Question

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

Answer 1

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Answer 2

Answer:

Given : $\sqrt{2}$ is irrational number.

To prove : $(5+3\sqrt{2})$ is an irrational number.

Assumption:

Let us assume $(5+3\sqrt{2})$ is a rational number.

Proof:

As $(5+3\sqrt{2})$ is rational.

They must be in the form of p/q where $q\neq 0$, and  p and 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,

$\sqrt{2}$ is irrational number

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

And $\text{Rational}\neq \text{Irrational}$

Therefore, our assumption is wrong.

$(5+3\sqrt{2})$ is an irrational number.