Why do exponents need to be the same

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An exponent is simply shorthand for multiplying that number of identical factors. One warning: Remember the order of operations. A negative exponent means to divide by that number of factors instead of multiplying. Instead, these expressions are evaluated using logarithms. If you memorize these three definitions, you can work everything else out by combining them and by counting:.

Let me make good on that promise, by showing you how all the other laws of exponents come from just the three definitions above. Suppose you have x 5 x 6 ; how do you simplify that? Why x 11? Five x factors from x 5 , and six x factors from x 6 , makes eleven x factors total.

Can you see that whenever you multiply any two powers of the same base, you end up with a number of factors equal to the total of the two powers? In other words, when the bases are the same , you find the new power by just adding the exponents :.

The rule above works only when multiplying powers of the same base. For instance,. Except in one case: If the bases are different but the exponents are the same , then you can combine them. And it works for any common power of two different bases:. What about dividing? Remember that dividing is just multiplying by 1-over-something. Then evaluate, using order of operations. In the example below, notice the how adding parentheses can change the outcome when you are simplifying terms with exponents.

The addition of parentheses made quite a difference! Parentheses allow you to apply an exponent to variables or numbers that are multiplied, divided, added, or subtracted to each other.

Whether to include a negative sign as part of a base or not often leads to confusion. To clarify whether a negative sign is applied before or after the exponent, here is an example. To evaluate 1 , you would apply the exponent to the three first, then apply the negative sign last, like this:. To evaluate 2 , you would apply the exponent to the 3 and the negative sign:.

The key to remembering this is to follow the order of operations. The first expression does not include parentheses so you would apply the exponent to the integer 3 first, then apply the negative sign.

The second expression includes parentheses, so hopefully you will remember that the negative sign also gets squared. In the next sections, you will learn how to simplify expressions that contain exponents. Come back to this page if you forget how to apply the order of operations to a term with exponents, or forget which is the base and which is the exponent!

In the following video you are provided with examples of evaluating exponential expressions for a given number.

Exponential notation was developed to write repeated multiplication more efficiently. There are times when it is easier or faster to leave the expressions in exponential notation when multiplying or dividing. What happens if you multiply two numbers in exponential form with the same base? Notice that 7 is the sum of the original two exponents, 3 and 4. And, once again, 8 is the sum of the original two exponents. This concept can be generalized in the following way:.

When you are reading mathematical rules, it is important to pay attention to the conditions on the rule. For example, when using the product rule, you may only apply it when the terms being multiplied have the same base and the exponents are integers.

When multiplying more complicated terms, multiply the coefficients and then multiply the variables. Therefore, you can only use this rule when the numbers inside the parentheses are being multiplied or divided, as we will see next.

What happens if you divide two numbers in exponential form with the same base? Consider the following expression. Notice that the exponent, 3, is the difference between the two exponents in the original expression, 5 and 2. When dividing terms that also contain coefficients, divide the coefficients and then divide variable powers with the same base by subtracting the exponents. Since the bases of the exponents are the same, you can apply the Quotient Rule. Divide the coefficients and subtract the exponents of matching variables.

In the following video we show another example of how to use the quotient rule to divide exponential expressions. Another word for exponent is power. In this section we will further expand our capabilities with exponents. We will learn what to do when a term with a power is raised to another power, and what to do when two numbers or variables are multiplied and both are raised to an exponent. We will also learn what to do when numbers or variables that are divided are raised to a power.

We will begin by raising powers to powers. It is the fourth power of 5 to the second power. This leads to another rule for exponents—the Power Rule for Exponents. To simplify a power of a power, you multiply the exponents, keeping the base the same. Take a moment to contrast how this is different from the product rule for exponents found on the previous page. Notice that the exponent is applied to each factor of 2 a. So, we can eliminate the middle steps. The product of two or more numbers raised to a power is equal to the product of each number raised to the same power.

How is this rule different from the power raised to a power rule? How is it different from the product rule for exponents on the previous page? Remember that quotient means divide.