Simplify:  square root of 3 plus 2 square root of 2 end root

Asked by Nandish | 7th May, 2016, 06:35: PM

Expert Answer:

C o n s i d e r space t h e space i r r a t i o n a l space n u m b e r space square root of 3 plus 2 square root of 2 end root
L e t space square root of 3 plus 2 square root of 2 end root equals a square root of 3 plus b square root of 2
W e space n e e d space t o space f i n d space t h e space v a l u e s space a space a n d space b
S q u a r i n g space b o t h space t h e space s i d e s comma space w e space h a v e
open parentheses square root of 3 plus 2 square root of 2 end root close parentheses squared equals open parentheses a square root of 3 plus b square root of 2 close parentheses squared
rightwards double arrow 3 plus 2 square root of 2 space equals 3 a squared space plus space 2 b squared space plus 2 a b square root of 6
E q u a t i n g space t h e space c o r r e s p o n d i n g space c o e f f i c i e n t s comma space w e space h a v e comma
3 a squared space plus space 2 b squared equals 3 space a n d space 2 a b square root of 6 equals 2 square root of 2
2 a b square root of 6 equals 2 square root of 2
rightwards double arrow 2 a b square root of 3 square root of 2 equals 2 square root of 2
rightwards double arrow a b square root of 3 equals 1
rightwards double arrow square root of 3 a equals 1 over b
S q u a r i n g space b o t h space t h e space s i d e s comma space w e space h a v e comma
3 a squared equals 1 over b squared
S u b s t i t u t i n g space t h e space v a l u e space 3 a squared equals 1 over b squared space i n space t h e space e q u a t i o n space 3 a squared plus 2 b squared equals 3 comma space w e space h a v e
1 over b squared space plus space 2 b squared equals 3
rightwards double arrow fraction numerator 1 plus 2 b to the power of 4 over denominator b squared end fraction space equals 3
rightwards double arrow 1 plus 2 b to the power of 4 space equals 3 b squared
rightwards double arrow 2 b to the power of 4 space minus 3 b squared plus 1 equals 0
L e t space t equals b squared
T h u s space t h e space a b o v e space e q u a t i o n space b e c o m e s
2 t squared minus 3 t plus 1 equals 0
S o l v i n g space t h e space a b o v e space q u a d r a t i c comma space w e space h a v e comma
t equals 1 half space o r space t equals 1
C a s e space t equals 1 colon
t equals 1 space a n d space t equals b squared
rightwards double arrow b squared equals 1
rightwards double arrow b equals plus-or-minus 1

A l s o space w e space h a v e
a b square root of 3 equals 1
rightwards double arrow I f space b equals 1 space rightwards double arrow a equals fraction numerator 1 over denominator square root of 3 end fraction
rightwards double arrow square root of 3 plus 2 square root of 2 end root equals a square root of 3 plus b square root of 2
rightwards double arrow square root of 3 plus 2 square root of 2 end root equals fraction numerator square root of 3 over denominator square root of 3 end fraction plus square root of 2
rightwards double arrow square root of 3 plus 2 square root of 2 end root equals 1 plus square root of 2

Answered by Vimala | 8th May, 2016, 05:40: PM