Formulario de álgebra - Matemóvil

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Formulario de álgebra 
Descarga y comparte 
1 
PROPIEDADES ARITMÉTICAS 
ASOCIATIVA 𝑎(𝑏𝑐) = (𝑎𝑏)𝑐 
CONMUTATIVA 𝑎 + 𝑏 = 𝑏 + 𝑎 𝑦 𝑎𝑏 = 𝑏𝑎 
DISTRIBUTIVA 𝑎(𝑏 + 𝑐) = 𝑎𝑏 + 𝑎𝑐 
 
LEY DE SIGNOS 
MULTIPLICACIÓN DIVISIÓN 
(+) × (+) = (+) (+) ÷ (+) = (+) 
(−) × (−) = (+) (−) ÷ (−) = (+) 
(+) × (−) = (−) (+) ÷ (−) = (−) 
(−) × (+) = (−) (−) ÷ (+) = (−) 
 
EJEMPLOS DE OPERACIONES ARITMÉTICAS 
𝑎𝑏 + 𝑎𝑐 = 𝑎(𝑏 + 𝑐) 
𝑎
𝑏
−
𝑐
𝑑
=
𝑎𝑑 − 𝑏𝑐
𝑏𝑑
 
𝑎 (
𝑏
𝑐
) =
𝑎𝑏
𝑐
 
𝑎 − 𝑏
𝑐 − 𝑑
=
𝑏 − 𝑎
𝑑 − 𝑐
 
(
𝑎
𝑏
)
𝑐
=
𝑎
𝑏𝑐
 
𝑎 + 𝑏
𝑐
=
𝑎
𝑐
+
𝑏
𝑐
 
𝑎
(
𝑏
𝑐)
=
𝑎𝑐
𝑏
 𝑎𝑏 + 𝑎𝑐
𝑎
= 𝑏 + 𝑐, 𝑎 ≠ 0 
𝑎
𝑏
+
𝑐
𝑑
=
𝑎𝑑 + 𝑏𝑐
𝑏𝑑
 
(
𝑎
𝑏
)
(
𝑐
𝑑
)
=
𝑎𝑑
𝑏𝑐
 
 
ECUACIÓN CUADRÁTICA 
𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 → 𝑥 =
−𝑏 ± √𝑏2 − 4𝑎𝑐
2𝑎
 
 
CURSO DE ÁLGEBRA 
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álgebra, dale un vistazo a nuestro 
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cientos de ejercicios resueltos. 
 
RADICALES 
√𝑎
𝑛
= 𝑏 ↔ 𝑎 = 𝑏𝑛 √𝑎
𝑛
= 𝑎
1
𝑛 
√ √𝑎
𝑛𝑚
= √𝑎
𝑚𝑛
 √𝑎𝑏
𝑛
= √𝑎
𝑛
⋅ √𝑏
𝑛
 
√
𝑎
𝑏
𝑛
=
√𝑎
𝑛
√𝑏
𝑛 √
𝑎𝑥
𝑏𝑦
𝑚
=
√𝑎𝑥
𝑚
√𝑏𝑦
𝑚 
√𝑎𝑛
𝑛
= 𝑎, 𝑠𝑖 𝑛 𝑒𝑠 𝑖𝑚𝑝𝑎𝑟 √𝑎𝑛
𝑛
= |𝑎|, 𝑠𝑖 𝑛 𝑒𝑠 𝑝𝑎𝑟 
 
 
 
LEYES DE EXPONENTES 
 
𝑎0 = 1; 𝑎 ≠ 0 𝑎−𝑚 =
1
𝑎𝑚
; 𝑎 ≠ 0 
𝑎𝑚 ⋅ 𝑎𝑛 = 𝑎𝑚+𝑛 
𝑎𝑚
𝑎𝑛
= 𝑎𝑚−𝑛 
(𝑎𝑚)𝑛 = 𝑎𝑚⋅𝑛 = 𝑎𝑛⋅𝑚 = (𝑎𝑛)𝑚 𝑎
𝑚
𝑛 = √𝑎𝑚
𝑛
 
(𝑎𝑚 ⋅ 𝑏𝑛 ⋅ 𝑐𝑝)𝑥 = 𝑎𝑚𝑥 ⋅ 𝑏𝑛𝑥 ⋅ 𝑐𝑝𝑥 
(
𝑎𝑚
𝑏𝑛
)
𝑥
=
𝑎𝑚⋅𝑥
𝑏𝑛⋅𝑥
 (
𝑎
𝑏
)
−𝑚
= (
𝑏
𝑎
)
𝑚
 
 
PRODUCTOS NOTABLES 
(𝑎 + 𝑏)2 = 𝑎2 + 2𝑎𝑏 + 𝑏2 (𝑎 − 𝑏)2 = 𝑎2 − 2𝑎𝑏 + 𝑏2 
(𝑎 + 𝑏)3 = 𝑎3 + 3𝑎2𝑏 + 3𝑎𝑏2 + 𝑏3 
(𝑎 + 𝑏)3 = 𝑎3 + 𝑏3 + 3𝑎𝑏(𝑎 + 𝑏) 
(𝑎 − 𝑏)3 = 𝑎3 − 3𝑎2𝑏 + 3𝑎𝑏2 − 𝑏3 
(𝑎 − 𝑏)3 = 𝑎3 − 𝑏3 − 3𝑎𝑏(𝑎 − 𝑏) 
𝑎2 − 𝑏2 = (𝑎 + 𝑏)(𝑎 − 𝑏) 
(𝑥 + 𝑎)(𝑥 + 𝑏) = 𝑥2 + (𝑎 + 𝑏)𝑥 + 𝑎𝑏 
(𝑎 + 𝑏)2 + (𝑎 − 𝑏)2 = 2(𝑎2 + 𝑏2) 
(𝑎 + 𝑏)2 − (𝑎 − 𝑏)2 = 4𝑎𝑏 
(𝑎 + 𝑏)(𝑎2 − 𝑎𝑏 + 𝑏2) = 𝑎3 + 𝑏3 
(𝑎 − 𝑏)(𝑎2 + 𝑎𝑏 + 𝑏2) = 𝑎3 − 𝑏3 
(𝑎 + 𝑏 + 𝑐)2 = 𝑎2 + 𝑏2 + 𝑐2 + 2𝑎𝑏 + 2𝑏𝑐 + 2𝑎𝑐 
(𝑎2 + 𝑎𝑏 + 𝑏2)(𝑎2 − 𝑎𝑏 + 𝑏2) = 𝑎4 + 𝑎2𝑏2 + 𝑎4 
(𝑎 + 𝑏 + 𝑐)3 = 𝑎3 + 𝑏3 + 𝑐3 + 3(𝑎 + 𝑏)(𝑎 + 𝑐)(𝑏 + 𝑐) 
 
FACTORIZACIÓN 
𝑎2𝑚 + 2𝑎𝑚𝑏𝑛 + 𝑏2𝑛 = (𝑎𝑚 + 𝑏𝑛)2 
𝑎2𝑚 − 2𝑎𝑚𝑏𝑛 + 𝑏2𝑛 = (𝑎𝑚 − 𝑏𝑛)2 
𝑎2𝑚 − 𝑏2𝑛 = (𝑎𝑚 + 𝑏𝑛)(𝑎𝑚 − 𝑏𝑛) 
𝑎3𝑚 + 𝑏3𝑛 = (𝑎𝑚 + 𝑏𝑛)(𝑎2𝑚 − 𝑎𝑚𝑏𝑛 + 𝑏2𝑛) 
𝑎3𝑚 − 𝑏3𝑛 = (𝑎𝑚 − 𝑏𝑛)(𝑎2𝑚 + 𝑎𝑚𝑏𝑛 + 𝑏2𝑛) 
𝑥2 + (𝑎 + 𝑏)𝑥 + 𝑎𝑏 = (𝑥 + 𝑎)(𝑥 + 𝑏) 
𝑎𝑥2𝑚 + 𝑏𝑥𝑚𝑦𝑛 + 𝑐𝑦𝑛 = (𝑎1𝑥
𝑚 + 𝑐1𝑦
𝑛)(𝑎2𝑥
𝑚 + 𝑐2𝑦
𝑛) 
 
𝑎1𝑥
𝑚
𝑎2𝑥
𝑚 
𝑐1𝑦
𝑛 ⇒ 𝑎2𝑐1𝑥
𝑚𝑦𝑛 
𝑐2𝑦
𝑛 ⇒ 𝑎1𝑐2𝑥
𝑚𝑦𝑛 
 (+) 
 
 𝑏𝑥𝑚𝑦𝑛 
 
DESIGUALDADES 
𝑆𝑖 𝑎 < 𝑏 → 𝑎 + 𝑐 < 𝑏 + 𝑐 𝑦 𝑎 − 𝑐 < 𝑏 − 𝑐 
𝑆𝑖 𝑎 < 𝑏 𝑦 𝑐 > 0 → 𝑎𝑐 < 𝑏𝑐 𝑦 𝑎/𝑐 < 𝑏/𝑐 
𝑆𝑖 𝑎 < 𝑏 𝑦 𝑐 < 0 → 𝑎𝑐 > 𝑏𝑐 𝑦 𝑎/𝑐 > 𝑏/𝑐 
 
 
 
 
 
Formulario de álgebra 
Descarga y comparte 
2 
FACTORIAL Y NÚMERO COMBINATORIO 
𝑛! = 1 × 2 × 3 × 4 × ⋯ × (𝑛 − 1) × 𝑛 ; 𝑛 ∈ ℕ; 𝑛 > 1 
1! = 1 0! = 1 
𝐶𝑘
𝑛 = (
𝑛
𝑘
) =
𝑛!
(𝑛 − 𝑘)! 𝑘!
 
𝐶0
𝑛 = 1 𝐶1
𝑛 = 𝑛 𝐶𝑛
𝑛 = 1 
 
NÚMEROS COMPLEJOS 
𝑖 = √−1 𝑖2 = −1 𝑖3 = −𝑖 𝑖4 = 1 
√−𝑎 = 𝑖√𝑎, 𝑎 ≥ 0 
(𝑎 + 𝑏𝑖) + (𝑐 + 𝑑𝑖) = 𝑎 + 𝑐 + (𝑏 + 𝑑)𝑖 
(𝑎 + 𝑏𝑖) − (𝑐 + 𝑑𝑖) = 𝑎 − 𝑐 + (𝑏 − 𝑑)𝑖 
(𝑎 + 𝑏𝑖)(𝑐 + 𝑑𝑖) = 𝑎𝑐 − 𝑏𝑑 + (𝑎𝑑 + 𝑏𝑐)𝑖 
(𝑎 + 𝑏𝑖)(𝑎 − 𝑏𝑖) = 𝑎2 + 𝑏2 
|𝑎 + 𝑏𝑖| = √𝑎2 + 𝑏2 (𝑎 + 𝑏𝑖)̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅ = 𝑎 − 𝑏𝑖 
(𝑎 + 𝑏𝑖)̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅(𝑎 + 𝑏𝑖) = |𝑎 + 𝑏𝑖|2 
1
𝑎 + 𝑏𝑖
=
𝑎 − 𝑏𝑖
(𝑎 + 𝑏𝑖)(𝑎 − 𝑏𝑖)
=
𝑎 − 𝑏𝑖
𝑎2 + 𝑏2
 
 
 VALOR ABSOLUTO 
|𝑎| = {
𝑎; 𝑠𝑖 𝑎 ≥ 0
−𝑎; 𝑠𝑖 𝑎 < 0
 |𝑎| = |−𝑎| 
|𝑎| ≥ 0 |𝑎𝑏| = |𝑎||𝑏| 
|
𝑎
𝑏
| =
|𝑎|
|𝑏|
 |𝑎 + 𝑏| ≤ |𝑎| + |𝑏| 
 
PROPIEDADES DE LOS LOGARITMOS 
𝑆𝑖 𝑙𝑜𝑔𝑏𝑎 = 𝑥 → 𝑎 = 𝑏
𝑥; 𝑎 > 0; 𝑏 > 0; 𝑏 ≠ 1 
𝑙𝑜𝑔10𝑎 = 𝑙𝑜𝑔𝑎 𝑙𝑜𝑔𝑒𝑎 = 𝑙𝑛𝑎 
𝑙𝑜𝑔𝑏𝑏 = 1 𝑙𝑜𝑔𝑏1 = 0 
𝑙𝑜𝑔𝑏(𝑥
𝑟) = 𝑟𝑙𝑜𝑔𝑏𝑥 𝑙𝑜𝑔𝑏𝑏
𝑥 = 𝑥 
𝑏𝑙𝑜𝑔𝑏𝑥 = 𝑥 𝑙𝑜𝑔𝑎𝑏 ⋅ 𝑙𝑜𝑔𝑏𝑐 ⋅ 𝑙𝑜𝑔𝑐𝑑 = 𝑙𝑜𝑔𝑎𝑑 
𝑙𝑜𝑔𝑎𝑥 =
𝑙𝑜𝑔𝑏𝑥
𝑙𝑜𝑔𝑏𝑎
 
𝑙𝑜𝑔𝑏𝑥
𝑙𝑜𝑔𝑎𝑥
= 𝑙𝑜𝑔𝑏𝑎 
𝑙𝑜𝑔𝑏(𝑥𝑦) = 𝑙𝑜𝑔𝑏𝑥 + 𝑙𝑜𝑔𝑏𝑦 𝑙𝑜𝑔𝑏 (
𝑥
𝑦
) = 𝑙𝑜𝑔𝑏𝑥 − 𝑙𝑜𝑔𝑏𝑦 
𝑐𝑜𝑙𝑜𝑔𝑏𝑥 = 𝑙𝑜𝑔𝑏 (
1
𝑥
) = 𝑙𝑜𝑔𝑏(1) − 𝑙𝑜𝑔𝑏𝑥 = −𝑙𝑜𝑔𝑏𝑥 
 
Versión 1.00 
Fórmulas: Jorge. 
Diseño: Pedro.