In Problems 43-54, write each expression in the standard form a+b i. [√(5)(cos(3 π)/(16)+i sin(3

step 1 Okay, so we have to take this to the power. So we have root 5 cosine of 3 pi over 16 plus i sine

In Problems 43-54, write each expression in the standard...

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Key Concepts

Complex Numbers

Complex numbers are numbers in the form a + bi, where a and b are real numbers and i is the imaginary unit satisfying i² = -1. They can be represented in different forms to facilitate various operations, including addition, multiplication, and exponentiation.

Standard Form of a Complex Number

The standard form, or rectangular form, of a complex number is written as a + bi, where a is the real part and b is the imaginary part. This representation is useful for performing arithmetic operations and for visualizing complex numbers on the complex plane.

Polar Form of a Complex Number

The polar form represents a complex number using a modulus and an argument, expressed as r(cos? + i sin?). This form is particularly beneficial when multiplying or raising complex numbers to powers because it simplifies these operations via the manipulation of the modulus and argument.

De Moivre's Theorem

De Moivre's Theorem provides a straightforward method to compute powers and roots of complex numbers in polar form. It states that (r(cos? + i sin?))^n equals r^n (cos(n?) + i sin(n?)), thereby converting exponentiation into simpler operations on the modulus and angle.

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