Question

Show that 7 − √5 is irrational, give that √5 is irrational​

Answer 1

hi mate

here is your answer

let

$7 + \sqrt{5}$

be any rational number.

such that it can be represented in the

p/q form.

Now,

$7 + \sqrt{5}$

=p/q

i.e

7+ root5 =a/b (where a and b are co primes)

i.e root5 = a/b - 7

i.e root5 = (a-7b)/b

here (a-7b)/b is rational as(a)and (b) are rational numbers.So root 5 is also rational number.

BUT THIS IS CONTRADICTION THE FACT THAT ROOT 5 IS IRRATIONAL.

HENCE OUR ASSUMPTION IS WRONG AND

$7 + \sqrt{5}$

is irrational number.

hope it helps

BE BRAINLY///

Answer 2

Answer:

7 - √5 = a/b                          ( where a and b are coprime and b ≠ 0)

√5 = a-7b/b

This contradicts the fact that a-7b/b is coprime

∴ a-7b/b is rational

But √5 is irrational

∴7-√5 is irrational