Question

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

Answer 1

Hey mate !!

Here's your answer !!

Fact: √2 is irrational.

To prove: 5 + 3√2 is irrational.

Assumption: Let as assume 5 + 3√2 as a rational number.

=> 5 + 3√2 = P / Q  ( Where P and Q are co-prime and Q ≠ 0 )

= 3√2 = P / Q - 5

= 3√2 = P - 5Q  / Q

= √2 = P - 5Q / 3Q 

Since P - 5Q / 3Q  is a rational number, it makes √2 also to be rational.

But it contradicts the fact that √2 is irrational. 

Hence our assumption was wrong. 

=> 5 + 3√2 is irrational

Hence proved.

Hope it helps !!

Cheers !!

Answer 2

Hey friend, Harish here.

Here is  your answer:

Given that,

√2 is irrational

To prove:

5 + 3√2 is irrational

Assumption:

Let us assume 5 + 3√2 is rational.

Proof:

As 5 + 3√2 is rational. (Assumed) They must be in the form of p/q where q≠0, and  p & q are co prime.

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,

$\sqrt{2}\ is\ irrational$

And,

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

And, Rational ≠ Irrational.

Therefore we contradict the statement that, 5+3√2 is rational.

Hence proved that 5 + 3√2 is irrational.
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Hope my answer is helpful to you.