SOLUTION: Find the cube roots of the following complex number; -5(square root of 2)+ 5(square root of 2i)

Question 1073428: Find the cube roots of the following complex number;
-5(square root of 2)+ 5(square root of 2i)

Found 2 solutions by ikleyn, KMST:
Answer by ikleyn(52805) (Show Source): You can put this solution on YOUR website!
.
Check if you correctly wrote parentheses.

Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
I believe what you wanted was the cube roots of
.
You can write that as
-5sqrt(2)+i5sqrt(2) or as -5*sqrt(2)+i*5*sqrt(2) .

The three cube roots will be numbers of the form
such that
,
meaning that for
.
So, gives us
, and .
You can get approximate values for
and for the trigonometric functions from a calculator.

Finding exact values is a little more involved,
and the resulting expressions may not look as nice as with approximate decimals.
The one answer with is easy,
because ,
so that the exact value for one of the three cube roots is
.

For the other two values of , it gets a little more complicated.
Since and ,
we can use the trigonometric functions for ,
knowing their relation to the trigonometric functions for and .
For second quadrant ,
and .
For fourth quadrant ,
and .

If we want exact values, we will need to use the half-angle trigonometric formulas:
and .
, so
and .
So, and
That makes the exact values for the other two cube roots
= and
= .
RELATED QUESTIONS
Can someone PLEASE help me with this problem?! THANK YOU! Find all the square roots... (answered by Alan3354)
What is the absolute value of the complex number -4-square root of... (answered by Solver92311)
please help to simplify - the square root of -25 - the square root of 2 times the... (answered by funmath)
find the cube root of the square root of n is... (answered by stanbon)
find the square root of (1-2 sq rt... (answered by Alan3354,MathLover1)
the square root of 2a - 5 square roots of 32a + 7 square roots of... (answered by adamchapman)
are the following numbers rational? -1/square root of 5, 13/square root of 5, square... (answered by Fombitz)
Multiply the following: Cube Root of Square Root of 3^3 * Cube Root of Square Root of... (answered by stanbon)
Between what two consecutive integers does each of the following lie? 1. square root... (answered by checkley77)