Based on material learned earlier in the course. The purpose...

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. If $z=6 e^{i \frac{7 \pi}{4}}$ and $w=2 e^{i \frac{5 \pi}{6}},$ find $z w$ and $\frac{z}{w} .$ Write the answers in polar form and in exponential form.

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
If $z=6 e^{i \frac{7 \pi}{4}}$ and $w=2 e^{i \frac{5 \pi}{6}},$ find $z w$ and $\frac{z}{w} .$ Write the answers in polar form and in exponential form.

Key Concepts

Division of Complex Numbers in Polar/Exponential Form

Similarly, dividing complex numbers in polar or exponential form involves dividing their magnitudes and subtracting the exponents (angles). This method streamlines division by handling the modulus and argument separately.

Multiplication of Complex Numbers in Polar/Exponential Form

When multiplying complex numbers expressed in polar or exponential form, the magnitudes are multiplied together and the angles are added. This property simplifies the calculation and analysis of complex products.

Polar and Exponential Forms

Polar form represents a complex number by its magnitude (modulus) and angle (argument) as r(cos? + i sin?), while exponential form, derived from Euler's formula, expresses it as re^(i?). These forms are particularly useful for operations such as multiplication and division.

Complex Numbers

Complex numbers extend the real number system by including imaginary numbers and are typically written in the form a + bi. They can be represented either in rectangular (Cartesian) form where a and b are the real and imaginary parts, or in polar/exponential form which expresses them in terms of a magnitude and an angle.

Transcript

00:01 The concept involved in this problem is to use operations with complex numbers that are written in exponential form to find the product and then the quotient of two given complex numbers.

00:16 Before we start the problem, let's review a couple of operations with complex numbers.

00:22 Let's say we have a complex number z sub 1 in exponential form, then it could be written in this word.

00:30 It would equal r1 e raised to the i times theta sub 1 and let's say there's a second complex number z sub 2 well in exponential form that could be written is r sub 2 e raised to the i theta sub 2 so there is a property that says if you are going to multiply z1 times z 2 then you can get the result by multiplying r1 times r2, e raised to the i times theta 1 plus theta 2.

01:23 And if you're going to get the quotient of those two complex numbers, z1 divided by z2, then that can be found by r1 divided by r sub 2, e raised to the i times theta 1 minus theta 2.

01:47 Okay, so we're going to use these two properties to find the product of the two given complex numbers, and then we will find the quotient.

01:58 Okay, so to find the product, we're going to use this first property.

02:03 You can think of the 6 is your r1, and the 7 pi over 4 is your theta 1.

02:10 The 2 is your r2, and the 5 pi over 6.

02:14 Is your theta so if i multiply r1 times r2 that will give me 12 e raised to the i and then we have to add these two together okay let's go ahead and do a little scratch work let me see i'll take this in red and i'll add these together 7 pi over 4 plus 5 pi over 6 let's say i'll need a common denominator of 12 so i'll be a 21 pi over 12 and a 10 pi over 12 which is going to give me a 31 pi over 12 which is that's bigger than 2 pi so i'm going to subtract 2 pi from it so if i subtract 2 pi that's subtracting 24 pi over 12 which will give us 7 pi over 12 okay who's 12 all right so my theta up here will be 7 pi over 12 okay now this is the exponential formula you're also asked in this problem to put your answer in a polar form so we can do that pretty easily that'll end up being 12 because that's a r cosine of 7 pi over 12 plus i sign of 7 power 12.

04:28 Okay so this is the polar form.

04:30 I'm sorry this is the exponential form and this is the polar form for the product of those two complex numbers...

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