Question

If m and n are prime positive integers,prove that (root m+ root n)
is an irrational

Answer 1

Answer:

√m + √n is irrational number, proved below

Step-by-step explanation:

To prove that √m + √n is irrational given that m and n are prime positive integers

We will prove by contradiction

let √m + √n is rational number

so √m + √n = a/b (a nd b are integers ans b ≠ 0)

(since rational number is a number which could be written in format a/b)

√m = a/b - √n

now square both sides

(√m)² = (a/b - √n)²

m = (a/b)² + n - 2(a/b)√n

2(a/b)√n =  (a/b)² + n - m

take b² as LCM

2(a/b)√n = (a² + nb² - mb²) / b²

cross multiply

√n = (a² + nb² - mb²) / 2ab

irrational = rational

    FALSE          

It's not possible

now as we know

• that n is prime number and any prime number in square root is irrational   number
• a number could be either a rational or irrational

So our assumption is wrong. Hence √m + √n is irrational number.