show that √3 + √5 rational √p √q irrational where √2 is irrational
Answers
Answer:
Step-by-step explanation:
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
2
−p
2
.
2p
q
⇒
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.