Here are the essential concepts you must grasp in order to answer the question correctly.
Complex Numbers
Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where 'a' is the real part and 'b' is the coefficient of the imaginary unit 'i', which is defined as the square root of -1. Understanding complex numbers is essential for performing operations involving square roots of negative numbers, as seen in this problem.
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Operations with Complex Numbers
To perform operations with complex numbers, such as addition, subtraction, multiplication, and division, one must apply algebraic rules while treating 'i' as a variable. In this exercise, squaring a complex number involves using the formula (a + bi)² = a² + 2abi + (bi)², which simplifies to a² - b² + 2abi, since (bi)² equals -b².
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Standard Form of Complex Numbers
The standard form of a complex number is expressed as a + bi, where 'a' and 'b' are real numbers. When performing operations on complex numbers, the final result should be presented in this form to clearly distinguish between the real and imaginary components, making it easier to interpret and use in further calculations.
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Complex Numbers In Polar Form