Question

prove that root 5 is an irrational number hence show that 7 - 3 root 5 is also an irrational number ​

Answer 1

Hey mate here is your answer

=> √5 is an irrational number.

Let us assume that √5 is rational number

so we can it can be written in the form a/b

where a and b are two co - primes number.

= √5 =a /b

=> Squaring both sides

= √5 = a²

b²

= 5b²=a²

Here we can see a is divisible by 5

b=5 c where c is an integer

Squaring both sides

b² = 5c²

b²=25c²

b²= 5c²

b is divisible by 5

We can see here a and b both have common factor 5 .

=> it contradicts are assumption a and b are co - primes

=> √5 is an irrational number..

Now,

2) Show that ( 7-3√5) is an irrational number

=> let us assume that 7-3√5 is an rational number

So it can be written in the form a /b

Where a and b are Two co - primes number

= 7-3√5 =a /b

= 7-ab =3√5

= 7b -a/3=√5

Here RHS in the form p/q where √5 is an irrational number..

Therefore, we can say that (7-3√5) is an irrational number ..

Hence Proved..

Hope it will help you ✌️