In this Series, Powers and Roots, we will begin by talking about rules of exponents. Check out the video, and remember, the transcript of the video is directly below!
Rules of Exponents: Multiplication
Now we’ll talk about powers and roots. In order to discuss the idea of an exponent let’s first think about multiplication. Multiplication is really a way of doing a whole lot of addition at once. So let’s think about this. If I were to ask you to add six fours together, no one in their right mind would sit there and add four plus four plus four plus four.
No one would do that. Of course what you would do is simply multiply four times six. It’s just important to keep in mind that in any act of multiplication, really what you’re doing is a whole lot of addition at once. Much in the same way exponents are a way of doing a whole lot of multiplication at once.
If I were to ask you to multiply seven 3’s together we wouldn’t write 3*3*3. We wouldn’t write up that long expression. Instead we would write 3^7th. Fundamentally, 3 to the seventh means that we multiply seven factors of 3 multiplied together. So it’s a very compact notation to express a lot of multiplication at once.
And I hasten to add the test will not expect you to compute that value. It’s not gonna be a test question, calculate 3 to the seventh. That’s not gonna be on the test. But, you’ll have to handle that quantity in relation to other quantities. For example, use the rules of exponents to figure out 3 to the seventh, and that whole thing squared, or multiplying by three to the fifth, or dividing it by something.
You have to use it, but you’re not gonna have to calculate its value. Symbolically we could say that b to the n means that n factors of b are multiplied together.
Fundamental Definition of Exponent
So this is the fundamental definition of what an exponent is. And right now I’ll just say b is the base, n is the exponent, and b to the n is the power.
Now this is a good definition for now, but as we’ll see this definition is ultimately somewhat naive. And we’re gonna have to expand it in later modules. And why is it naive? Well, if you think about it, how many factors of b that are multiplied together? This means that n is a counting number.
That is to say, it is a positive integer. And so this definition, this way of thinking about exponents, is perfectly good as long as the exponents are positive integers. But as we will see in upcoming modules, there are all kinds of exponents that are not positive integers. We’ll talk about negative exponents and fraction exponents, all that.
Rules of Exponents: Positive Integers
Let’s not worry about that in this module. In this module, we’ll just stick with the positive integers, so we can stick with this very intuitive definition of what an exponent is. First of all, notice that we can give exponents to either numbers or variables. We have already seen variables with powers in the algebra module, especially in the videos on quadratics where you have x squared.
Notice that we can read that expression either as 7 to the power of eight, or 7 to the eighth. Either one of those is perfectly correct. Notice that we have a different way of talking about exponents of 2 or 3. Something to the power of 2 is squared, and something to the power of 3 is cubed. So where we would rarely say something to the power of 3, and we would never say something to the power of 2, that just sounds awkward.
We would always say that thing’s squared. If one is the base, then the exponent doesn’t matter. One to any power is one. And in fact, that expression one to the n equals one, that works for all n. That’s not restricted to positive integers. That actually works for every single number on the number line, so every single number on the number line, if you put it in for n, 1 to the n equals 1.
Important to Remember
So that’s an important thing to remember. If 0’s the base then 0 to any positive exponent is 0. So 0 to the n equals 0 as long as 0 is positive. And in fact this is true not only to positive integers, it’s also true of positive fractions. It’s true of everything to the right of zero on the number line.
So don’t worry about zero to the power of zero or zero to the power of negatives. You will not have to deal with that on the test. That gets into either illegal mathematics or other forms of mathematics that we don’t need to worry about. That’s just something we can ignore. An idea we have already discussed in the Integer Properties and Algebra lessons, if an exponent is not written, we can assume that the exponent is one.
We talked a little about this in prime factorizations and we talked about this again in the algebra module. Another way to say that is any base of the power of 1 means that we have only one factor of that base. So 2 to the 1 is 2. 2 squared is 4, 2 cubed is 3 factors, so that’s 8.
So again we’re using the exponent as a way to count the number of factors we have in the total product.
When the Base Is Negative
What happens if the base is negative? What if we start raising a negative number to powers? Well, -2 to the one would of course be -2. -2 squared, that’s negative times negative, that would be +4.
If we multiply another factor of -2, positive times negative gives us a -8. Multiply another factor of 2, we get -8 times -2 = +16. Multiply another factor of 2 we get -32. And notice we have, it’s kind of an alternating pattern here. We’re going from negative to positive, negative to positive, negative to positive.
So we get a negative to any even power is a positive number, and a negative to any odd power is negative. We’ll talk more about this in the next video. This has implications for solving algebraic equations. For example, the equation x squared equals 4 has two solutions, x = 2 and x = -2, because either of those squared equals 4.
By contrast, the equation x cubed equals 8 has only one solution, x = +2. If we cube +2, we get +8, but if we cube -2 we get -8. Notice also that an equation of the form something squared equals a negative has no solution. So for example, (x- 1) squared equals -4, well, there’s no way that we can square anything and get -4.
So that’s an equation that has no solution. But we could have something cubed equals a negative, that’s perfectly fine. If something cubed equals negative 1, then that thing must equal negative 1, and then we can solve for x. Finally, just as it is important to know your times tables, so it is important to know some of the basic powers of single-digit numbers.
Rules of Exponents: Recommendations for Now
So here’s what I’m gonna recommend memorizing and knowing. And it’s helpful actually to multiply these out step by steps to help you remember them. First of all, I’ll recommend knowing the powers of two up to at least 2 to the ninth. And why all the way up to 2 to the ninth? Well, we’ll be talking about this more when we talk about some of the rules for exponents.
But again, very good actually to practice once in a while, just keep on multiplying by 2 and get all these numbers just so that you verify for yourself where they come from. Know the powers of 3 up to at least 3 to the fourth. The powers of 4 up to the 4th, the powers of 5 up to the 4th. Again, multiply all these out from time to time, just to remind yourself of all these so that you really can remember them very well.
And then you should know, of course, the squares and the cubes of everything from 6 to 9. And why would you need to know all these? Well, again, we’ll talk about these more when we talk about some of the rules of exponents. And of course, know all the powers of 10.
That was discussed in the Multiples of 10 lesson. It’s very easy to figure out powers of 10, you’re just adding zeros or for negative powers you’re putting it behind the decimal point. Fundamentally, b to the n means n factors of b multiplied together. That is the fundamental definition of an exponent. And it’s very good, as we move through the laws of exponents, to keep in mind that fundamental definition of an exponent.
1e to any power is 1, 0 to any positive power is 0. A negative to an even power is positive, and negative to an odd power is odd. An equation with an expression to an even power equal to a negative has no solution, but an odd power can equal a negative. And finally, know the basic powers of the single-digit numbers.