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 4's together, no one in the right mind would sit there and add, 4+4+4+4+4+4.

No one would do that, of course, what you would do is simply multiply, 4 times 6. Is just important to keep in mind that in any act of multiplication, really what you're doing is a whole lot of addition that one. 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 times 3 times 3, we wouldn't write out that long expression, instead, we would write 3 to the seventh.

Fundamentally, 3 to the seventh means, that we multiply seven factors of 3 multiplied together. So is a very compact notation to express a lot of multiplication at once. Now, I hasten to add the test will not expect you to compute that value. It's not going to be a test question calculate 3 to the seventh, that's not going to be on the test.

But you will have to handle that quantity in relation to other quantities. For example, use the laws of exponents to figure out 3 to the seventh, and that whole thing squared, or multiplying it by 3 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.

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, 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, infraction, exponents, all that.

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 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 squared.

If 1 is the base, then the exponent doesn't matter, 1 to any power, is 1, and in fact that expression, 1 to the n equals 1, 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, so that's an important thing to remember. If zero is the base, then zero to any positive exponent is zero, so zero to the n equals zero as long as n was positive.

And in fact, this is true not only a 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 this on the test that gets into either illegal mathematics or other forms of mathematics that we don't need to worry about.

So that's just going to be 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 1. We talked a little bit about this in prime factorization, 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 one is 2, 2 squared is 4, 2 cubed is three factors, that's eight. So again, we're using the exponent as a way to count the number of factors we have in the total product. What happens if the base is negative? What if we start raising a negative number to powers? Well, negative 2 to the 1 of course will be negative 2.

Negative 2 squared, that's negative times negative that would be positive 4. If we multiply another factor of negative 2, positive times negative gives us a negative 8. Multiply another factor 2, we get negative 8 times negative 2 gives us positive 16. Multiply another factor 2, we get negative 32. And notice we have 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 equals 2 and x equals negative 2, because either those squared equals four. By contrast the equation x cubed equals 8 has only one solution, x equals positive 2. If we cube +2, we get +8, but if we -2, we get -8. Notice also that equation of the form something squared equals a negative has no solution.

So for example, (x-1) squared = -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. So here's what I'm going to 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 2 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 fourth, the powers of 5 up to the fourth, again, multiply all these out from time to time just to remind yourself of all this so that you really can remember them very well.

And then you should know, of course, the squares and the cubes of everything from six to nine. 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.

And 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.

One to any power is one. Zero to any positive power is zero. A negative to an even power is positive, a negative to an odd power is negative. 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.

Read full transcriptNo one would do that, of course, what you would do is simply multiply, 4 times 6. Is just important to keep in mind that in any act of multiplication, really what you're doing is a whole lot of addition that one. 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 times 3 times 3, we wouldn't write out that long expression, instead, we would write 3 to the seventh.

Fundamentally, 3 to the seventh means, that we multiply seven factors of 3 multiplied together. So is a very compact notation to express a lot of multiplication at once. Now, I hasten to add the test will not expect you to compute that value. It's not going to be a test question calculate 3 to the seventh, that's not going to be on the test.

But you will have to handle that quantity in relation to other quantities. For example, use the laws of exponents to figure out 3 to the seventh, and that whole thing squared, or multiplying it by 3 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.

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, 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, infraction, exponents, all that.

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 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 squared.

If 1 is the base, then the exponent doesn't matter, 1 to any power, is 1, and in fact that expression, 1 to the n equals 1, 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, so that's an important thing to remember. If zero is the base, then zero to any positive exponent is zero, so zero to the n equals zero as long as n was positive.

And in fact, this is true not only a 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 this on the test that gets into either illegal mathematics or other forms of mathematics that we don't need to worry about.

So that's just going to be 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 1. We talked a little bit about this in prime factorization, 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 one is 2, 2 squared is 4, 2 cubed is three factors, that's eight. So again, we're using the exponent as a way to count the number of factors we have in the total product. What happens if the base is negative? What if we start raising a negative number to powers? Well, negative 2 to the 1 of course will be negative 2.

Negative 2 squared, that's negative times negative that would be positive 4. If we multiply another factor of negative 2, positive times negative gives us a negative 8. Multiply another factor 2, we get negative 8 times negative 2 gives us positive 16. Multiply another factor 2, we get negative 32. And notice we have 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 equals 2 and x equals negative 2, because either those squared equals four. By contrast the equation x cubed equals 8 has only one solution, x equals positive 2. If we cube +2, we get +8, but if we -2, we get -8. Notice also that equation of the form something squared equals a negative has no solution.

So for example, (x-1) squared = -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. So here's what I'm going to 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 2 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 fourth, the powers of 5 up to the fourth, again, multiply all these out from time to time just to remind yourself of all this so that you really can remember them very well.

And then you should know, of course, the squares and the cubes of everything from six to nine. 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.

And 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.

One to any power is one. Zero to any positive power is zero. A negative to an even power is positive, a negative to an odd power is negative. 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.