}{y^8a^6}}$$
$$\dfrac{\sqrt[3]{-8x^9a^3}}{\sqrt[3]{125y^6}} = \mathbf{\dfrac{-2x^3a}{5y^2}}$$
$$\begin{aligned}
\frac{2x}{3} - \frac{100}{x^8} &= \mathbf{\dfrac{2x^9-300}{3x^8}}
\end{aligned}$$
$$\begin{aligned}
\left(\tfrac{6x^3}{2}\right)^{\!2} + 2\!\left(\tfrac{6x^3}{2}\right)\!\left(\tfrac{-4x^2}{6}\right) + \left(\tfrac{-4x^2}{6}\right)^{\!2} \\[14pt]
= \mathbf{9x^6 - 4x^5 + \dfrac{4x^4}{9}}
\end{aligned}$$
$$\frac{(x-1)(x+3)}{(x-2)(x-1)} = \mathbf{\dfrac{x+3}{x-2}}$$
Bloque 3
$$\frac{4x^6a^3b^3}{20xb^4} = \mathbf{\dfrac{x^5a^3}{5b}}$$
$$\frac{-24a^4x^3y}{5b^3x} = \mathbf{\dfrac{-24a^4x^2y}{5b^3}}$$
$$\frac{16x^6y^4a^2}{36x^{14}a^2} = \mathbf{\dfrac{4y^4}{9x^8}}$$
$$\mathbf{\dfrac{5xb^4}{2y^3}}$$
$$\begin{aligned}
\frac{2x}{3} - \frac{54}{x^8} &= \mathbf{\dfrac{2x^9-162}{3x^8}}
\end{aligned}$$
$$\mathbf{\dfrac{25x^6}{4} + \dfrac{20x^4}{3} + \dfrac{16x^2}{9}}$$
$$\frac{(x-4)(x-2)}{(x-4)(x+5)} = \mathbf{\dfrac{x-2}{x+5}}$$
Bloque 4
$$\frac{-15x^{12}a^3}{2a^8y} = \mathbf{\dfrac{-15x^{12}}{2a^5y}}$$
$$\frac{-40x^{14}y^7a}{10ax^7} = \mathbf{-4x^7y^7}$$
$$\frac{-64x^{18}y^9}{125x^{21}b^9} = \mathbf{\dfrac{-64y^9}{125x^3b^9}}$$
$$\dfrac{\sqrt[3]{64x^{12}a^6}}{\sqrt[3]{1000a^3}} = \dfrac{4x^4a^2}{10a} = \mathbf{\dfrac{2x^4a}{5}}$$
$$\begin{aligned}
\frac{3x}{5} - \frac{16}{x^{10}} &= \mathbf{\dfrac{3x^{11}-80}{5x^{10}}}
\end{aligned}$$
$$\mathbf{\dfrac{9x^{16}}{4} - 5x^{12} + \dfrac{25x^8}{9}}$$
$$\frac{(x-4)(x-2)}{(x+6)(x-2)} = \mathbf{\dfrac{x-4}{x+6}}$$
Ejercicios extra
▼- $\left(\dfrac{7a^8}{6a^3c}\right)\!\left(\dfrac{-9x^4a}{c^7}\right)$
- $\left(\dfrac{5a^8}{12a^2c}\right)\!\left(\dfrac{-6c^3a}{c^5}\right)$
- $\left(\dfrac{4x^3}{5a^4b}\right)\!\left(\dfrac{-6x^2b}{b^6}\right)$
- $\left(\dfrac{5x^2}{6a^2b}\right)\!\left(\dfrac{-4x^3a}{b^5}\right)$
- $\dfrac{\dfrac{6x^8y^3}{c^5}}{\dfrac{8xc^4}{-7xc}}$
- $\dfrac{\dfrac{-x^4y^3}{3c^4}}{\dfrac{7xc^3}{-5yc}}$
- $\dfrac{\dfrac{5x^6y^2}{b^5}}{\dfrac{7xb^3}{-6y^3}}$
- $\dfrac{\dfrac{7x^7y^2}{b^4}}{\dfrac{3xb^3}{-5xy}}$
- $\left(\dfrac{2x^5y^3c^6}{-5y^9a}\right)^{\!3}$
- $\left(\dfrac{6x^3y^4c^5}{-5y^8c}\right)^{\!3}$
- $\left(\dfrac{3x^3y^5b^6}{-2y^7a}\right)^{\!3}$
- $\left(\dfrac{4x^4y^2b^5}{-3y^6a}\right)^{\!3}$
- $\sqrt{\dfrac{81x^{14}c^6}{36c^6a^8}}$
- $$\dfrac{\sqrt[3]{-10x^{12}c^6}}{\sqrt[3]{64y^3a^9}}$$
- $\sqrt{\dfrac{36x^{14}b^8}{25y^4a^6}}$
- $$\dfrac{\sqrt[3]{-48x^{15}a^9}}{\sqrt[3]{27y^6a^3}}$$
- $-\dfrac{10x}{6} + 2\left(\dfrac{4}{x^7}\right)^{\!3}$
- $-\dfrac{7x}{4} - 5\left(\dfrac{2}{x^4}\right)^{\!2}$
- $\dfrac{11x}{5} - 3\left(\dfrac{5}{x^5}\right)^{\!3}$
- $\dfrac{9x}{6} - 4\left(\dfrac{6}{x^6}\right)^{\!2}$
- $\left(\dfrac{5x^2}{7} + \dfrac{3x}{4}\right)^{\!2}$
- $\left(\dfrac{4x^3}{5} - \dfrac{7x}{6}\right)^{\!2}$
- $\left(\dfrac{7x^2}{8} + \dfrac{5x}{6}\right)^{\!2}$
- $\left(\dfrac{5x^2}{7} - \dfrac{4x}{3}\right)^{\!2}$
- $\dfrac{x^2-8x+15}{x^2-x-20}$
- $\dfrac{x^2-4x+5}{x^2-3x-10}$
- $\dfrac{x^2+6x+9}{x^2-x-12}$
- $\dfrac{x^2-7x+10}{x^2+6x-16}$