In Problems 43-54, write each expression in the standard form a+b i. (√(2)-i)^6

Name: Problem 53, Chapter 8: Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry
Uploaded: 2020-06-19T14:20:19-08:00
Duration: 3 min 9 s
Channel: Joseph Lentino
Description: In Problems 43-54, write each expression in the standard form a+b i. (√(2)-i)^6

step 1 So we want to expand root 2 minus i to the sixth power. And the easiest way to handle that is to

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

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Key Concepts

Standard Form

The standard form of a complex number is written as a + bi, where a represents the real part and b represents the imaginary part. Expressing complex numbers in standard form simplifies performing arithmetic operations and comparing numbers by clearly delineating their real and imaginary components.

De Moivre's Theorem

De Moivre's Theorem is a fundamental principle in complex number theory that allows one to raise complex numbers to integer powers by converting them to polar form. It states that for a complex number in polar form, (r(cos? + i sin?))? equals r?(cos(n?) + i sin(n?)), making exponentiation more manageable.

Complex Numbers

Complex numbers are numbers that include a real part and an imaginary part, typically expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying i² = -1. They provide a way to extend the one-dimensional number line into the complex plane for solving equations that have no real solutions.

Polar Form Conversion

Polar form expresses a complex number in terms of its magnitude (or modulus) and angle (or argument) using the representations r(cos? + i sin?). Converting a complex number from standard form to polar form can simplify operations like multiplication, division, and exponentiation.

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