Question

prove that 5+3√2is an irrational number
​

Answer 1

step by step explanation

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}5+3

2

=

q

p

⇒ 3 \sqrt{2}= \frac{p}{q} - 53

2

=

q

p

−5

⇒ 3 \sqrt{2} = \frac{p - 5q}{q}3

2

=

q

p−5q

⇒ \sqrt{2}= \frac{p-5q}{3q}

2

=

3q

p−5q

We know that,

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

2

is irrational (given)

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

3q

p−5q

is rational

And, Rational ≠ Irrational.

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

Hence proved that 5 + 3√2 is irrational

Answer 2

Answer:

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