Prove that 6+ 2 is irrational. - Vidyadip

Question

Prove that $6+\sqrt { 2 }$ is irrational.

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Answer

Let us assume that 6 + √2 is a rational number.
So we can write this number as
6 + √2 = a/b
Here a and b are two co-prime numbers and b is not equal to 0
Subtract 6 both side we get
√2 = a/b – 6
√2 = (a-6b)/b
Here a and b are integers so (a-6b)/b is a rational number. So √2 should be a rational number. But √2 is an irrational number. It is a contradiction.
Hence result is 6 + √2 is a irrational number

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