Okay today, we are going to continue our discussion in Algebra on the topic of exponents and the order of operations, so let us dive right in to it and start talking about exponents.
Take a look at something like this, four to the power of two. This is also known as four squared. And, basically, the only thing you have to note to the code how to do this thing is the number on the top indicates how many times I multiplied the bottom number together. So, just go on a through a few quick examples, four squared is simply four times four, four to the third is four times four times four, 59 to the fourth is 59 times 59 times 59. All this is a shorthand way of compacting this down. We just write four squared and four times four, four to the third is four times four times four. Make sure that you do not say that this is equal to four times two. This is not equal to one another. And to see that, let us go ahead and just do that real fast, so it raised these other examples and proceed.
Four times four is 16, four times two is eight, and you can see that those are definitely not equal to each other, so this is a simply a shorthand way of writing four times four. Do not get into the habit of thinking as four times two. You may have to force yourself to think twice that make sure that it is that way. And, some people might say as well “who cares about writing it like this, I will just write it out all the time” well see, pretty soon that you are not going to be able to do that. The equation will get just massive unless you start to write in exponential form.
Okay, let us take another example, something a little different but very similar. Six squared, okay, that is simply equal to six times six, which is equal to 36, that is very simple as long as you know how to do it.
Let us do something a little bit different this time. One over 10 to the fourth, it is a fraction right? And you are like ‘Oh! My goodness what do I do with that?” Well, you just write it out, you got one over 10 times one over 10 times one over 10 times one over 10 okay. Whatever is inside the parenthesis, notice that the entire body of whatever you have written inside the parenthesis, the entire thing is raised to the fourth power so whatever is in here is multiplied four times, and so we have introduce the parenthesis here. All you do is you treat whatever is inside there as a complete unit as a single thing that you cannot break it part and you multiply it together.
So, what is this equal to over here, one times one times one times one is one, remember we multiply the top of fractions and we multiply the bottom of the fractions. 10 times 10 times 10 times 10, you do that in your calculator, you will write that as 10, 000. Okay, that is the answer. Now, one quick short way of doing this. You can write this all out, no problem, but I tell you another thing about fractions. When you are raising fractions to a power like this, all you need to do, you can raise the top part of the fraction independently of the bottom. So you can also write this as one to the fourth power over 10 to the fourth power, and you just know that you know one to the fourth is one times one times one times one. That is one. 10 to the fourth is 10 times 10 times 10 times 10. So you get the same thing 10,000. And really, we have done exactly same thing, nothing is even different here. It is just in one case, we have explicitly written all the fractions out and multiply it in the other. I can just tell you that you can raise top to the fourth power and the bottoms of the fourth power, and you do not really have to extend all of it out explicitly unless you just really want to.
Okay, let us do another example of how to deal with fractions, and exponents, and they will be on our way, like I said the important thing with fractions raising them with exponents is just make sure that you do the top and the bottom. You can raise the top and the bottom of the fraction dependently you just your keep your fraction bar there, and that is what is easy about fractions. Fractions are really are—they are difficult.
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