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Which of the following is irrational?

A. \(\sqrt\frac{4}{9}\)

B. \(\frac{4}5\)

C. \(\sqrt7\)

D. \(\sqrt{81}\)

1 Answer

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Proof: let us assume that √7 be rational. 

then it must in the form of p / q [q ≠ 0] [p and q are co-prime] 

√7 = p / q 

√7 x q = p 

squaring on both sides 

7q= p2 (i) 

p 2 is divisible by 7 

p is divisible by 7 

p = 7c [c is a positive integer] [squaring on both sides ] 

p 2 = 49 c2 (ii) 

Substitute p2 in eq (i), we get, 

7q2 = 49 c2 

q 2 = 7c2 

q is divisible by 7 

Thus q and p have a common factor 7. 

There is a contradiction 

As our assumption p & q are co - prime but it has a common factor. 

So that √7 is an irrational.

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