prove that under root 3 + under root 5 is irrational
Prove that root 3 plus root 5 is irrational
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Solution: Let √3+√5 be any rational number x
x=√3+√5
squaring both sides
x²=(√3+√5)²
x²=3+5+2√15
x²=8+2√15
x²-8=2√15
(x²-8)/2=√15
As x is a rational number
so x² is also a rational number, 8 and 2 are rational numbers. ,
so √15 must also be a rational number as quotient of two rational numbers is rational
But this contradicts the fact that √5 is irrational
This contradiction arose because of our false assumption
so √3+√5 is not a rational number.
So √3+√5 is irrational.
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