Question

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

Answer 1

Answer:

Step-by-step explanation:

Answer:

7-√5 is irrational. The proof is given below

Step-by-step explanation:

Let us assume, to the contrary, that 7-√5 is rational

That is, we can find Coprime a  and b (b≠ 0) such that  

7-√5 = a/b

Therefore, 7 - a/b = √5

Rearranging this equation √5 = (7b -a)/b

since a and b are integers,so (7b -a)/b is an rational.

And so √5 is rational

But this contradicts the fact that √5 is irrational.

This  contradiction  has  arisen  because  of  our  incorrect  assumption  that  7-√5 is rational.

Therefore  we can conclude that

7-√5 is irrational.

2nd Method

We have the sum or difference of one rational and one irrarional numbber is also an irrational number.

Here 7-√5  have 7 is rational number and √5 is irrational number

Therefore 7-√5  is also irrational number.

Answer 2

Answer:

Step-by-step explanation:

Let , 7-root5 is a rational no. is equals to r, where r is a rational no.

So,

7-root 5 = r

7-r = root 5.

Now,

LHS is a rational but RHS is an irrational.

So, our assumption was wrong; 7-root5 is not a rational no.

Therefore, 7-root5 is a irrational no.

Hence, proved.