hi plsease help me i cant understand the rationalising the denominators

Asked by preethi singaram | 26th Apr, 2015, 07:09: PM

Expert Answer:

L e t space u s space c o n s i d e r space o f space e v a l u a t i n g space t h e space e x p r e s s i o n comma space fraction numerator 15 over denominator square root of 10 plus square root of 20 plus square root of 40 minus square root of 5 minus square root of 80 end fraction L e t space u s space o b s e r v e space t h e space d e n o m i n a t o r. T h u s comma space d e n o m i n a t o r space i s space h a v i n g space s q u a r e space r o o t s space o f space n o n minus p e r f e c t space s q u a r e space n u m b e r s. A d d i t i o n comma space s u b t r a c t i o n comma space m u l t i p l i c a t i o n space a n d space d i v i s i o n space o f space s u c h space e x p r e s s i o n s space i s space e a s y space i f space t h e i r space d e n o m i n a t o r s a r e space f r e e space f r o m space s q u a r e space r o o t s. space T o space m a k e space t h e space d e n o m i n a t o r s space f r e e space f r o m space s q u a r e space r o o t s comma space w e space m u l t i p l y space t h e space n u m e r a t o r space a n d space d e n o m i n a t o r space b y space a s u i t a b l e space i r r a t i o n a l space n u m b e r comma N o w space c o n s i d e r space t h e space d e n o m i n a t o r space o f space t h e space g i v e n space e x p r e s s i o n. T h u s comma square root of 10 plus square root of 20 plus square root of 40 minus square root of 5 minus square root of 80 equals square root of 10 plus square root of 4 cross times 5 end root plus square root of 4 cross times 10 end root minus square root of 5 minus square root of 16 cross times 5 end root equals square root of 10 plus square root of 2 squared cross times 5 end root plus square root of 2 squared cross times 10 end root minus square root of 5 minus square root of 2 to the power of 4 cross times 5 end root equals square root of 10 plus 2 square root of 5 plus 2 square root of 10 minus square root of 5 minus 4 square root of 5 equals square root of 10 plus 2 square root of 10 minus square root of 5 minus 4 square root of 5 plus 2 square root of 5 equals open parentheses 1 plus 2 close parentheses square root of 10 plus open parentheses 2 minus 1 minus 4 close parentheses square root of 5 equals 3 square root of 10 minus 3 square root of 5 equals 3 open parentheses square root of 10 minus square root of 5 close parentheses H e n c e space t h e space g i v e n space e x p r e s s i o n space b e c o m e s fraction numerator 15 over denominator square root of 10 plus square root of 20 plus square root of 40 minus square root of 5 minus square root of 80 end fraction equals fraction numerator 15 over denominator 3 open parentheses square root of 10 minus square root of 5 close parentheses end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 5 over denominator open parentheses square root of 10 minus square root of 5 close parentheses end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 5 over denominator open parentheses square root of 10 minus square root of 5 close parentheses end fraction cross times fraction numerator open parentheses square root of 10 plus square root of 5 close parentheses over denominator open parentheses square root of 10 plus square root of 5 close parentheses end fraction space space space space open square brackets sin c e space r a t i o n a l i sin g space f a c t o r space o f space square root of 10 minus square root of 5 space i s space square root of 10 plus square root of 5 close square brackets space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 5 open parentheses square root of 10 plus square root of 5 close parentheses over denominator open parentheses square root of 10 minus square root of 5 close parentheses open parentheses square root of 10 plus square root of 5 close parentheses end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 5 open parentheses square root of 10 plus square root of 5 close parentheses over denominator open parentheses square root of 10 close parentheses squared minus open parentheses square root of 5 close parentheses squared end fraction space space space space space space open square brackets because open parentheses a plus b close parentheses open parentheses a minus b close parentheses equals a squared minus b squared close square brackets space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 5 open parentheses square root of 10 plus square root of 5 close parentheses over denominator 10 minus 5 end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 5 open parentheses square root of 10 plus square root of 5 close parentheses over denominator 5 end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals square root of 10 plus square root of 5

Answered by Vimala Ramamurthy | 27th Apr, 2015, 09:12: AM

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