Question

Prove that
$5 + 3 \sqrt{2}$
if irrational​

Answer 1

Answer:

If possible let 5+3√2 be rational. Then,

( 5+3√2 is rational, 5 is rational

{(5+3√2)-5} is rational ( since, difference of two rational is rational)

3√2 is rational

(1/3*3√2) is rational ( since, products of two rational is rational)

√2 is rational

This contradicts the fact that √2 is irrational

Since the contradiction arrises by assuming that 5+3√2 it rational .

Hence,5+3√2 is irrational

Step-by-step explanation:

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Answer 2

Step-by-step explaination

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+3root2=p/q

=3root2=p/q-5

=3root2=p-5q/q

=root2=p-5q/3q

We know that

root2 is irrational [given]

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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