Prove that 4+3root2 is rational
Answers
Answer:
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.
Answered By
toppr
134 Views
How satisfied are you with the answer?
This will help us to improve better
answr
Get Instant Solutions, 24x7
No Signup required
girl
More Questions by difficulty
EASY
MEDIUM
HARD
Prev Question
Next Question