Powers, Indices, and Exponents Starring Exponents and the Three Rules
We’re gonna take a look at the logic and conventions around powers, indices, and exponents. In fact, they’re all the same thing; but depending on where you live, you’ll know them by one of those names. We’re also gonna talk about the three rules of working with powers: the addition rule, the subtraction rule, and the multiplication rule. So we have powers, indices, or exponents. But I’ll just call them powers in this video. They consist of a base number — here five is the base number — and a superscript, which is this small number just above it — called the power or index or exponent.
In simple cases, the power tells you how many times to write out the base number and then put multiplication signs between them and carry out the calculation. So our base of five is written out two times because the power is two. So write out the five twice, put the multiplication signs in between, and then carry out the calculations: five times five is equal to twenty-five. Now with three to the power of four, three is the base number; four is the power. So we’re gonna write out three four times, we’re gonna put multiplication signs between them, and then we’re going to carry out the calculation. So three times three is nine; three times three is nine. So we got nine times nine which is equal to eighty-one. So three to the power of four, we’ve written out three four times, multiplied them together, and got an answer of eighty-one.
It also works in exactly the same way with fractions: the base here is a half and the power is four. So we’ve written out a half four times, put our multiplication signs in between them, and I’m gonna multiply them together: one times one times one times one on the top is one, and two times two times two times two on the bottom — two, four, eight — is sixteen. So the answer is one over sixteen. Now it’s also worth mentioning that there’s a slightly different way to write this because we have one times one times one on the numerator, we can write one to the power of four on the numerator and we had two times two times two times two on the denominator, we can write two to the power of four on the denominator.
These two things are interchangeable and exactly the same thing. So you may have noticed I used a bracket around my original fraction and that tells us that’s it’s everything in the bracket that is to the power of four: the one on the top and the two on the bottom. If I’d had written my fraction like this: a half to the power of four, it would look more like it was just the one that was to the power of four and the two would not be included in that. This is why our top tip is use brackets to make things absolutely clear. If you want the whole fraction to be raised to the power of four, put it in a bracket to make it nice and clear.
Powers can also be applied to negative number bases, but again brackets are highly recommended for clarity. Try typing negative three to the power of two on your calculator like it’s written here. It’s a bit of a test of how accurate your calculator is. Technically, the power of two has a higher precedence than the negative sign. This means that the calculator should square the three first and then apply the negative sign afterwards to get an answer of minus nine; that’s the correct answer. If you wanted to square negative three, then you should put the negative three in brackets. Negative three to all to the power of two, this means negative three times negative three. And in this case, negative times negative makes a positive. So this answer is gonna be positive nine.
The format is very important; so use brackets for clarity. Now you also have to be quite careful with the negative signs and think about whether the power is odd or even. So let’s look at a few examples. Negative two all to the power of four: we’ve written our negative two out four times, we’ve put our multiplication signs in between, and now we’re going to do the calculation. So we can pair these up: negative two times negative two makes positive four; negative two times negative two also makes positive four. Now multiplying those together, positive four times positive four is equal to positive sixteen. So with even powers, our negatives paired up to make positive numbers. And when we multiplied all the positive numbers together, we got a positive answer.
Now let’s look at an odd power: negative two to the power of five. We’ve written negative two out five times and put the multiplication signs in between and now we’re gonna do the calculation. Again pairing these up, negative two times negative two is positive four; negative two times negative two is positive four. But now we’ve got one negative two left on its own. So with an odd power, we pair up the negatives to make positives. And this is gonna work all the way through, but we will be left with one negative number on the end. So we’re gonna have positive four times positive four times negative two. And in this case sixteen times negative two is negative thirty-two. So with an odd power, we’re going to get a negative answer.
So the general rule: if you have a negative base with an even power, the result will be positive because all of the negatives pair up to make positives when you multiply them together. If you have a negative base to an odd power, the result is going to be negative because all but one of the negative base numbers are going to pair up to cancel out to make positive numbers and then you’ll be left with one other negative number to multiply that by at the end. And the result will be negative.
So that’s the basic stuff you need to know about the most simple cases of powers. When the power is a positive integer, it corresponds to repeated multiplication of the base. So 𝑎 to the power of 𝑥 is 𝑎 times 𝑎 times 𝑎 times 𝑎 times 𝑎. And we write out the 𝑎 𝑥 times and then multiply them altogether. You do have to be a little bit careful about how you describe it though because it’s often said that the power tells you how many times you multiply the base by itself, but that’s not quite right. For example, with five to the power of two, you don’t multiply five by itself twice; you write down the number five twice and then you multiply those numbers together. So you multiply it just once. So this shorthand method of writing a number, multiply it by itself lots of times is a convention that’s been used by mathematicians for hundreds of years. And it’s even been extended from positive integer powers to negative powers and even fractional powers, but that’s a story for another day.
Now let’s look what happens when we multiply together two numbers in power form with the same base. We’ve got three to the power of two times three to the power of four. So we’ve got two threes written out here multiplied together and we’ve got four threes written out here multiplied together. So in total, we’ve got six of them. So three times three times three times three times three times three is three to the power of six. So what we did is we added these two and these four all to the page, so two plus four which gave us six of these threes multiplied together.
And let’s do another one: seven to the power of three times seven to the power of five. I’ve got three sevens multiplied together here and five sevens multiplied together here. If I multiply them altogether, I’ve got one great big long string of three plus five is eight sevens. So that is gonna be seven to the power of eight. It’s seven written down eight times and all of those multiplied together.
Now it’s very important that we had the same base; otherwise, you can’t just add the powers. For example, two to the power of three times three to the power of four. We’ve got three twos multiplied together and we’ve got four threes multiplied together, but we can’t really combine them; we can’t simplify this any further. It’s not two to the power of seven and it’s not three to the power of seven; it’s just two to the power of three times three to the power of four. So when you don’t have the same base, you can’t just add the powers. To describe this process, we could say when multiplying two numbers in power form with the same base, we simply add the powers or 𝑎 to the power of 𝑥 times 𝑎 to the power of 𝑦 gives us 𝑎 to the power of 𝑥 plus 𝑦.
Now let’s look at what happens when we divide two numbers in power form with the same base. We’ve got three to the power of four divided by three to the power of two. So that means three times three times three times three on the top — that’s three to the power of four — and three times three on the bottom — that’s our three squared. Now I can divide the top by three and I can divide the bottom by three, so I cancel out those two threes. I can divide the top by three and I can divide the bottom by three, so I cancel out those threes. But now there’s nothing else that I can cancel, so I’m just left with three times three on the top and one on the bottom. Well dividing by one, we can just leave that which means I’ve just left with three squared. So looking back at that we started off with, four threes all multiplied together on the top because we had three to the power of four and we effectively took away two of those — we crossed out two of those, cancelled two of those — because we had two on the denominator that we’re gonna cancel out. So we started off with four threes; we took away two of them, which left us with two.
One more quick example, five to the power of six divided by five to the power of three. We’ve got six fives on the top and three fives on the bottom. We can cancel out three of the sets of fives. So we can take this three away from this six, which leaves us with three left. So we’re subtracting the powers; start off with six fives, take away three fives leaves us with five to the power of three. To describe this process, we could say when dividing two numbers in power form with the same base, we simply subtract the powers. And we’ve got 𝑎 to the 𝑥 divided by 𝑎 to the 𝑦; we simply subtract the powers 𝑎 to the power of 𝑥 take away 𝑦. Again, it’s very important that you have to have the same base for this to work. If we had five to the power of three divided by seven to the power of four, we’ve got fives on the top, sevens on the bottom. Nothing cancels; we can’t simplify that. The powers can’t be subtracted in this case.
Lastly then, let’s take a look at what happens when we raise a base to a power and then we raise that whole number to another power. So how many twos are we ending up multiplying together here? Well, two to the power of three means we got three twos multiplied together. But because that whole thing is to the power of four, we’re gonna do that four times. So we’ve got four lots of three twos, so we’re gonna basically multiply those two powers together to give us two to the power of twelve.
Another quick example, five to the power of two to the power of seven. Well, five to the power of two is just two fives written out and multiplied together, and we’ve done that seven times; we take this whole bracket here and repeat it seven times — multiply it by itself seven times. So how many fives have I multiplied together in total? Well, it’s seven lots of two; that’s fourteen fives. So by multiplying the powers together, that gives me the answer that I’m looking for provided I’ve got the same base.
To describe that process, we could say when raising a base number to a power and then raising that result to another power, we simply multiply the powers together or 𝑎 to the power of 𝑥 all to the power of 𝑦 is equal to 𝑎 to the power of 𝑥 times 𝑦. We do need to be quite careful when writing this; so again brackets are king. Because if we just write 𝑎 to the 𝑥 to the 𝑦 like we have here, the convention is to work out the 𝑥 to the 𝑦 first and then raise 𝑎 to that power; that’s not the same as 𝑎 to the power of 𝑥 to the power of 𝑦. So for example, if we wrote two to the power of three to the power of four as we have here, that means two to the power of three to the power of four which is two to the power of eighty-one. And the answer is over two septillion; this is a number bigger than my calculator can handle. so I don’t know what all these digits are, but it’s basically a two with twenty-four other digits after it. But if I did two to the power of three, all to the power of four, that means two to the power of three is eight, eight to the power of four, and eight to the power of four is only four thousand and ninety-six. So clearly these two things are very, very different answers. So use brackets for clarity; otherwise, you could end up doing massively the wrong calculation.
So just a quick summary then, 𝑎 to the power of 𝑥: the 𝑎 is the base and the power there 𝑥 — sometimes called the power, sometimes called the index, and sometimes called the exponent. When we write our number in that format, we call it power form. When we do the actual calculation and come up with a number, we just call it an ordinary number. When we apply this to fractions, by using the bracket we indicate that the whole fraction needs to be raised to the power that we specify. And when we unpacked that, we can see that we’ve got the two to the power of three and the three to the power of three in this particular case. When we’re dealing with negative bases if we have an odd power, we’re going to end up with a negative answer; if we have an even power, we’re going to end up with a positive answer. And very finally, we learned about the addition rule where we add the powers, the subtraction rule where we subtract the powers, and the multiplication rule where we have to multiply the powers. And remember if in doubt, use brackets.