Question

given that root two is irrational prove that (5+3root two)is an irrational

Answer 1

Hey friend, Harish here.

Here is your answer:

Given that,

√2 is an irrational number.

To  prove,

5 + 3√2 is an irrational number.

Assumption:

Let 5 + 3√2 be a rational  number.

Proof:

As 5 + 3√2 is assumed to be rational , then it must be of the form p/q, Where q≠0.

Then,

$5+3 \sqrt{2} = \frac{p}{q}$

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

⇒ $\sqrt{2}= \frac{p-5q}{3q}$

We know that,

$\sqrt{2}\ is \ irrational\ (Given)$

$\frac{p-5q}{3q} \ is \ rational$

As rational ≠ irrational.

We contradict the statement that 5+3√2 is rational.

Therefore it is irrational.
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Hope my answer is helpful to you.

Answer 2

heya 
let us assume to the contrary that 5+3√2 is rational .
then 5+3√2 = a/b where a and b are two coprime numbers
then rearranging , we get
√2 = (a+5b)/3b
then we know that a , b , 3, 5 are rational numbers so (a+5b)/3b is also a rational number which makes √2 a rational number but this contradicts the fact that √2 is irrational.
this contradiction gas arisen due to our wrong assumption that 5+3√2 is rational

thus 5+3√2 is irrational