Question

given that root 2 is irrational prove that ( 5+3 root2) is an irrational number

Answer 1

Hey.. I think this can be ur answer!!

Given:root2 is irrational
Now,let us assume that 5 +3root2 is rational
So,
5+3root2 = a/b
3root2 = a/b-5 =a-5b/b
Root2 = a-5b/3b

But we know that root2 is irrational
So, our assumption was wrong.
Hence 5+3root2 is irrational.

Hence proved

Hope it helps you dear ☺️☺️

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,

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

⇒ $3 \sqrt{2}= \frac{p}{q} - 5$

⇒ $3 \sqrt{2} = \frac{p - 5q}{q}$

⇒ $\sqrt{2}= \frac{p-5q}{3q}$

We know that,

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

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

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.