Exponent Laws

We have reached the end of the second week of math 10. This week we learned a very essential skill, that skill being applying multiple exponent laws in one equation. I personally believe this was the most important skill we learned this week as this is a key aspect of evaluating of simplifying many equations. Applying different skills may be confusing at times as it might not be obvious which one to use first or where to start. However, in reality it isn’t that complicated. You simply have to choose what you like best and go with that.

There are four laws we learned were: the multiplication (product), division (quotient), power, and negative exponent law. These rules can be done in any order, however in this example I am going to being going from the multiplication – the division – the power – the negative exponent law (M-D-P-N).

Here is the example question we are going to be working with:

Step 1 – the Multiplication Law: The multiplication law says that the exponents of at least two bases need to be added. In order for this to work they need to be like bases (x and x or y and y). In addition to this, the bases are only multiplied if they are on the same side of the fraction. So in this case the two “y’s” exponents in the numerator of the fraction will be added together. Other than this there isn’t anywhere else we can apply the multiplication law.

Note: Remember bases without exponents have an “invisible” exponent of one.

Step 2 – the Division Law: The division law says that exponents of like bases that are on opposite sides of the fraction need to be subtracted. In this case the “x” exponent in the denominator will be subtracted from the “x” exponent in the numerator. The same thing will occur with the negative exponents. After this step is completed the fraction will be gone which makes things simpler.

Step 3 – the Power Law: the power law says to multiply the exponent outside the brackets with all the exponents inside the brackets. The exponent outside of the brackets will be distributed throughout the brackets. In this case the “2” will be multiplied with exponent of the “x”, the “y”, and the “4” which has an invisible exponent of 1. This is a common mistake to not multiply by the exponent of the coefficient, it is very easy to forget about.

Step 4 – Evaluating the Exponents: this step is only possible for the exponents on actual numbers (coefficients) not on bases such as “x’s” and “y’s”. If the number has an exponent of “1” this step can be skipped. Here all that we need to do is raise 4 to the power of 2 which is equal to 16.

Step 5 – the Negative Exponent Law: this is final step that is necessary for this equation. Typically a final answer is not left with negative exponents, they are usually reversed to become positive. In order to do this, the negative exponent law needs to be used. The “16” and “” can stay where they are since they have positive exponents. However, the “y” has a negative exponent so it needs to be moved. To fix this you need to simply reciprocate (flip the fraction) and put the “y” on the other side of the fraction and change its exponent to a positive number.

After this step has been completed you will have your final answer.

 

Summary:

That is how you use multiple exponent laws in an equation. This skill can be very useful in many equations and once it is learned correctly it can be used simply and very effectively. And like I said before, I completed the equation in a certain order, but this does not actually matter. The order can be chosen according to your liking.