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4m ____________________ 4m ______________ __ 4 __ 2 4 
E = √[(√2 - 1) (√2 + 1)] = √[(√2 ) - 1]
______ __
E = 
4m
√(2 - 1)4 = 
m
√1 
E = 1 valor aritmético
2.- Efectuar:
3 
______ 8 __________ __
√√2 - 1 √3 + 2√ 2 
R = ––––––––––––––––––6 ______ 12 __________ __
√√2 + 1 √5√2 - 7
Solución:
Transformando el segundo factor del numerador:
4 
__________
8 _________ __________ 4 _________ __ __
√3 + 2 √2 = √√3 + 2 √2 = √√2 + 1
sustituyendo:
3 ______ 4 ________ __ __
√√2 - 1 √√ 2 + 1 
R = ––––––––––––––––––6 ______ 12 ________ __ __
√√2 + 1 √5√2 - 7
homogenizando los radicales:
12 ________________ 12 ________________
__ 4 __ 3 __ 4 __ 3
(√2 - 1) (√2 + 1) (√2 - 1)(√2 + 1)
R = –––––––––––––––– = ––––––––––––––––__ 2 __ __ __√(√2 + 1)(5√2 - 7) √(3+2√2 )(5√2 - 7)
Efectuando el producto en el denominador:
12
________________
__ 4 __ 3
(√2 - 1) (√2 + 1)
R = –––––––––––––––––__√ (√2 - 1)
12 ________________ 4 _________________ 3 __ 3 __ __
R = √(√2 - 1) (√2 + 1) = √(√2 - 1)(√2 + 1)
4 ________ 2 ____ __ 
R = √(√2 ) - 1 = 4√2 - 1 = 4√1 
R = 1
3.- Efectuar:
__________________ ______________________ _____
√x3 - 3x+(x2 - 1)√x2 - 4 - √x3 - 3x - (x2 - 1)√x2 - 4
E = –––––––––––––––––––––––––––––––––––––––––––___________ ______________ _____
√x + √x2 - 4 - √x - √x2 - 4
Solución:
Elevando al cuadrado la expresión dada:
__________________ ______________________ _____
2(√x3 - 3x+(x2 - 1)√x2 - 4 -√x3 - 3x - (x2 - 1)√x2 - 4))
E2 = –––––––––––––––––––––––––––––––––––––––––––__________ ______________ _____
2(√x + √x2 - 4 - √x - √x2 - 4 )
efectuando:
____ _______________
x3 - 3x + (x2 - 1)√x2- 4 - 2√[(x3 - 3x) + (x2-1)
E2 = –––––––––––––––––––––––––––––––––––––...________________________________ ____ ____ ____
x +√x2 - 4 - 2√(x +√x2-4)(x -√x2- 4)+x -√x2- 4
______________________________ ____ ____ 
.√x2-4][(x3- 3x) - (x2 - 1)√x2 - 4]+x3 -3x -(x2 - 1)√x2- 4... –––––––––––––––––––––––––––––––––––––––––––––––––––
reduciendo:
______________________
2x3 - 6x - 2 √(x3 - 3x)2 - (x2 - 1)2(x2 - 4) 
E2 = –––––––––––––––––––––––––––––––––––_________
2x - 2 √x2 - (x2 - 4)
___________________________
2x3 - 6x -2 √x6 - 6x4 + 9x2 - (x6 - 6x4 +9x2 - 4)
E2 = ––––––––––––––––––––––––––––––––––––––__
2x - 2√4
__
2x3 - 6x - 2√4 x3 - 3x - 2
E2 = ––––––––––––––– = –––––––––
2x - 4 x - 2
dividiendo por Ruffini:
1 0 -3 -2
↓
2 2 4 +2
1 2 +1 0
el cociente es:
x2 + 2x + 1 = (x + 1)2
Por lo que:
E2 = (x + 1)2
∴ E = x + 1
4.- Hallar el valor de:
3 __________________ 3 _________________________ ________
__ __ __ __
3
4
√a 9√a 3
4
√a 9√a 
–––––– + –––––– - 1 + –––––– - –––––– -1 √ 2 √ 4 √ 2 √ 4
E = ––––––––––––––––––––––––––––––––––––––––
3
_______________________________________
3
_________________ _____________________________ __________ __ __ __
__
3
4
√a 9√a 3
4
√a 9√a√ 4√a +√––––– + √––––– -1+√–––––– - √––––– -1
2 4 2 4 
Á L G E B R A
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Algebra 27/7/05 16:30 Página 229