Question

prove that 4+3 root 2 is irrational if root 2 is irrational​

Answer 1

Answer:

,

If possible, let 4−3

2

be rational.

Then,

4−3

2

is rational and 4 is rational.

[(4−3

2

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

−3

2

is rational.

2

is rational.

Let the simplest form of

2

be

b

a

.

Then, a and b are integers having no common factor other than 1, and b



= 0.

Now,

2

=

b

a

2b

2

=a

2

2 divides a

2

. [2 divides 2b

2

]

2 divides a

Let a=2c for some integer c.

Therefore,

2b

2

=4c

2

b

2

=2c

2

2 divides b

2

[2 divides 2c

2

]

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, 4−3

2

is irrational

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