Squarring on both sides, we get

2=a2/b2+3-2 a/b√3

Rearranging

2a/b√3=a2 b2+3-2

=a2/b2+1

√3=a2+b2/2ab

Since a, b are integers,a2+b2/2ab is rational,and so √3 is rational.

This contradicts the fact that √3 is rational.Hence √2+√3 is irrational.

note:

1. The sum of two irrational numbers need not be irrational

for example,if a=√2 and b=√2 then both a nd b are irrational but a + b=0 which is rational.

2. The product of two irrational numbers need not be irrational.

For example, a =√ 2 and b =3√ 2 , where both a and b are irrational, but ab = 6 which is rational.

Exercize-1.4

1. prove that the following are irrational.

(i) 1/√2      (ii) √3+√5    (iii)6+√2     (iv)√5     (v)3+2√5  

2. prove that √p+√q is an irrational, where p, q are primes.

1.4 EXPONENTIALS REVISTED

we know the power of a number 'a' with natural exponent 'n' is the product of 'n' factors each of which is equal to 'a'

20,21,22,23, ..............are powers of 2 30,31,32,33. ..............are powers of 3.


page no:17

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