Multiplying and Dividing Exponents

When working with exponents, there are a few rules and methods that can help you with your tasks. These can make your work much easier and produce less complicated results. Multiplying exponential expressions where the base numbers are the same. Let's say we have an expression like this, two to the third power times two to the fifth power. One way we could simplify this, since we know all the values in this expression is to simply expand the exponents into their actual values. We know through the associative property of multiplication that we can multiply these two groups of numbers together in any order we want. So we can drop the parentheses and just multiply all the numbers together. Now, I could multiply that moderately large swarm of twos together, but that would be tedious. So instead, I will simply count the number of twos and stick them all into the exponent. I mean, that's what an exponent is, multiplying a number by itself a certain number of times. Two to the third power times two to the fifth power is really the same as two to the eighth power. Cool. According to my trusty calculator, two to the eighth power is 256. Now, wait a second. That original expression two third times two the fifth, that equals two to the eighth. Two third times two to the fifth equals two to the eighth, isn't three plus five equal to eight? Yeah, and that's the rule for multiplying exponential expressions when the base numbers are the same to multiply exponential expressions, when the base numbers are the same. You just add up all the exponents. That's pretty easy, right. But what if we didn't know the base number? What if we had an expression like this? In this expression, the base is a variable. Now, these bases are the same variable. So we can simplify this using the same rule. Let's simplify this expression by just adding the exponents. X to the fourth times x to the six equals x to the tenth. We can now turn our attention to finding out x and getting the full answer if we are solving an equation or just leave it like this if x is a mystery. Either way, we've simplified the expression. Y. We've learned how to multiply exponential expressions when the base numbers are the same. You just add the exponents. Uh oh. Division. Rats. Wait. Don't worry. It's not that bad. We can do this. Let's say we have an expression like this. The rule here is so simple, it's almost silly. Spoiler alert, to divide exponential expressions when the base numbers are the same, just subtract the exponents. But it's important which exponent gets subtracted and which exponent gets subtracted from. Specifically, you'll subtract the exponent in the denominator from the exponent in the numerator. Isn't that great? To divide exponential expressions when the base numbers are the same, subtract the exponent in the denominator from the exponent in the numerator. To understand why this works, let's expand this expression. Using algebra, the three twos on the bottom cancel out three of the twos on top. Leaving us with five twos on the top or two to the fifth power. Piece of cake, right? And that's how you can multiply or divide exponential expressions with the same base. Just add the powers when multiplying and take the exponent in the numerator minus the exponent and the denominator when dividing. However, if the base numbers are not the same, then you can't multiply or divide the exponential expressions by adding or subtracting the exponents.