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Byju's Answer
Standard XII
Mathematics
Square Root of a Complex Number
If 1, ω, ω2...
Question
If
1
,
ω
,
ω
2
are the cube roots of unity, prove that:
(
a
+
b
)
(
a
ω
+
b
ω
2
)
(
a
ω
2
+
b
ω
)
=
a
3
+
b
3
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Solution
Given:-
1
,
ω
,
ω
2
are the cube roots of unity
To prove:-
(
a
+
b
)
(
a
ω
+
b
ω
2
)
(
a
ω
2
+
b
ω
)
=
a
3
+
b
3
Proof:-
Since
1
,
ω
and
ω
2
are the cube roots of unity.
Therefore,
1
+
ω
+
ω
2
=
0
ω
3
=
1
ω
3
n
+
1
=
ω
Now,
(
a
+
b
)
(
a
ω
+
b
ω
2
)
(
a
ω
2
+
b
ω
)
=
(
a
2
ω
+
b
2
ω
2
+
a
b
(
ω
+
ω
2
)
)
(
a
ω
2
+
b
ω
)
=
(
a
2
ω
+
b
2
ω
2
−
a
b
)
(
a
ω
2
+
b
ω
)
=
a
3
ω
3
+
a
2
b
ω
2
+
b
2
a
ω
4
+
b
3
ω
3
−
a
2
b
ω
2
−
a
b
2
ω
=
a
3
−
a
2
b
ω
2
+
a
b
2
ω
+
b
3
−
a
2
b
ω
2
−
a
b
2
ω
=
a
3
+
b
3
Hence proved.
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Q.
1
,
ω
,
ω
2
are cube roots of unity, then the value of
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a
+
b
+
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,
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If
1
,
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,
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are the cube roots of unity, then prove that
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9
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