prove that root3 +root4 is a irrational number
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let √3+√4 be a rational number such that it can be written in the form of a/b where a and b are positive integers and co primes as well
therefore, √3+√4=a/b
squaring both sides
(√3+√4)²=(a/b)²
3+4+2√12=a²/b². (using identity (a+b)²=a²+b²
+2ab)
7+2√12=a²/b²
2√12=a²-7b²/b². (taking LCM here)
√12=a²-7b²/2b²
here according to the condition √12 is rational as it is eaqual to something which is in the form of a/b
but as we know that √12 is irrational
so, this contradiction is arisen bcoz of our wrong assumption that √3+√4 is rational
so.. we conclude that √3+√4 is irrational
therefore, √3+√4=a/b
squaring both sides
(√3+√4)²=(a/b)²
3+4+2√12=a²/b². (using identity (a+b)²=a²+b²
+2ab)
7+2√12=a²/b²
2√12=a²-7b²/b². (taking LCM here)
√12=a²-7b²/2b²
here according to the condition √12 is rational as it is eaqual to something which is in the form of a/b
but as we know that √12 is irrational
so, this contradiction is arisen bcoz of our wrong assumption that √3+√4 is rational
so.. we conclude that √3+√4 is irrational
kashish192:
thanks simuna for marking me as a brainleast
mention not
but if we take root3+root4 =x
and both sides square
can we do this
so x can be written in the form of a/b that is x/1 so.. √3+√4 =x , √3+2=x √3=x/1-2 here √3 become rational which is not so... we conclude that √3+√4 is irrational
here we don't meed to square
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