Use a calculator to help answer the questions. Show that a solution of x^6+1=0 is (√(3))/(2)+(1)

step 1 Okay, in this problem we have x to the 6 plus 1 equals 0. And we're trying to show that one of t

Use a calculator to help answer the questions. Show that a...

Key Concepts

Complex Numbers

Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit with the property that i² = -1. They extend the real number system and allow for solutions to equations that have no real solutions.

Polar and Exponential Form

The polar form of a complex number represents it by its magnitude and an angle, typically written as r(cos? + i sin?). Using Euler’s formula, this can be expressed in exponential form as re^(i?). This representation is particularly useful for multiplying, dividing, and finding powers and roots of complex numbers.

Euler’s Formula

Euler’s formula, e^(i?) = cos? + i sin?, establishes a fundamental relationship between the exponential function and trigonometric functions. This formula allows complex numbers to be easily converted between their rectangular (a + bi) and polar (or exponential) forms.

De Moivre’s Theorem

De Moivre’s Theorem states that for any real number ? and integer n, (cos? + i sin?)^n = cos(n?) + i sin(n?). This theorem facilitates finding the nth roots of complex numbers, which is essential when solving polynomial equations like x? + 1 = 0.

Transcript

00:01 Okay, in this problem we have x to the 6 plus 1 equals 0.

00:09 And we're trying to show that one of the solutions to this is that root 3 over 2 plus 1 half i, when we plug it in for x, it gives us 0.

00:20 So that means our expression to the 6 power plus 1 equals 0.

00:27 Really, we just have to show that this expression to the 6 power is equal to negative 1.

00:35 So let's work on that a little bit.

00:36 Like expanding it out to the 6 power is going to take a little bit of work and hopefully you have a calculator with you to follow along.

00:45 So i don't like to raise things immediately to the six power, especially with these large fractions.

00:52 It's just going to get unwieldy.

00:55 So we use a little bit of algebra to help simplify things down.

01:00 We can pull this a one -half out.

01:05 So it'll be one -half to the six -power times root three plus i to the six -pac.

01:17 And again, we're trying to show that that equals negative one.

01:20 One half to the six power is one over 64, so we can multiply both sides by 64.

01:27 And get a little bit easier expression to work with.

01:31 So root 3 plus i to the six power is negative 64.

01:41 Okay, and again, to avoid the polynomial expansion, even with the square roots, i'm just going to do it one part at a time.

01:47 So let's just find out what root 3 plus i.

01:49 Squared is so root 3 plus i squared we can just do a binomial expansion on this so root 3 squared is just 3 then we're going to do 2 times both of our terms plus the square of our last term which since i squared is negative 1 that will give us 2 plus 2 i root 3 okay and that's just the square of it so if we have this and we raise it to the third power, we would hope that that would give us negative 64...

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