Question

prove that 7 minus 6 root 5 is an irrational number

Answer 1

Hello friend ..... ☺

Here is your solution...✴✴✴

♦ let ( 7 - 6√5 ) = a/b ( i.e; a form of rational no.)

=> 7 - a/b = 6√5

=> 7b - a / b = 6√5

=> 7b - a / 6b = √5

Here , LHS is in the form of a rational number whereas RHS i.e ; √5 is an irrational number.

Hence, our assumption is wrong.

Therefore, we can say that (7 - 6√5) is an irrational number....

Hence, proved... ☺

Thanks....

Answer 2

↑ Here is your answer ↓
_____________________________________________________________
_____________________________________________________________

Lets assume that it is a rational number,

Then,

$7 - 6 \sqrt 5 = p/q$

{ p & q are co-primes, q ≠ 0 }

$6 \sqrt 5 = 7 - p/q$

$6 \sqrt 5 = (7q - p)/q$

$\sqrt 5 = (7q-p)/6q$

Here,

7, 6, q and p are integers. 

Then, √5 is rational

But this contradicts the fact that √5 is irrational

So, our assumption is wrong

Therefore, 7 - 6√5 is irrational