Question

Proof that 5 minus 3 by 7 root 3 is irrational rational number

Answer 1

5-3/7root is rational number

do p and q are co primes

such that 5-3/7root3=p/q

5-3=p/q 7root3

2p/7q=root3

where 2p/7q is a rational and root 3 is aorrational number

therefore 5-3/7rot 3 is a irrational number

Answer 2

$5-\frac{3}{7\sqrt{3}}$ is an irrational number.

Step-by-step explanation:

To proof : $5-\frac{3}{7\sqrt{3}}$ is a irrational number ?

Proof :

Let us assume that,

$5-\frac{3}{7\sqrt{3}}$ is a rational number.

Then it can be written in p/q form where p and q are co-prime.

$\frac{3}{7\sqrt{3}}=5-\frac{p}{q}$

$\frac{3}{7\sqrt{3}}=\frac{5q-p}{q}$

$\frac{3q}{5q-p}=7\sqrt{3}$

$\frac{3q}{7(5q-p)}=\sqrt{3}$

Here, 7,5,3,q,and p are integers.

i.e. LHS is rational number.

We know, $\sqrt{3}$ is an irrational number.

A rational number cannot be equal to irrational number.

This contradict the assumption.

Therefore, $5-\frac{3}{7\sqrt{3}}$ is an irrational number.

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