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Show that \frac{2}{\sqrt{7}} is irrational.
CBSE | Class 10 | Exercise 1D | Maths | Real Numbers | RS Aggarwal
Show that \frac{2}{\sqrt{7}} is irrational.

Answer:

\begin{array}{l} \frac{2}{{\sqrt 7 }} = \frac{2}{{\sqrt 7 }} \times \frac{{\sqrt 7 }}{{\sqrt 7 }}\\ = > \frac{2}{7}\sqrt 7 \end{array} is a rational number.

p and q are some integers and HCF(p,q) = 1

2√7q = 7p

(2√7q)2 = (7p) 2

7(4q2) = 49p2

4q2 = 7p2

q2 is divisible by 7

q is divisible by 7

q = 7m, where m is some integer.

∴2√7q = 7p

[2√7 (7m)] 2 = (7p)2

343(4m2) = 49p2

7(4m2) = p2

p2 is divisible by 7

p is divisible by 7

The assumption is wrong.

Hence, 2/√7 is irrational.

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