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Byju's Answer
Standard XI
Mathematics
Cube Root of a Complex Number
Let α,β are...
Question
Let
α
,
β
are the imaginary cube roots of unity then find the value of
α
4
.
β
4
−
1
α
.
β
.
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Solution
Given
α
,
β
are imaginary cube roots of unity.
Then either
α
=
w
or,
β
=
w
2
and
α
=
w
2
or
β
=
w
; where
w
3
=
1
.
In both cases
(
α
β
)
=
w
3
=
1
.
Now,
α
4
β
4
−
1
α
β
=
(
α
/
β
)
4
−
1
(
α
β
)
=
1
4
−
1
/
1
=
1
−
1
=
0
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Similar questions
Q.
If
α
and
β
are imaginary cube roots of unity, then
α
4
+
β
4
+
1
α
β
[IIT 1977]
Q.
If
α
,
β
are non-real complex cube roots of unity, then
α
4
+
β
4
+
α
−
1
β
−
1
Q.
If
α
and
β
are the complex cube roots of unity, then
α
4
+
β
4
+
α
−
1
+
β
−
1
.
Q.
If
α
and
β
are the complex cube-roots of unity, then
α
4
+
β
4
+
α
−
1
β
−
1
=
Q.
Let
α
,
β
denote the cube roots of unity other than
1
and
α
≠
β
. Let
S
=
302
∑
n
=
0
(
−
1
)
n
(
α
β
)
n
. Then, the value of
S
is
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