Question

Prove that. 4 - 3√2÷5 is irrational

Answer 1

Let (4-3√2)/5 be a rational number.

A rational number can be written in the form of p/q.

(4-3√2)/5 = p/q

4-3√2 = 5p/q

3√2 = 4-5p/q

3√2 = (4q-5p)/q

√2 = (4q-5p)/3q

p, q are integers then (4q-5p)/3q is a rational number.

Then √2 is also a rational number.

But this contradicts the fact that √2 is an irrational number.

Hence our supposition is false.

Therefore, (4-3√2)/5 is an irrational number.

Hence proved.

Answer 2

remember this property that

when we (add, subtract, multiply, divide) a rational no. with an irrational no. we always get an irrational no. and vice versa.

here 4 is a rational no. and 32^1/2is an irrational no. so we are subtracting a rational with an irrational so this will be an irrational no.

now 4-32^1/2 is an irrational no. when it is divided by a rational no. ie. 5
we will get

4-32^1/2/5 is an irrational no.

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