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 4+4+4+4. Read full transcript
No one would do that. Of course, what you would do is simply multiply 4x6. 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 three's together, we wouldn't write 3x3x3. We wouldn't write out that long expression instead we would write three to the seventh. Fundamentally, three to the seventh means that we multiply seven factors of three multiplied together. So, it's a very compact notation to express a lot of multiplication at once.
I hasten to add, the test will not expect you to compute that value. It's not gonna be a test question, calculate three 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 laws of exponents to figure out three 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. 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 for 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've already seen variables with powers in the Algebra modules, especially in the videos on quadratics, where you have x squared. Notice that we can read that expression either as seven to the power of eight or seven to the eighth.
Either one of those is perfectly correct. Notice that we have a different way of talking about exponents of two or three. Something to the power of two is squared and something to the power of three is cubed. So we would rarely say, something to the power of three, and we would never something say something to the power of two.
That just sounds awkward. We would always say that thing 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, one to the n equals one. 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 zero 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 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 gonna be something we can ignore. An idea we have already discussed in the integer properties in 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 to the power of one means that we have only one factor of that base, so two to the one is two. 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.
What happens if the base is negative? What if we start raising a negative number to powers? Well, negative two to the one, of course, would be negative two. Negative two squared, that's negative times a negative, that would be positive four. If we multiply another factor of negative two, positive times negative gives us a negative eight.
Multiply another factor of 2, we get negative 8 times negative 2 gives us positive 16. Multiple 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 implication for solving algebraic equations. For example, the equation x squared equals four has two solutions, x equals two and x equals negative two, because either of those squared equals four.
By contrast, the equation x cubed equals eight has only one solution x equals positive two. If we cube positive two, we get positive eight, but if we cube negative two, we get negative eight. Notice also that an equation of the form something squared equals a negative has no solution.
So, for example, x minus one squared equals negative four. Well, there's no way that we can square anything and get negative four. 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 one than that thing must equal negative one 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'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 two to the ninth.
And why all the way up to two 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 awhile. Just keep on multiplying by two and get all these numbers, just so that you verify for yourself where they come from.
Know the powers of three up to at least three to the fourth. The powers of four up to the fourth, the powers of five up to the fourth. 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. 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 ten, and that was discussed in the multiples at ten lesson is very easy to figure out powers of ten here. 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 odd. An equation with an expression with to an even power equal to a negative is no solution, but an odd power can equal a negative. And finally, know the basic powers of the single digit numbers.