(cube root of 8+3rt21 )+ (cube root of 8-3rt21)
Answers
Answered by
0
Answer:
The cube roots of
8
are
2
,
2
ω
and
2
ω
2
where
ω
=
−
1
2
+
√
3
2
i
is the primitive Complex cube root of
1
.
Explanation:
Here are the cube roots of
8
plotted in the Complex plane on the circle of radius
2
:
graph{(x^2+y^2-4)((x-2)^2+y^2-0.01)((x+1)^2+(y-sqrt(3))^2-0.01)((x+1)^2+(y+sqrt(3))^2-0.01) = 0 [-5, 5, -2.5, 2.5]}
They can be written as:
2
(
cos
(
0
)
+
i
sin
(
0
)
)
=
2
2
(
cos
(
2
π
3
)
+
i
sin
(
2
π
3
)
)
=
−
1
+
√
3
i
=
2
ω
2
(
cos
(
4
π
3
)
+
i
sin
(
4
π
3
)
)
=
−
1
−
√
3
i
=
2
ω
2
One way of finding these cube roots of
8
is to find all of the roots of
x
3
−
8
=
0
.
x
3
−
8
=
(
x
−
2
)
(
x
2
+
2
x
+
4
)
The quadratic factor can be solved using the quadratic formula:
x
=
−
b
±
√
b
2
−
4
a
c
2
a
=
−
2
±
√
2
2
−
(
4
×
1
×
4
)
2
⋅
1
=
−
2
±
√
−
12
2
=
−
1
±
√
3
i
Answer link
Similar questions