Question

Given root5 is irrational .prove that 3 - 5 root2 is also irrational.

Answer 1

Given:- Root2 is an irrational number.

To prove = 3-5root2 as irrational number.

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Let $3-5\sqrt{2}$ be a rational number.

this means hat it can be expressed in he form of p/q where p and q are integers and q is not equal to zero.

also,

p and q are co-prime numbers having no common factor other than one(1)

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$3-5\sqrt{2} =\dfrac{p}{q}\\ \\ \\5\sqrt{2}=3- \dfrac{p}{q}\\ \\ \\ 5\sqrt{2}=\dfrac{3q-p}{q}\\ \\ \\ \sqrt{2}=\dfrac{3q-p}{5q}$

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We know that root two is irrational as it is given in the question.

we assumed a as well as b as an integer .Hence,a nd b are rational numbers.

this means that the RHS of the above equation is a rational number while the LHS of the above equation is an irrational number.

both of them can never be equal to each other.

hence,

It is a contradiction which has arose because we took 3-5 root 2 as a rational number.

This implies that 3-5root2 is an irrational number.

hence proved!

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