Question
Practice
- Convert each number to polar form and evaluate:
- Find the cube roots of:
(a) 125 j
(a) 125 j
Ask by Willis Horton. in South Africa
Mar 21,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
To convert each complex number to polar form, use
Solution
To convert a complex number to polar form, we use the formula
Given the complex numbers:
(a)
(b)
©
(d)
(a)
(b)
©
(d)
We will convert each complex number to polar form and then evaluate the given expressions.
Let’s start with the first complex number:
(a)
(a)
The magnitude
The angle
Now, we can express
Let’s calculate the values of
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Simplify this solution The Deep Dive
To convert numbers to polar form, you first find the magnitude and angle (argument) of the complex numbers involved. For instance, for
Now let’s dive into the cube roots of
for
Now, for a bit more context: The history of polar form stems from the desire to simplify complex number operations, particularly multiplication and division, which can be done through magnitudes and angles instead of wrestling with real and imaginary parts.
And if you’re looking for practical advice, always remember: when adding or subtracting complex numbers, it’s best to combine like terms first. Many mistakes arise when people forget to handle the real and imaginary parts separately!
