The Dreaded Exponent
Exponents have long been the bane of many students. And that should come as no surprise—after all, there are negative exponents, fractions as exponents, and the terrible sounding exponents of exponents. Feeling comfortable with exponents in all their various (and nefarious) guises will definitely help you test day. But, for right now, let’s stick to the fundamentals of exponents.
Meet the Base
The base is the number underneath the exponent. In , two is the base and three is the exponent. means that you are multiplying (the base) three times: .
That’s the easy part. Now, what about when we are combining bases? Take a look:
The rule is as follows: if the bases are the same and you are multiplying them, add the exponents. So . That is I keep the base the same (I don’t multiply it) and then I add the exponents.
What about ? Well, the bases are different so we cannot combine them as we did before. In this case, you would just have to multiply the long way, .
There is one exception to this different base rule: if the exponents are the same but the base is different, you can multiply the bases. In , the bases are different but both are to the fifth power. In this case, we keep the exponent the same and we multiply the bases. Therefore, .
Let’s try a few more examples:
What about when you are adding similar bases?
Can we add the and the ? The answer is an unequivocal no. The only way you can change the above number is by factoring: . This last step is relatively advanced, and I wouldn’t worry about it too much unless you are going for a high score. Just remember, if you are adding the bases—whether the same or different—then you cannot add the exponents.
Finally, what happens when you take an exponent to an exponent?
In this case, you multiply the exponents while keeping the base, 4, the same.
Therefore, not .
Think you got it? Okay see if you can answer the next three questions correctly!
- Cannot mix bases; leave as is