Math Problem Statement
Solution
The expression given in the image is a complex number. First, we'll simplify this expression and then find the modulus of its cube.
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Simplifying : where is the imaginary unit with .
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Express :
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Calculating : The modulus of a complex number is calculated as: Applying this to our : Then, the cube of the modulus is:
Thus, .
Would you like to know more about complex numbers or any other related topics? Here are some additional questions you could ask:
- How do I represent complex numbers in the complex plane?
- What are the properties of modulus and argument in complex numbers?
- How can I multiply and divide complex numbers?
- What is the significance of Euler's formula in relation to complex numbers?
- Could you show how to solve quadratic equations that have complex solutions?
Tip: Remember, the modulus of a complex number gives you the distance of the point from the origin in the complex plane, which can be visualized as the hypotenuse of a right triangle formed by the real and imaginary parts.
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Math Problem Analysis
Mathematical Concepts
Complex Numbers
Modulus
Powers of Complex Numbers
Formulas
Modulus of complex number |z| = √(a^2 + b^2), where z = a + bi
Cube of the modulus |z^3| = |z|^3
Theorems
Modulus theorem for complex numbers
Suitable Grade Level
Grades 10-12