Question

Which of the following is irrational? *

1 point

√81

√(12/3)

√7

√(9/4)

​

Answer 1

$\huge\underbrace\mathfrak\color{pink}7 \: Is \: Irrational$

$\huge\underbrace\mathfrak\color{pink}prove$

Let we assume that 7 is rational. Then, there exist co-prime positive integers a and b such that

$\sqrt{7} = \frac{a}{b}$

$\implies{a=b \sqrt{7} }$

Squaring on both sides, we get

A² = 7B²

Therefore, A² is divisible by 7 and hence, a is also divisible by7

so, we can write a=7p, for some integer p.

Substituting for a, we get 49p²=7b²⟹b² = 7p²

This means, b² is also divisible by 7 and so, b is also divisible by 7.

Therefore, a and b have at least one common factor, i.e., 7.

But, this contradicts the fact that a and b are co-prime.

Thus, our supposition is wrong.

Hence, √ 7 is irrational.

HØPÈ ÏT HÊLPS ♠♦♠