Hello,

I was reading through the book Fundamentals of Electromagnetics and came across the two expression simplification below. Can someone help me understand how both these are being simplified please?

 

Do you mean how do you go step by step from the original expression to the final result? OR
Do you mean how is the second expression a simplification?

 

When I first looked at the problem I thought that the variable X, Y, Z with the ^ symbol above them followed by a number meant raising the Variable by the power of the following number. I decided that was not the case and that the variable was X with the ^ above it And the following number was just multiplying it. When I went to school (Over 60 years ago.) The number was put before the variable not after it. When I realised this it was just a matter of using algebra that that I was taught in school.
The first step was to make the denominator the same for both parts of the formula inside the square brackets the same. to do this multiply the top and bottom lines of the first part by 8 (to change the denominator to 216) I don't know how to create characters of letters with the ^ symbol above it so I will leave the the ^ out. So after this step the formula inside the square brackets becomes
16(X2 - Y2 - Z) 4(X6)
________________ - ___________
216 216

so if we expand this we get

X32 - Y32 - Z16 - X24
_____________________________
216
So if we combine the number of Xs we get

X8 - Y32 -Z16
____________________
216
This can be simplified by dividing top and bottom lines by 8 which gives

X - Y4 - Z2
_______________
27

If we now multiply this by the value outside the square bracket we get the same as shown in your question.
There are other sequences of step you could use to get the same result.
It is not really a question about magnetics it is just algebra.
EDIT The forum software removes multiple spaces which makes the above formula look wrong.
Les.

 

Bold type with a hat over it means that it is a unit vector for the three coordinate axes. Yes the ordinary rules of algebra apply. What you end up with should be an expression with three terms where each term has a clearly defined magnitude.