Question

Prove that 5+2√3 be irrational

Answer 1

Step-by-step explanation:

Let 5 + 2√3 be rational.

Then,

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

where p and q are integers and HCF(p,q) = 1.

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

$\frac{p}{q} - 5 \: is \: rational \: \\ because \: difference \: of \: 2 \: rational \: numbers \: is \: rational$

$2 \sqrt{3} \: is \: irrational$

A rational number cannot be equal to an irrational number.

Therefore our assumption is wrong.

Hence 5 + 2√3 is irrational.

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Answer 2

GIVEN:

• 5 + 2√3

TO FIND:

• Prove that 5 + 2√3 is an irrational number.

PROOF:

Let 5 + 2√3 be a rational number, it can be written in the form of p/q (q ≠ 0), where p and q are coprimes

$\rm{\dashrightarrow 5 + 2 \sqrt{3} = \dfrac{p}{q} }$

$\rm{\dashrightarrow 2\sqrt{3} = \dfrac{p}{q} -5}$

$\rm{\dashrightarrow \sqrt{3} = \dfrac{p- 5q}{2q}}$

Since, p and q are integers so p–5q/2q is rational, and so √3 is rational.

But this contradicts this fact that √3 is Irrational.

So, we conclude that 5 + 2√3 is an irrational number.

Hence 5 + 2√3 is an irrational number.

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