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Lesson by
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Mike McGarry
**

Magoosh Expert

Magoosh Expert

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 4s together, no one in their right mind would sit there and add 4 plus 4 plus 4 plus 4.

No one would do that. Of course, what you would do is simply multiply 4 times 6. 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 times 3 times 3, we wouldn't write out that long expression. Instead, we would write 3 to the 7th. Fundamentally, 3 to the 7th 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 7th, 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 laws of exponents to figure out 3 to the 7th and that whole thing squared or multiplying it by 3 to the 5th 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 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. 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 8 or 7 to the 8th.

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 0 is 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 of positive integers, it's also true of positive fractions. It's true of everything to the right of 0 on the number line. So don't worry about 0 to the power of 0 or 0 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 gonna 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 about this in prime factorizations, and we talked about this again in the algebra module. Another way to say that is, that any base to the power of 1 means that we have only factor of that base.

So 2 to the 1 is 2. 2 squared is 4. 2 cubed is 3 factor, so that's 8. So again, we're, 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 of 2 we get negative 8 times negative 2 gives us positive 16.

Multiply another factor of 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 of those squared equals 4. By contrast, the equation x cubed equals 8 has only one solution, x equals positive 2.

If we cube positive 2, we get positive 8. But if we cube negative 2, we get negative 8. Notice also that an equation of the form something squared equals a negative has no solution. So, for example, x minus 1 squared equals negative 4. Well, there's no way that we can square anything and get negative 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 than 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 gonna recommend memorizing and knowing. And it, it's helpful, actually, to multiply these out step-by-step, to help you remember them. First of all, I'll recommend knowing the powers of 2 up to at least 2 to the 9th. And why all the way up to 2 to the 9th? 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 4th, 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, 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 was very easy to figure out powers of 10, you're just adding zeros or, 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.

1 to any power is 1. 0 to any positive power is 0. A negative to an, 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.

Show TranscriptNo one would do that. Of course, what you would do is simply multiply 4 times 6. 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 times 3 times 3, we wouldn't write out that long expression. Instead, we would write 3 to the 7th. Fundamentally, 3 to the 7th 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 7th, 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 laws of exponents to figure out 3 to the 7th and that whole thing squared or multiplying it by 3 to the 5th 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 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. 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 8 or 7 to the 8th.

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 0 is 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 of positive integers, it's also true of positive fractions. It's true of everything to the right of 0 on the number line. So don't worry about 0 to the power of 0 or 0 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 gonna 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 about this in prime factorizations, and we talked about this again in the algebra module. Another way to say that is, that any base to the power of 1 means that we have only factor of that base.

So 2 to the 1 is 2. 2 squared is 4. 2 cubed is 3 factor, so that's 8. So again, we're, 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 of 2 we get negative 8 times negative 2 gives us positive 16.

Multiply another factor of 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 of those squared equals 4. By contrast, the equation x cubed equals 8 has only one solution, x equals positive 2.

If we cube positive 2, we get positive 8. But if we cube negative 2, we get negative 8. Notice also that an equation of the form something squared equals a negative has no solution. So, for example, x minus 1 squared equals negative 4. Well, there's no way that we can square anything and get negative 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 than 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 gonna recommend memorizing and knowing. And it, it's helpful, actually, to multiply these out step-by-step, to help you remember them. First of all, I'll recommend knowing the powers of 2 up to at least 2 to the 9th. And why all the way up to 2 to the 9th? 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 4th, 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, 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 was very easy to figure out powers of 10, you're just adding zeros or, 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.

1 to any power is 1. 0 to any positive power is 0. A negative to an, 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.