Question

Given+that+√2+is+irrational+,+prove+that+(+5+3√2)+is+an+irrational+number

Answer 1

Let us assume that 5+3√2 is a rational number i .e

5+2√3 =p/q where p and q are co-prime numbers and q is not equal to zero

5+2√3=p/q

> 2√3=p/q-5

>√3=p-5q/2q

So,p-5q/2q is a rational no.

Therefore,√3 is a rational no.

This contradicts the fact that √3 is an irrational no.

Since a rational number can never be equal to an irrational number

Therefore,our assumption is wrong and 5+2√3 is a rational no.

Hence Proved

Answer 2

Let us assume that 5+3√2 is a rational number.

5+3√2 =a/b where a and b are co-prime numbers and b is not=0.

5+3√2=a/b

3√2=a/b-5

√2=a/3b-5

So,a/3b-5 is a rational number.

Therefore,√2 is a rational number.

But,we know that √2 is an irrational number.

Therefore,our assumption is wrong and 5+3√2 is a irrational number.

Hence Proved

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