Prove that (213+V5) is an irrational number. Also check whether
(213 + V5)(23 – V5) is rational or irrational.
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Let us assume that 2
3
+
5
is rational number.
Let P=2
3
+
5
is rational
on squaring both sides we get
P
2
=(2
3
+
5
)
2
=(2
3
)
2
+(
5
)
2
+2×2
3
×
5
P
2
=12+5+4
15
P
2
=17+4
15
4
P
2
−17
=
15
………..(1)
Since P is rational no. therefore P
2
is also rational &
4
P
2
−17
is also rational.
But
15
is irrational & in equation(1)
4
P
2
−17
=
15
Rational
= irrational
Hence our assumption is incorrect & 2
3
+
5
is irrational number.
b) P=(2
3
+
5
)(2
3
−
5
)
P=12−5=7
Hence P is rational as
q
p
=
1
7
& both p & q are coprime numbers.
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