Level 1 Exponents Basic Exponents
Level 1 Exponents
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- Welcome to the presentation on exponents.
- If I were to ask you what
- -- let me make sure this pen is the right width--
- if I were to ask you what two times three is,
- well I think at this point this should be pretty easy for you.
- That's the same thing as two plus two plus two, which equals six.
- You didn't have to do this.
- You all know that two times three is equal to six.
- So what we're going to do now is
- we're going to learn exponents which are the same thing
- that multiplication is to addition, exponents are to multiplication.
- I'll explain that in a second.
- I know that probably confused you.
- If I were to say what two to the third power is,
- instead of two plus two plus two,
- this is equal to two times two times two, which equals eight.
- Or if I were to say three the second power,
- that is equal to three times three.
- Remember, I said three times two would be three plus three.
- So the first three times three this equals nine,
- and three plus three equals six.
- The reason why I'm doing this is because there's always this temptation
- when you first learn exponents is to multiply.
- When I first learned it and I saw three to the second power or three squared,
- I'd always be like oh, that's six.
- But you also remember it's three times three,
- which equals nine.
- Let's do some more problems.
- If I were to tell you negative four squared.
- Once again, that's the same thing as
- negative four times negative four.
- Well, we all learn from multiplying negative numbers
- that a negative times a negative is a positive.
- And then four times four,
- well this equals positive sixteen.
- You don't have to write the positive,
- I'm just doing that for emphasis.
- If I would ask you what negative four to the third power is,
- well that equals negative four times negative four
- times negative four.
- Well we know already that negative four times negative four,
- that this equals sixteen, positive sixteen,
- and then we multiply that times negative four.
- And then that equals minus sixty-four.
- So something very interesting here to observe.
- When I took a negative number,
- and we call this the base,
- when the base is negative,
- in this case, negative four,
- and I raise it to an even power,
- I got a positive number, right?
- Negative four to an even power is positive sixteen,
- and when I took a negative number to an odd power, to three,
- I got a negative number.
- And that makes sense because every time
- you multiply by a negative number again,
- it switches signs.
- I'll show you the, I guess you'd call it
- the simplest example.
- Negative one to the one power is equal to negative one, right?
- Because that's just negative one times itself one time.
- And if I said negative one squared,
- well that's negative one times negative one,
- well that equals positive one.
- But if I said negative one to the third power,
- once again that's negative one times negative one times negative one.
- Well now this equals negative one times negative one is positive one
- times negative one equals negative one again.
- So I could tell you
- what negative one to the fifty-first power is.
- Because fifty-one is odd,
- we know that that is equal to negative one.
- If it was a fifty then it would be a positive one.
- Hope I didn't confuse you too much.
- Let's do a couple more problems.
- If I asked you what five to the third power is,
- well that will go into five times five times five,
- which equals one hundred and twenty-five.
- So really if I were ask to ask
- what negative five to the third power is,
- that would be negative five times negative five times negative five,
- which would be negative one hundred and twenty-five.
- Now one principle of exponents
- that might not seem completely intuitive to you at first
- is when I raise something to the zero power.
- So let's say I had two to the zero power.
- It turns out that anything to the zero power is equal to one.
- So two to the zero power is one,
- three to the zero power is equal to one,
- negative nine hundred to the zero power is equal to one.
- And let me see if I can give you a little bit of intuition of
- why that is actually the case.
- So if I were to ask you,
- let's do three to the fourth power.
- That equals three times three times three times three,
- which equals eighty-one.
- three to the third power is equal to twenty-seven,
- this three times three times three.
- three to second power is equal to nine.
- three to the first power is equal to three.
- Now we're going to say what's three to zero power?
- Well we already know, I already told you the rule,
- anything with the zero power is equal to one,
- but this will hopefully give you some intuition.
- When we went from the fourth power to the third power,
- we divided by three, right?
- eighty-one divided by three is twenty-seven.
- When you went from the third power to the second power,
- we divided by three.
- When we went from the second power from nine to three,
- we divided by three.
- So it kind of makes logical sense that
- when we go from the first power to the zero power,
- we'll just divide by three again.
- So three divided by three is one.
- Hopefully that gives you a little bit of an intuitive sense.
- You might want to replay that and think about why that is.
- And there's actually other aspects of exponents that
- why this also makes sense,
- why something to the zero power
- is equal to one.
- But let's just do some more problems in the time we have.
- I don't want to get you too confused.
- So if I were to ask you seven squared,
- well that's seven times seven, that's forty-nine.
- If I asked you negative six to the third power
- -- parentheses around here so you know it's a whole negative six to the third power--
- that equals negative six times negative six times negative six.
- Negative six times negative six is positive thirty-six times negative six,
- and that equals what?
- It's one hundred and eighty and thirty-six,
- that's minus two hundred and sixteen,
- if my mental math is correct.
- You could have actually multiplied it out.
- I think you're getting the point at this point.
- Oh, and another thing,
- if I told you zero to the hundredth power,
- well, that's pretty easy.
- That's zero times itself one hundred times,
- which is still equal to zero.
- If I were to ask you one to the thousandth power,
- well that's just equal to one, right?
- You can multiply one by itself as many times as you want
- and you're still going to get one.
- And remember, if I had negative one to the one thousandth power,
- well, this is an even exponent so you're still going to get one.
- If it was negative one to the one thousand and one,
- then it would be negative one.
- I think you remember why this is,
- because when you multiply a negative times itself an even number of times,
- the negatives cancel out.
- And then if you multiply it by a negative one more time
- it becomes a negative number again.
- Well let's just do some normal problems.
- I just want to make sure you get the basics of exponents down.
- If I were to tell you
- -- let me think of a good one--
- eight squared,
- that equals eight times eight equals sixty-four.
- If I were to tell you twenty-five squared,
- that's twenty-five times twenty-five, which equals six hundred and twenty-five.
- Powers of two is always very interesting.
- It's especially interesting
- if you one day go into computer science.
- So two to the fourth power,
- that's two times two times two times two.
- So two times two is four so this equals sixteen.
- And I did something very interesting here
- kind of on purpose.
- Notice that two to the fourth is equal to four times four, right?
- Because we did four times four here.
- I'm going to detail this more later on,
- but I want to think about what that means.
- Because four itself is the same thing as two squared.
- So we learned, just real fast,
- two to the fourth is the same thing
- as two squared times two squared.
- So I'll let you sit and think about that,
- but other than that I think you have the general idea
- of how basic exponents work,
- and I think you're ready to try the level one exponent module.
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