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Question

Examine, whether the following numbers are rational or irrational:

(i) 7 (ii) 4

(iii) 2+3 (iv) 3+2

(v) 3+5 (vi) (22)2

(vii) (22)(2+2) (viii) 2+3)2

(ix) 52 (x) 23

(xi) 225 (xii) 0.3796

(xiii) 7.478478..... (xiv) 1.101001000100001.....

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Solution

(i) 7

It is an irrational as 7 is not a perfect square.

(ii) 4

It is a rational number as 4 is a perfect square of 2.

(iii) 2+3

It is an irrational number as sum of a rational number and an irrational number is also an irrational number.

(iv) 3+2

Irrational as sum of two irrational numbers is also an irrational number.

(v) 3+5

is an irrational number as sum of two irrational numbers is also an irrational.

(vi) (22)2 = 2 + 4 + 2 2 × 2 = 6 + 4 2

{(ab)2=a2+b22ab)}

it is an irrational number as sum of a rational and an irrational number is an irrational number.

(vii) (22)(2+2) = (2)2(2)2
(a+b)(ab)=a2b2=42=2
Which a rational number

(viii) 2+3)2=2+3+223

{ (a+b)2=a2+b2+2ab}

= 5 + 2 6

Which is an irrational number as sum of a rational and an irrational number is an irrational number.

(ix) 52 is an irrational number as difference of an irrational number and a rational number is also an irrational number.

(x) 23 is an irrational number as 23 is not a perfect square.

(xi) 225 = 15

(xii) 0.3796 is a rational number its decimal is terminating.

(xiii) 7.478478..... = 7. ¯¯¯¯¯¯¯¯478

Which is non-terminating recurring decimal. Therefore it is a rational number.

(xiv) 1.101001000100001.....

It is an irrational number as its decimal is non-terminating non-recurring decimal.


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