Math, asked by vasudha08, 6 months ago

Prove that root 3 is an irrational number hence check whether 2-5 root 3 is rational or irrational.
Pls help me !!!

Answers

Answered by avanichoudharymps
1

Let us consider  

3

​  

 be a rational number, then

3

​  

=p/q, where ‘p’ and ‘q’ are integers, q

=0 and p, q have no common factors (except 1).

So,

3=p  

2

/q  

2

 

p  

2

=3q  

2

 …. (1)

As we know, ‘3’ divides 3q  

2

, so ‘3’ divides p  

2

 as well. Hence, ‘3’ is prime.

So 3 divides p

Now, let p=3k, where ‘k’ is an integer

Square on both sides, we get

p  

2

=9k  

2

 

3q  

2

=9k  

2

 [Since, p  

2

=3q  

2

, from equation (1)]

q  

2

=3k  

2

 

As we know, ‘3’ divides 3k  

2

, so ‘3’ divides q  

2

 as well. But ‘3’ is prime.

So 3 divides q

Thus, p and q have a common factor of 3. This statement contradicts that ‘p’ and ‘q’ has no common factors (except 1).

We can say that  

3

​  

 is not a rational number.

3

​  

 is an irrational number.

Now, let us assume (2/5)  

3

​  

 be a rational number, ‘r’

So, (2/5)  

3

​  

=r

5r/2=  

3

​  

 

We know that, ‘r’ is rational, ‘5r/2’ is rational, so ‘  

3

​  

’ is also rational.

This contradicts the statement that  

3

​  

 is irrational.

So, (2/5)  3  is an irrational number.

Hence proved.

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