Mastering Algebra - Working With Exponents - Part II

In Part I of this article, we discussed how to work with exponents, specifically how to simplify expressions which involved multiplying like bases, raising an exponent to another power, and the property of any expression to the 0th and the 1st powers. Here we examine the distributive and quotient properties of exponents.

When you have an expression that involves one or more variables and numbers, and each of these may itself have an exponent, and then we have this expression enclosed in parentheses and raised to a power, you must use the distributive property of exponents to simplify: thus (3x^2y^3)^3 would qualify as such an expression. To simplify this expression, we simply multiply the exponent of each term by the exponent outside parentheses. Recall that the number 3 term has the invisible demon exponent 1, and this was covered in the previous article. Thus we obtain 3^3x^6y^9, and simplifying for the number term, 27x^6y^9. Observe that we are distributing the exponent 3 outside the parentheses to each of the exponents inside parentheses, thus the name distributive property. Looking at another example, take (2^2x^4y^6z^3)^4. Distributing the 4 over the inside exponent terms and multiplying we have 2^8x^16y^24z^12 and simplifying for the number term we have 256x^16y^24z^12.

The quotient property comes into play when we divide one expression containing like bases by another. For example, take the expression (x^6y^3)/(x^2y^2). To simplify this expression, we subtract the exponents of like bases: thus x^4y^1 or more simply x^4y is the resulting expression. Again to understand why this property works the way it does, let us return to the analogy of pearls on a string, which we employed in Part I of this article. If we write out (x^6y^3)/(x^2y^2) we have xxxxxxyyy/xxyy. Now using the cancellation property, we can strike 2 x-pearls and 2 y-pearls from the numerator to end up with our answer of 4 x-pearls and 1 y-pearl, namely x^4y.

That is all there really is to these two properties. To make anything more out of them would simply be complicating something unnecessarily. Remember: mathematics is hard in itself; yet there is a lot to this subject which is readily understandable, such as the properties outlined in these two articles. Learn these rules and become familiar with their uses, as then mastery to algebra will be right around the corner.