prove that√3+√2 irrational number
Answers
Answer:
Let √3 - √2 = (a/b) is a rational no. So,5 - 2√6 = (a2/b2) a rational no. Since, 2√6 is an irrational no.
Step-by-step explanation:
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Answer:
Let as assume that √2 + √3 is a rational number
Then , there exists co - prime positive integers p and q such that
⇒
3
+
2
=
q
p
, where p,q∈z,q
=0
⇒
3
=
q
p
−
2
By squaring on both sodes, (
3
)
2
=(
q
p
−
2
)
2
3=
q
2
p
2
−2.
2
.
q
p
+2
2
2
.
q
p
=
q
2
p
2
+2−3
⇒2
2
.
q
p
=
q
2
p
2
−1
2(
2
)
q
p
=
q
2
p
2
−q
2
2
=(
q
2
p
2
−q
2
)(
2p
q
)
2
=
2pq
p
2
−q
2
⇒
2
is a rational number ∵
2pq
p
2
−q
2
is rational.
But
2
is not a rational number. This leads us to a contradiction.
∴ our assumption that
3
+
2
, is a ab be rational number is wrong
⇒
3
+
2
is an irrational number
Step-by-step explanation: