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Byju's Answer
Standard X
Mathematics
Irrational Numbers
Prove that ...
Question
Prove that
3
+
2
√
5
is irrational.
Open in App
Solution
Let us assume
3
+
2
√
5
+ is rational.
So we can write this number as
3
+
2
√
5
=
a
b
---- (1)
Here a and b are two co-prime number and b is not equal to zero.
Simplify the equation (1) subtract 3 both sides, we get
2
√
5
=
a
b
−
3
2
√
5
=
a
−
3
b
b
Now divide by 2 we get
√
5
=
a
−
3
b
2
b
Here a and b are integer so
a
−
3
b
2
b
is a rational number, so
√
5
should be a rational number.
But
√
5
is a irrational number, so it is contradict.
Therefore,
3
+
2
√
5
is irrational number.
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Similar questions
Q.
Given that
√
2
is irrational, prove that
(
5
+
3
√
2
)
is an irrational number.
Q.
Prove that
3
+
2
√
5
is an irrational.
Q.
Prove that
√
2
is irrational and hence prove that
5
−
3
√
2
7
is irrational.
Q.
Prove that
√
2
is on irrational number and also prove that
3
+
5
√
2
is irrational number.
Q.
Question 2
Prove that
3
+
2
√
5
is irrational.
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