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Equivalent forms of exponential expressions

Sal rewrites (1/32)*2^t as 32*1024^(t/10-1).

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  • blobby green style avatar for user lew.lehmann
    Look, I've come a long way in my maths learning thanks to Sal, but this particular module on exponents is honestly pretty bad. The practice questions involve techniques not covered in the videos, and the videos gloss over important details.

    The only reason I've been able to follow up until this point is because I recently did a small course on exponents, and learned the properties of exponents formulas. Here, the formulas aren't explicitly explained. They're just mentioned as side notes. I found I developed far better intuition when starting with the properties, then going into examples of why they work the way they do.

    Perhaps it's just my learning style, but I found this module to be badly thought out.
    (75 votes)
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  • aqualine seed style avatar for user Marc Joe
    at what does "hairy" mean? does it mean complicated?
    (8 votes)
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  • piceratops ultimate style avatar for user M. McManus
    Can I get some help with ((2/3)^x+4) -(2/3)^x? This is one of the exercise problems, and I don't quite understand the hint. I understand everything up to the point where 2/3^x * 15/16 -2/3^x. The next hint shows (2/3^x) * (15/16-1). I don't know where the -1 came from nor what happened to the other 2/3^x. I understand that you can factor a -1 out of -2/3^x.
    (15 votes)
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    • aqualine sapling style avatar for user Pancake
      Here is why 2/3^x * 15/16 -2/3^x becomes (2/3^x) * (15/16-1):

      Let's use a different example. Look at the expression (9*3)-9 (which equals 18). Notice, we multiply 9 by 3 to get a total of three "nines" (which equal 27), then subtract a nine to get only two "nines" (which equal 18). Nine times two equals eighteen. Thus, (9*3)-9 is the same thing as 9*(3-1), or 9*2.

      In the example above, we have 15/16 "2/3^x"s and are subtracting one "2/3^x". Thus, the expression is equal to (2/3^x) * ((15/16)-1). I hope I explained it well!
      (2 votes)
  • primosaur ultimate style avatar for user Malek
    i've been doing these practice problems for over a combined 8 hours and still cant get more than 2 out of 4 questions right.. I can usually almost complete a full unit in this timeframe.. I never know when to use what.. how can i get better?
    (12 votes)
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  • blobby green style avatar for user avery3024
    hello, i am very confused on this practice section. I don't understand how you find which expressions are similar. ive watched the video like 20 times but it doesnt cover what is asked in the practice.
    (3 votes)
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  • duskpin tree style avatar for user Cynthis Lu
    At , I'm confused at why 1034 is over 32. How did that happen?
    (5 votes)
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    • boggle purple style avatar for user lily J
      First, it's 1/32*(1024^(t/10-1))*1024, and sal said that he want it in the A*B^x form. So he want to multiply 1/32 and 1024 together to form A then multiply (1024^(t/10-1)) to get that A*B^x form.

      And 1/32*1024 = 1024/32

      Hop that helps!
      (2 votes)
  • blobby green style avatar for user Akshay
    Wait... isn't a^1 the same as if it was a^(-1)? Just a random thought while watching this video. Logically it makes sense. The root of one is the number itself as the regular exponent, thus it sould be the same.
    (2 votes)
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  • blobby green style avatar for user Maple
    why is this needed?
    (3 votes)
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  • aqualine ultimate style avatar for user veniamin.zaloginov
    what grade level is this?
    (2 votes)
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  • marcimus pink style avatar for user Tyshuana White
    How would you write -3^2 in expanded form?
    How would you write (-3)^2 in expanded form?
    (1 vote)
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    • duskpin sapling style avatar for user Iris Nogueroles Langa
      Because of the order of operations BEDMAS (or whichever letters you have been taught), you need to do the brackets first and then the exponents.

      For -3^2, there are no brackets so the exponent has priority. That means you do 3^2 first and then add the minus. This gives an answer of -9. If you were to add brackets, this would have been written as -(3^2).

      For (-3)^2, you need to realise that there are brackets. This shows that the entire -3 (including the minus sign) is being squared. When you square a number, you multiply it by itself. (-3)x(-3) is equal to 9 because multiplying a negative number with another negative number always gives a positive number.

      Therefore, in expanded form:
      -3^2 = -(3x3) = -9
      (-3)^2 = (-3)x(-3) = 9

      This is a great example to illustrate the importance of adding brackets to avoid getting different answers! :)

      Hope that helps!
      (4 votes)

Video transcript

- [Voiceover] What I hope to do in this video is, start with a exponential expression that's in a fairly straightforward form and then turn it into one in a hairier form. And let's actually just do it. I'll show the initial expression and then the form that I want it in, or that we want it in. And then we can talk a little bit about, why would ever actually wanna do that? So let's say my expression is 1/32 times two to the t power. So this is fairly straightforward exponential expression. But let's say we want it, we want it in the form A time B, and this is where it's gonna get hairy, A time B to the t over 10 power, minus one. And so, you're probably immediately saying, "Why would I ever wanna take something "nice and simple like this "and turn it into this beastly thing right over here?" And the answer is, when you get into higher math and you start doing your physics and your chemistry, you're going to see, maybe, you know, you got a result like this, but then you look in your textbook, or your professor, it has a result like this. Just wanna know, "How do I transfer from this to this?" Or actually, sometimes when you transfer to a form like this ... Obviously, I've just arbitrarily written this form here. But sometimes when you write it in another form that might even be a little hairier, it can give you an intuition on the underlying processes that that expression is trying to describe. So if you can take that on a leap of faith, let's actually try to do it. And, at a minimum, it's gonna make you a lot better at exponent properties. So see if you can rewrite this in this form. So I'm assuming you took a go at it. So let's try to do it together. So the first thing, the first thing that I might wanna do is, well, let's see if we can ... Let's see. What would I wanna do? The first thing I wanna do is, take this t and get it into a t ... let's get it into a t over 10. So, to do that, we essentially just need to multiply by 10 and divide by 10. So let's multiply by 10, and then also divide by 10. Then we haven't changed the value up here. So we can rewrite this, we can rewrite this as 1/32 times two to the ... So let me circle t ... Do this in a different color. T over 10, t over 10 times 10. Times 10. All right. So we got a t over 10 over here. But then I have this times 10. So how do I deal with this? Well, one thing that I can do ... Let me actually just write this the other way around. Let me write it as, let me write it as 10 times t over 10. 10 times t over 10. So hopefully what I just did here isn't a huge stretch here. I just literally multiplied and divide by, divided by 10, times this t over 10. But when I write it this way, an exponent property might jump out at you. If I have, if I have a to the b, and then I raise that to the c, that's going to be a to the bc. Or, another way around, a to the bc is going to be a to the b to the c. And so, this piece right over here, I can rewrite it as two to the 10th, and then raise that to the t over 10 power. To the t over 10 power. Once again, two to the 10th and then raise that to the t over 10, that's gonna be the same thing as two to the 10 times t over 10. And of course, we still have the one over 32 over here. One over, one over 32. I'm tempted to write that as two to the negative fifth power, but I won't do that just yet. So let's see. What's two to the 10th power? Actually, let's just, let's just keep it, let's just keep it as two to the 10th power, just for simplicity right now. Later we can, you might know that that's gonna be 1,024. But let's just, let's see what else we can do. So we know this is going to be some num- ... Actually, let me just write out as 1,024. So we have one over 32 times 1,024 to the t over 10, to the t over 10 power. So it seems like we're getting close. If there was no minus one here, we're essentially done. But now there's this minus one. So how do we deal with that? Well, we can do a similar type of strategy. We can subtract one, and then we could add, and then we could add one. Then we're not actually changing the value. Just as we multiplied by 10 and divide by 10, we're not changing the value up here. If you subtract one and add one to the exponent, you're not changing its value. And so, what is this going to be? We wanna leave this minus one here. But we wanna get rid of, we wanna get rid of this plus one somehow. And here, we just have to remind ourselves that, if we have a to the b times a to the c, that's going to be equal to a to the b plus c. If you have the same base, multiplied, same base raised to different exponents and you multiply them, you could just add the exponents. And so you could also go the other way around. If you have a to the b plus c, you could break it up into a to the b times a to the c. So, this business right over here, this business right over here, this is 1,024 to the t over 10 minus one, plus one. So we can break this up as, we can break this up as, 1,024, 1,024 to the t over 10 minus one, that's this part here, and then times 1,024 to the one. Times ... Let me make this in a different color. So, let's see green. So this right over here. So times 1,024 to the one power. That's this one power right over here. And of course, we still have the one over, the one over 32. All right. So now we're really close. We have the 1,024 to the t over 10 minus one, we have t over 10 minus 1. And now we just have to simplify. We can rewrite. This is going to be equal to, this is going to be equal to, we can just bring the 1,024, 1,024 to the first power, that's just 1,024. So that's going to be 1,024 over ... I can put all these commas here if I like. 1,024 over this 32. Let me do that magenta color. Over this 32 times ... home stretch. Times, times 1,024 to the t over, to the t over 10 minus one power. And now we could just simplify this. You might recognize 1,024, we already saw, that was the same thing as two to the 10th power. 32 is the same thing as two to the fifth power. So two to the 10th divided by two to the fifth ... Actually, this is another exponent property at play here, although you could just divide the numbers. If you have a to the b over a to the c, this is going to be equal to a to the b minus c. So this is going to be two to the 10 minus five, over this whole thing. This whole thing right over here. And this thing ... See, I wanted to do that in a different color ... This thing is just going to be two to the fifth power, or 32. So this is going to be 32 times 1,024. We were in the home stretch here. 1,024 to the t over 10 minus one. So once again, normally in our lives, we like to make things simpler, and I'm a big advocate of that. It's a good life philosophy. But this is a case where we really did make it more complicated. We started with 1/32 times two to the t ... Actually, we could have ... Well, there's other ways we could have written that ... And we turned it into this thing with this somewhat hairier exponent. But it's a useful skill to have because you might get a result like what we originally started with, and then someone else might get a result like this, and it's very important to realize, "Hey, you actually go the same result. "They're just different ways of expressing "the same exponential expression."