Question

Prove that root 5 - root 3 is an irrational number

Answer 1

here is your answer .

Hi!!!

Answer 2

We can prove this by "Proof by contradiction", and we can find a contradiction through arithmetic basis.


`\text{Assume } √(5) - √(3) \text{ is rational.}`

`\text{That is, } √(5) - √(3) = (a)/(b)\text{, where a and b do not have any common factors.}`


`\text{Squaring both sides: } (√(5) - √(3))^(2) = (a^(2))/(b^(2))`

`5 - 2√(15) + 3 = (a^(2))/(b^(2))`

`8 - 2√(15) = (a^(2))/(b^(2))`

`2(4 - √(15)) = (a^(2))/(b^(2))`

We can prove by contradiction based on the fact that the square root of 15 is irrational. We've made our assumption that we can write √5 - √3 in fraction form. By making √15 the subject, we would have contradicted our original assumption.


`4 - √(15) = (a^(2))/(2b^(2))`

`√(15) = 4 - (a^(2))/(2b^(2))`

Now we've hit our jackpot. Since we can write √15 in a rational form, we've contradicted ourselves. This implies that our original assumption was wrong, which was that we can write √5 - √3 in fraction form. This further implicates that √5 - √3 cannot be rewritten in simplified fraction form, which means that √5 - √3 is irrational.

Thus, our proof is complete.