Video: Finding the Power of a Complex Number in Polar Form

Given that 𝑧 = 3√2 (cos 225Β° βˆ’ 𝑖 sin 225Β°), find 𝑧², giving your answer in exponential form.

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Video Transcript

Given that 𝑧 equals three root two multiplied by cos of 225 minus 𝑖 sin of 225, find 𝑧 squared, giving your answer in exponential form.

Euler’s formula says that 𝑒 to the plus or minus π‘–πœƒ is equal to cos πœƒ plus or minus 𝑖 sin πœƒ. We can extend this and say that π‘Ÿ multiplied by 𝑒 to the plus or minus 𝑖 πœƒ is equal to π‘Ÿ multiplied by cos πœƒ plus or minus 𝑖 sin πœƒ. We can use this to help us convert between polar and exponential form of a complex number.

The modulus β€” that’s the value of π‘Ÿ in our complex number β€” is three root two. And the argument is given as 225 degrees. However, when we write a complex number in exponential form, we do so using radians. Remember to change a number from degrees to radians, we multiply it by πœ‹ over 180.

So the argument is in fact 225 multiplied by πœ‹ over 180, which is five πœ‹ over four. And since the coefficient of 𝑖 sin πœƒ is negative, we can rewrite our complex number as three root two multiplied by 𝑒 to the negative five πœ‹ over four 𝑖.

We actually need to work out 𝑧 squared. So we’re going to square this entire expression. 𝑧 squared is, therefore, equal to three root two 𝑒 to the negative five πœ‹ over four 𝑖 all squared. Let’s begin by squaring three root two. Three squared is nine and root two squared is two. So this becomes a nine multiplied by two which is 18. For the exponent, we know that we need to multiply negative five πœ‹ over four 𝑖 by two. And that gives us 𝑒 to the power of negative five πœ‹ over two 𝑖.

And our expression for the complex number 𝑧 squared in exponential form is 18𝑒 to the negative five πœ‹ over two 𝑖. Notice though that each of the exponents in the possible answers to our question have a positive coefficient of 𝑖.

In fact, the imaginary exponential is periodic with a period of two πœ‹. So we can add two πœ‹ to negative five πœ‹ over two repeatedly until we find a positive value. Negative five πœ‹ over two plus two πœ‹ is negative one-half πœ‹. Negative a half of πœ‹ plus two πœ‹ is three πœ‹ over two.

And our expression for 𝑧 squared in exponential form is 18𝑒 to the three πœ‹ over two 𝑖.

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