# prove that root 5-3 root 2 is an irrational number

to prove that 5-3√2  is an irrational number:
Assume that 5-3√2 is a rational number
i.e. 5-3 √2 = p/q (p and q are integers, q ≠0 , p and q are co-primes)
Now take all the rational no. from R.H.S and L.H.S to 1 side
5/1-p/q = 3√2
5q-p/q = 3√2 (i.e 3*√2)
5q/3 – p/3q = √2
Since p and q are integers (rational no.) ,5q/3 – p/3q  should also be a rational no.
But √2 is irrational
Which contradicts our assumption is wrong
Therefore 5-3√2  is an irrational no.

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to prove that 5-3√2  is an irrational number:
Assume that 5-3√2 is a rational number
i.e. 5-3 √2 = p/q (p and q are integers, q ≠0 , p and q are co-primes)
Now take all the rational no. from R.H.S and L.H.S to 1 side
5/1-p/q = 3√2
5q-p/q = 3√2 (i.e 3*√2)
5q – p/3q = √2
Since p and q are integers (rational no.) ,5q – p/3q  should also be a rational no.
But √2 is irrational
Which contradicts our assumption is wrong
Therefore 5-3√2  is an irrational no.
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