Express the following complex numbers in the form x+y i for...

Express the following complex numbers in the form $x+y i$ for $x, y \in \mathbb{R}$ :
(a) $(2+3 i)+(4+i)$
(b) $(2+3 i)^{2}(4+i)$
(c) $\frac{2+3 i}{4+i}$
(d) $\frac{1}{i}+\frac{3}{1+i}$
(e) $(-i)^{-1}$
(f) $(-1+i \sqrt{3})^{3}$

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Key Concepts

Standard Form Representation of Complex Numbers

This concept involves writing a complex number in the form x + yi, where x represents the real part and y represents the imaginary part. It is the primary way of expressing complex numbers to clearly distinguish between their real and imaginary components.

Arithmetic Operations with Complex Numbers

This concept includes performing addition, subtraction, and multiplication on complex numbers. By operating on the real parts independently from the imaginary parts and using the property i² = -1, one can simplify expressions and combine like terms to obtain results in standard form.

Division of Complex Numbers and the Use of the Complex Conjugate

Division of complex numbers typically requires eliminating the imaginary component from the denominator. This is achieved by multiplying both numerator and denominator by the conjugate of the denominator, thereby rationalizing the expression into standard form with a real denominator.

Exponentiation and Powers of Complex Numbers

This concept deals with raising complex numbers to integer powers. Techniques such as binomial expansion for small powers or employing De Moivre’s theorem for general cases help in efficiently computing powers and expressing the results in the standard x + yi form.

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