Question

prove that 4_3root2 is an irrational number

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Answer 1

bro it is in the 1st chapter bro

Answer 2

If possible, let 4−32​ be rational. 

Then,

4−32​ is rational and 4 is rational.

[(4−32​)−4] is rational.         [Difference of two rationals is rational]

−32​ is rational.

2​ is rational.

Let the simplest form of 2​ be ba​.

Then, a and b are integers having no common factor other than 1, and b = 0.

Now, 2​=ba​

2b2=a2

2 divides a2.       [2 divides 2b2]

2 divides a

Let a=2c for some integer c.

Therefore,

2b2=4c2

b2=2c2

2 divides b2            [2 divides 2c2]

2 divides b

Thus, 2 is a common factor of a and b.

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

So, 2​ is irrational.

Hence, 4−32​ is irrational.