Question

Show that 5 - root 3 is an irrational.

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Answer 1

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step-by-step explanation:

To prove :-

5-√3 is irrational

Proof:-

Let us assume that 5 -√3 is rational.

now,

Let ,

 5 - √3 = r ,

where "r" is rational

therefore,

5 - r = √3

Here,

LHS is purely rational.

But,on the other hand ,

RHS is irrational.

This leads to a contradiction.

Hence,

5-√3 is irrational

Thus,

proved✍️✍️

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Answer 2

Answer:

$5\;-\;\sqrt3$ is irrational.

Step-by-step explanation:

Given :-

$5\;-\;\sqrt3$

To find :-

To prove that $5\;-\;\sqrt3$ is irrational.

Solution :-

Assume $5\;-\;\sqrt3$ is rational.

So,

$5\;-\;\sqrt3\;=\;\frac{a}{b}$

Transposing,

$5\;-\;\frac{a}{b}\;=\;\sqrt3$

$\sqrt3\;=\;5\;-\;\frac{a}{b}$

$\sqrt3\;=\;\frac{5b-a}{b}$

Since a and b are integers, $5\;-\;\frac{a}{b}$ is rational, and we know $\sqrt3$ is irrational.

We also know that rational number cannot be equal to an irrational number.

So our assumption is incorrect, so we proved that $5\;-\;\sqrt3$ is irrational.