Question

if p is a prime no. then lrove that under root p is irrational. ​

Answer 1

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step-by-step explanation:

It is given that,

p is a prime number,

so,

it will satisfy any prime number.

now,

for convenience,

let,

p = 3

Now,

we have to prove that,

√3 is an irrational number.

PROOF:-

suppose root 3 is a rational no.

such that

root 3 = a/b ,

where a and b both r integers and b = nonzero integer

niw,

'a ' and 'b ' have no common factor other than 1

therefore,

root 3 = a/b .......where a nad b r coprime nos.

therefore ,

b root 3 = a

therefore,

3b^2 = a^2 (squaring both sides)..... (1)

therefore ,

b^2 = a^2/3

therefore,

3 divides a^2 ,

so

3 divides a

so

we write,

a = 3c..........where 'c ' is an integer.....(2)

a^2 = (3c)2 .....squaring both sides

therefore,

3b^2 = 9c^2 .........substuting (2) in (1)

therefore,

b^2 = 3 c^2

therefore,

c^2 = b^2 / 3

3 divides b^2 ,

means 3 divides b

therefore

'a ' and 'b ' have at least 3 as common factor

but

it was stated before that it was stated that a and b had no common factors other than 1.

this contradiction arises because we have assumed that root 3 is rational.

therefore,

root 3 is irritional no.

Hence,

p is also an irrational number.

Thus,

Proved✍️✍️