Question

prove that root 3 is an irrational number . hence show that 5-7root3 is also an irrational number​

Answer 1

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ANSWER

Solution:

If possible , let 3 be a rational number and its simplest form be 

ba then, a and b are integers having no common factor 

other than 1 and b=0.

Now, 3=ba⟹3=b2a2    (On squaring both sides )

or, 3b2=a2         .......(i)

⟹3 divides a2   (∵3 divides 3b2)

⟹3 divides a

Let a=3c for some integer c

Putting a=3c in (i), we get

or, 3b2=9c2⟹b2=3c2

⟹3 divides b2   (∵3 divides 3c2)

⟹3 divides a

Thus 3 is a common factor of a and b

This contradicts the fact that a and b have no common factor other than 1.

The contradiction arises by assuming 3 is a rational.

Hence, 3 is irrational.

2nd part

If possible, Let (7+2