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

# Question Video: Finding the Power of a Complex Number in Polar Form Mathematics • Third Year of Secondary School

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Given that π§ = 3β2 (cos 225Β° β π sin 225Β°), find π§Β², giving your answer in exponential form.

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### Video Transcript

Given that π§ equals three root two multiplied by cos of 225 minus π sin of 225, find π§ squared, giving your answer in exponential form.

Eulerβs formula says that π to the plus or minus ππ is equal to cos π plus or minus π sin π. We can extend this and say that π multiplied by π to the plus or minus π π is equal to π multiplied by cos π plus or minus π sin π. We can use this to help us convert between polar and exponential form of a complex number.

The modulus β thatβs the value of π 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 over 180.

So the argument is in fact 225 multiplied by π over 180, which is five π over four. And since the coefficient of π sin π is negative, we can rewrite our complex number as three root two multiplied by π to the negative five π over four π.

We actually need to work out π§ squared. So weβre going to square this entire expression. π§ squared is, therefore, equal to three root two π to the negative five π over four π all squared. Letβ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 over four π by two. And that gives us π to the power of negative five π over two π.

And our expression for the complex number π§ squared in exponential form is 18π to the negative five π over two π. Notice though that each of the exponents in the possible answers to our question have a positive coefficient of π.

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

And our expression for π§ squared in exponential form is 18π to the three π over two π.

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