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If π§ one is equal to cos of 210 degrees plus π sin of 210 degrees, π§ two is equal to three cos of 135 degrees plus π sin of 135 degrees, and π§ three is equal to four cos of 135 degrees plus π sin of 135 degrees, what is the exponential form of π§ one multiplied by π§ two multiplied by π§ three all to the power of four.
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We have been given three complex numbers in polar, or trigonometric form.
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Weβre being asked to find their product, and then we need to raise that to the power of four.
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And weβre also being asked to give our answer in exponential form.
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There are two ways we can go about this.
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We could begin by finding their product in polar form.
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We could then raise that to the power of four, and then convert it to exponential form.
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Alternatively, we could convert each complex number into exponential form first.
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We could then find their product, and then raise that number to the power of four.
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In fact, weβre going to apply DeMoivreβs theorem no matter which method we use.
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So, letβs look at finding their product in polar form first.
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To find the product of complex numbers, itβs fairly straightforward.
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We multiply their moduli, and we add their arguments.
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Now Iβve written this list of complex numbers in polar form, but this also works for complex numbers in exponential form.
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Letβs begin by multiplying the moduli of these three complex numbers.
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The modulus of our first complex number is one.
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The modulus of our second complex number is three.
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And the modulus of our third complex number is four.
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So, the modulus of the product of these three complex numbers is one multiplied by three multiplied by four, which is, of course, 12.
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Next, weβre going to add their arguments.
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Now we generally like to represent our arguments in radians, but actually itβs easier to add them in degrees and convert them in a moment.
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The argument of our first complex number is 210.
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For our second complex number, itβs 135.
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And for our third complex number, itβs also 135.
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And so, the argument for the product of our complex numbers is going to be 210 plus 135 plus 135, which is 480 degrees.
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Now it doesnβt really matter if we carry on working with degrees at this stage.
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But we may as well turn four 480 degrees into radians, as weβre going to need to do that represent it in exponential form.
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Here, we need to recall that two π radians is equal to 360 degrees.
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And if we divide through by 360, we see that one degree is equal to two π by 360 radians, or π by 180 radians.
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And this means we can convert from degrees to radians by multiplying by π by 180.
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480 multiplied by π by 180 is eight π by three.
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So, we can say that the argument of the product of these three complex numbers is eight π by three radians.
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And we can now represent the product of these three complex numbers in polar form.
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Itβs 12 multiplied by cos of eight π by three plus π sin of eight π by three.
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Next, we need to raise the sum to the power of four.
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Again, we can do this before or after changing it into exponential form.
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Either way weβre going to use DeMoivreβs theorem, so it doesnβt really matter.
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DeMoivreβs theorem says that for a complex number π§ in the form π cos π plus π sin π, π§ to the power of π is equal to π to the power of π multiplied by cos of ππ plus π sin of ππ.
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And thatβs when π is a natural number.
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So, this means we can find the modulus of π§ one π§ two π§ three to the power of four by finding 12 to the power of four, which is 20736.
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And we need to multiply the argument by that power; itβs four.
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So, the argument of π§ one π§ two π§ three to the power of four is four multiplied by eight π by three.
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Four multiplied by eight π by three is 32π by three.
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So, at this stage, we can see that our complex number is 20736 multiplied by cos of 32π by three plus π sin of 32π by three.
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So, how do we represent this in exponential form?
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Well, for a complex number π§ in polar form π cos π plus π sin π, we can represent in exponential form as ππ to the ππ.
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Now itβs important at this stage that the argument is in radians.
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And weβve already done that conversion, so weβre good to go.
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We can take the modulus of 20736 and the argument of 32π by three and substitute it into that formula.
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And we see that our complex number is 20736π to the power of 32π by three π.
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And thereβs one thing left to do.
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We prefer to write our numbers in terms of their principal argument.
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Thatβs greater than negative π and less than or equal to π.
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We can see the 32π by three is considerably larger than π.
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And we achieve the principal argument by adding or subtracting multiples of two π.
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In this case, weβre going to have to subtract five lots of two π.
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And when we do, we get two π by three, or two-thirds π as our principal argument.
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And now we really are done.
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We can say that the exponential form of π§ one π§ two π§ three to the power of four is 20736π to the power of two π by three π.