Problem statement: Given the complex numbers z and w, determine:

Let:

we know:

Let (in phasor form):

we know:

since:

the machine and Mathcad yield:

so we know what the answer should be.

We are given in literature:

so substituting...

We get the same answer yielded by the machine and Mathcad. But this is NO FUN!

Let's use the theorems, laws, and identities we know to develop the solution shown in literature.

Given:

(1st law of exponents)

Breaking

into two parts:

and

Solving z^c using de Moivre's identity:

(where c is a real number)

remembering:

substituting:

Solving z^di using Euler's formula: (twice)

Euler's formula:

do you understand this!

(3rd law of exponents)

simplifying each of these two parts individually:

part 1

(3rd law of exponents)

what to do now?

part 2

From http://en.wikipedia.org/wiki/Exponentiation#Summary,:

thinking about it and applying it to our 5^di problem yields ----->

do you understand this from logarithms!

using Euler's formula (again):

So putting it all together and using 5 decimal places to maintain accuracy but presenting the answer with 1 decimal place accuracy:

(answer)

Just for kicks, lets have the machine confirm our solution using the formula in literature:

Remembering:

substituting yields:

which is the same as the machine when it computes everything itself:

A QUICK REVIEW of complex number basics:

(rectangular form)

(the complex conjugate)

by observation we see the phasor form for z:

the real part:

imaginary part:

Complex numbers are easily added, subtracted, multiplied, and divided

For complex numbers to a real power (n), we use de Moivre's identity:

(for real values of n, NOT when n is complex)

Also study next page 299 to know Euler's Formula:

For complex exponentiation (a complex number to a complex exponent), know the derivation of the following:

Let:

Euler's Formula derivation/proof using Taylor Series expansions (one of many ways) of e^x, sine and cosine:

basic facts about the powers of i:

etc.

For complex z we define each of these functions by the above series, replacing x with z (an imaginary number). We find that:

just for kicks:

since: