prove that √3 + √15 is irrational
Answers
Answer:
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Step-by-step explanation:
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Answer:
To prove :
3
+
5
is irrational.
Let us assume it to be a rational number.
Rational numbers are the ones that can be expressed in
q
p
form where p,q are integers and q isn't equal to zero.
3
+
5
=
q
p
3
=
q
p
−
5
squaring on both sides,
3=
q
2
p
2
−2.
5
(
q
p
)+5
⇒
q
(2
5
p)
=5−3+(
q
2
p
2
)
⇒
q
(2
5
p)
=
q
2
2q
2
−p
2
⇒
5
=
q
2
2q
22 .
2pq
⇒ 5 = 2pq (2q 2−p 2)
As p and q are integers RHS is also rational.
As RHS is rational LHS is also rational i.e
5 is rational.
But this contradicts the fact that
5 is irrational.
This contradiction arose because of our false assumption.
so,
√3+ √5 irrational.