Question

Prove that root 7 minus root 5is an irrational number?

Answer 1

Step-by-step explanation:

let √ 7 - √ 5 is a rational number then it can be represented as a/b

so

√7- √5 = a/b

√ 7 = √ 5 + a /b

squaring both sides

7 = 5 + ( a/b ) ^ 2 + 2 * √ 5 * a / b

taking LCM

7 b ^ 2 = a^ 2 + 5 b ^2 + 2 √ 5 ab

2 √ 5 ab = 2b ^2 - a^ 2

√ 5 = (2 b ^ 2 - a^2 )/ 2ab

we know that √ 5 is an irrational number where as ( 2b ^2 - a^2) / 2ab will be rational

so our assumption is wrong and √ 7 - √ 5 is an irrational number