PROPIEDADES DE LAS POTENCIAS
Enviado por Erick Pedraza • 2 de Marzo de 2021 • Resúmenes • 3.428 Palabras (14 Páginas) • 386 Visitas
PROPIEDADES DE LAS POTENCIAS
Para cualesquier números reales 𝑎 𝑦 𝑏
𝑎0 = 1; 𝑎 ≠ 0; 𝑎1 = 𝑎
𝑎𝑥2 + 𝑏𝑥 + 𝑐 = (𝑎𝑥 + 𝑚)(𝑎𝑥 + 𝑛) =
𝑚𝑛 𝑎[pic 1]
𝑎𝑥2 + (𝑚 + 𝑛)𝑥 +[pic 2]
𝑎−𝑛 = 1[pic 3]
𝑎𝑛
, 𝑛 ∈ ℤ+
𝑏 = 𝑚 + 𝑛, 𝑐 =
𝑚𝑛 𝑎
[pic 4]
𝑎𝑚 • 𝑎𝑛 = 𝑎𝑚+𝑛 𝑎
𝑎𝑚
𝑎𝑚 ÷ 𝑎𝑛 =[pic 5]
𝑎𝑛
= 𝑎𝑚−𝑛; 𝑎 ≠ 0
PROPIEDADES DEL VALOR ABSOLUTO
Sean a y b números reales:
(𝑎𝑚)𝑛 = 𝑎𝑚𝑛
(𝑎 • 𝑏)𝑛 = 𝑎𝑛 • 𝑏𝑛
|𝑎| = {
𝑎 𝑠𝑖 𝑎 ≥ 0
𝑎 𝑛[pic 6]
(𝑎 ÷ 𝑏) = ( )[pic 7]
𝑏
= 𝑎𝑛 ÷ 𝑏𝑛 =
𝑎𝑛
[pic 8]
𝑏𝑛
; 𝑏 ≠ 0
−𝑎 𝑠𝑖 𝑎 ≤ 0
|𝑎| ≥ 0 𝑦 |𝑎| = 0 ↔ 𝑎 = 0
|𝑎| < 𝑏 ↔ −𝑏 < 𝑎 < 𝑏
PROPIEDADES DE LOS RADICALES
𝑎𝑚/𝑛 = 𝑛√𝑎𝑚[pic 9][pic 10][pic 11]
[pic 12] [pic 13] [pic 14] [pic 15]
𝑛√𝑎 + 𝑏 ≠ 𝑛√𝑎 + 𝑛√𝑏, 𝑛√𝑎 − 𝑏 ≠ 𝑛√𝑎 − 𝑛√𝑏
|𝑎| > 𝑏 ↔ 𝑎 > 𝑏 ó 𝑎 < −𝑏
GEOMETRIA ANALITICA
Distancia entre 𝑃(𝑥 , 𝑦 )𝑦 𝑄(𝑥
, 𝑦 )
1 1 2 2[pic 16]
[pic 17]
𝑛 𝑛 𝑛
[pic 18] [pic 19][pic 20]
𝑛 𝑎
[pic 21]
[pic 22]
𝑛 𝑎
[pic 23]
[pic 24]
𝑑 = ̅𝑃̅̅𝑄̅ = √(
− 𝑥
)2 + (𝑦
− 𝑦 )2
√𝑎. 𝑏 = √𝑎. √𝑏, √
𝑏
= √ , 𝑏 ≠ 0
𝑏
2 1 2 1
𝑛 𝑚
√
𝑚𝑛[pic 25]
[pic 26]
Circunferencia con centro en 𝐶(ℎ, 𝑘)𝑦 𝑟𝑎𝑑𝑖𝑜 𝑟
(𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟2
√ √𝑎 = √𝑎[pic 27]
Pendiente de la recta que pasa por los puntos
𝑛√𝑎𝑛 = 𝑎 𝑦 (𝑛√𝑎)𝑛 = [pic 28][pic 29]
𝑃(𝑥 , 𝑦 )𝑦 𝑄(𝑥 , 𝑦 ): 𝑚 = 𝑦2−𝑦1
𝑥 −𝑥[pic 30]
PRODUCTOS NOTABLES
(𝑎 ± 𝑏)2 = 𝑎2 ± 2𝑎𝑏 + 𝑏2
(𝑎 ± 𝑏)3 = 𝑎3 ± 3𝑎2𝑏 + 3𝑎𝑏2 ± 𝑏3 (𝑎 + 𝑏)(𝑎 − 𝑏) = 𝑎2 − 𝑏2
(𝑥 + 𝑚)(𝑥 + 𝑛) = 𝑥2 + (𝑚 + 𝑛)𝑥 + 𝑚𝑛
(𝑎𝑥 + 𝑚)(𝑏𝑥 + 𝑛) = 𝑎𝑏𝑥2 + (𝑎𝑛 + 𝑏𝑚)𝑥 + 𝑚𝑛
𝑘 = 𝑛[pic 31]
2 1
Rectas paralelas 𝑚1 = 𝑚2
Rectas perpendiculares 𝑚1𝑚2 = – 1
[pic 32]
(𝑎 ± 𝑏)𝑛 = ∑ 𝑛
𝑘[pic 33]
) (−1) 𝑘𝑎𝑛−𝑘𝑏𝑘
𝑘 = 0
CASOS DE FACTORIZACION
𝑎𝑥 + 𝑏𝑥 − 𝑐𝑥 = 𝑥(𝑎 + 𝑏 − 𝑐)
𝑎𝑥 + 𝑏𝑥 − 𝑎𝑦 − 𝑏𝑦 = (𝑥 − 𝑦)(𝑎 + 𝑏)
𝑎2 − 𝑏2 = (𝑎 + 𝑏)(𝑎 − 𝑏)
𝑎3 − 𝑏3 = (𝑎 − 𝑏)(𝑎2 + 𝑎𝑏 + 𝑏2)
𝑎3 + 𝑏3 = (𝑎 + 𝑏)(𝑎2 − 𝑎𝑏 + 𝑏2)
𝑎𝑛 − 𝑏𝑛 = (𝑎 − 𝑏)(𝑎𝑛−1 + 𝑎𝑛−2𝑏 + ⋯ + 𝑎𝑏𝑛−1
+ 𝑏𝑛), 𝑛 𝑝𝑎𝑟 𝑜 𝑖𝑚𝑝𝑎𝑟
𝑎𝑛 + 𝑏𝑛 = (𝑎 + 𝑏)(𝑎𝑛−1 − 𝑎𝑛−2𝑏 + ⋯ − 𝑎𝑏𝑛−1
+ 𝑏𝑛), 𝑛 𝑖𝑚𝑝𝑎𝑟
[pic 34]
[pic 35][pic 36]
PROPIEDADES DE LOS LOGARITMOS
𝐿𝑜𝑔𝑏 1 = 0
𝐿𝑜𝑔𝑏 (𝑚𝑛) = 𝐿𝑜𝑔𝑏 𝑚 + 𝐿𝑜𝑔𝑏 𝑛
𝑚
[pic 37]
𝑥2 ± 2𝑥𝑦 + 𝑦2 = (𝑥 ± 𝑦)2
𝐿𝑜𝑔𝑏 ( 𝑛 ) = 𝐿𝑜𝑔𝑏 𝑚 − 𝐿𝑜𝑔𝑏 𝑛
𝐿𝑜𝑔𝑏 (𝑚𝐾) = 𝐾 𝐿𝑜𝑔𝑏 𝑚
...