SOLUTION: simplify by rationalising the denominator [(sqrt10)+(2sqrt5)]/[(sqrt10)+(sqrt5)]

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Question 407910: simplify by rationalising the denominator
[(sqrt10)+(2sqrt5)]/[(sqrt10)+(sqrt5)]
Found 2 solutions by Edwin McCravy, graphmatics:
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
%28%28sqrt%2810%29%2B2sqrt%285%29%29%29%2F%28%28sqrt%2810%29%2Bsqrt%285%29%29%29


The conjugate of the denominator is red%28%28sqrt%2810%29-sqrt%285%29%29%29.
Put that over itself like this %28red%28%28sqrt%2810%29-sqrt%285%29%29%29%29%2Fred%28%28sqrt%2810%29-sqrt%285%29%29%29.

Then multiply:



Multiply the tops and the bottoms:



Simplify:





%0D%0A%2810%2Bsqrt%2850%29-2%2A5%29%2F%0D%0A%2810-5%29+

%0D%0A%2810%2Bsqrt%2850%29-10%29%2F%0D%0A5+

%0D%0A%28cross%2810%29%2Bsqrt%2850%29-cross%2810%29%29%2F%0D%0A5+

%0D%0A%28sqrt%2850%29%29%2F%0D%0A5+

%0D%0A%28sqrt%2825%2A2%29%29%2F%0D%0A5+

%0D%0A%28sqrt%2825%29sqrt%282%29%29%2F5+

%0D%0A%285sqrt%282%29%29%2F5+

%0D%0A%28cross%285%29sqrt%282%29%29%2Fcross%285%29+

sqrt%282%29

Edwin

Answer by graphmatics(170) About Me  (Show Source):
You can put this solution on YOUR website!
We are given the expression %28sqrt%2810%29%2B2%2Asqrt%285%29%29%2F%28sqrt%2810%29%2Bsqrt%285%29%29+. In general when there are radicals %28sqrt%28a%29%2Bsqrt%28c%29%29%2F%28sqrt%28a%29%2Bsqrt%28b%29%29+ in the denominator of a fraction
we multiply the fraction by %28sqrt%28a%29-sqrt%28b%29%29%2F%28sqrt%28a%29-sqrt%28b%29%29+
(example: 1%2F%28sqrt%28a%29%2Bsqrt%28b%29%29+ =

So for the given expression we have that

%28sqrt%2810%29%2B2%2Asqrt%285%29%29%2F%28sqrt%2810%29%2Bsqrt%285%29%29+


%28%28sqrt%2810%29%2B2%2Asqrt%285%29%29%2A%28sqrt%2810%29-sqrt%285%29%29%29%2F%2810-5%29%29+
%28%28sqrt%2810%29%2B2%2Asqrt%285%29%29%2A%28sqrt%2810%29-sqrt%285%29%29%29%2F%285%29%29+
We will simplify the numerator

%28sqrt%2810%29%2Asqrt%285%29%29%2F%285%29%29+
%28sqrt%282%2A5%29%2Asqrt%285%29%29%2F%285%29%29+
%28sqrt%282%29%2A5%29%2F%285%29%29+
sqrt%282%29+