Variables and Multiplication

        Multiplication between variables, especially when there are many different variables involved, can be very complicated and challenging. Variables being multiplied by numbers is very simple. When multiplying variables that are the same variable, the resultant is the same variable but with a different power. When you multiply two, or more, of the same variable all you have to do to figur out the resultant variable is to add all of the powers of the variables. For example: to multiply "X3" by "X 2", since both of them are the same variable all you have to do is add together their powers. Therefore the resultant would be "X(3+2)" or in a better form "X5". It is not any different if one of or both of the numbers are already multiplied by a constant(a number) such as "3*X2". If you were to multiply "3*X" by "4X3", the resultant would be found by multiplying the like-terms together first. Therefore, you could multiply the constants first (as in "3*4") so the resulting number multiplier would be 12. Next, you would multiply the variables. "X * X3" would equal "X4". So the overall resultant would be "12X4".

Examples:
Set 1
Set 2
Set 3

Application Problems:
Problem 1
Problem 2

        Multiplication when it involves different variables still has the same principles but it can sometimes get more complicated because you need to keep track of each different variable. Each different variable must each by kept seperate, there is no way to combine X and Y and get only one variable so you must handle each variable seperately. That means if you were to do "X * Y" the resultant could simply be put as "XY". When two variables are next to each other there is an implied multiplication between them. For practice, if you were to multiply "X2 Y5" by "X3 Y2" you would take it one variable at a time. Therefore, the resultant would be "X(2+3)Y(5+2) "="X5Y7". Just like with a single variable, there is only a minor difference when there is a constant with the variables during the multiplication. What is meant by that is when you are doing "4X7 * 5Y4". Simply do the multiplication of the constants first ("4*5"), which would result in 20. This is the constant part of the result. Then you move onto the variables. Since there are no like variables all you can do is keep them multiplied by each, meaning that the variable part of the result would be "X7 Y4". Now, combine the two parts and then the resultant would be "20X7 Y4". It get more complicated when each term contains several variables each, such as if you were to do "2X7Y 3N2 * 7Y5N3". Simply take it one step at a time. First, look at the constants. "2*7"=14. Therefore, the constant part of your result would be "14". Now when looking at the variables, look for the variables that appear more than once, and those that only appear once. The "X" only appears once which means that it can be left alone, so it stays as "X7". Both the "Y" variable and the "N" variable appear twice, once in each term. Now multiply the variables that appear more than once through. The simplest way to do this is to take it one variable at a time. To Multiply the "Y" variable through you would be looking at "Y3 * Y5" which equals "Y8". Now looking at the "N" variable. The two "N" variables are "N2" and "N3". Now you must multiply them "N2 * N3" = "N5". Now, simply combine all of the terms to get "14 * X7 * Y8 * N5" or "14X7Y8N5".