Which of the following is irrational?A. \(\sqrt\frac{4}{9}\)B. \(\fr

1.	Which of the following is irrational?A. \(\sqrt\frac{4}{9}\)B. \(\frac{4}5\)C. \(\sqrt7\)D. \(\sqrt{81}\)
Answer» 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 7q²= 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.

Which of the following is irrational?A. \(\sqrt\frac{4}{9}\)B. \(\frac{4}5\)C. \(\sqrt7\)D. \(\sqrt{81}\)

Answer»

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

7q²= 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.