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Intro to exponents

How do we interpret exponents? 2 to the 3rd power (2^3) is the same as multiplying three 2's together: 2 x 2 x 2 = 8. So, in this case, the exponent (3) tells you how many times to multiply the base number (2) by itself. Created by Sal Khan.

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  • hopper cool style avatar for user ABHISAB
    can you have an exponent that has a decimal or fraction like 5^4.8?
    (51 votes)
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  • blobby green style avatar for user aguirre33618
    what is 23 to the 42 power
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  • blobby green style avatar for user scarlett
    when your multiplying like 2 x 2 x 2 do u always have to put the dot for multiplication in that situation?
    (6 votes)
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  • starky seedling style avatar for user senpai Alphyy
    I need help with finding the answer was g and the family had it but the only one in your room with me it will work it would help you in any way please the attached document as per my conversation in any way possible in any way please you have the attached document as per my conversation you in anyway possible the family are the attached resume as you are able it but I'm in any way please it would take for me in your room or in any way possible it would have taken my meds to it would help you are for exponent 100.
    (6 votes)
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  • blobby green style avatar for user Maria Soriano
    How do I divide exsponents I don't get it
    (5 votes)
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  • aqualine ultimate style avatar for user Liny Tripathy
    Can I do any number to the power of a negative number?
    Like, 3^ -2 ??
    (6 votes)
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  • aqualine ultimate style avatar for user ashaaz.sharif
    When will you teach about square roots?
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  • leaf blue style avatar for user mendoncasarah43
    How is exponents like basically than Addition, Subtraction, Multiplication, and Division?
    (2 votes)
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    • leaf red style avatar for user Ralph Schraven
      Important to realize is that exponentiation is not simply repeated multiplication. Otherwise, what is 3^1.258? I can't multiply something 1.258 times, but this definitely has an answer! Similarly, if multiplication is repeated addition, then what is 6.2*5.5? I can't add a number half a time! If you think you can simply add half of 3.2 the last time, then let me ask you this. What would pi*pi be? Now, you have an infinite amount of decimals to worry about! You can never do this algebraically in terms of addition!

      Multiplication is perhaps closer related to computing the area of a surface or (potentially hyperdimensional) volume based on its sides. Now pi * pi is the area of a surface formed by edges whose length is a rolled-out perimeter of a circle with unit diameter. There, we can understand pi * pi now!

      Exponents are harder to understand in a simple, conceptual way. We can understand rational numbers quite easily using roots, but that leaves the question of irrational reals in between the rationals.

      Although you could just write this off as voodoo-stuff that churns out a somewhat sensible
      solution (perhaps using some higher-level math such as making sure the differentials all make sense). But
      we usually define the value of raising bases to real-number or even complex-number exponents using the natural exponent using the fact that e^(ln(x)) = x which means x^z = (e^ln(x))^z = e^(ln(x)*z). Why this is convenient will become clear later when you work more with e and complex numbers!

      Conclusion? Maths is fun, and it can be "easy" when you look for functional shortcuts such as "exponentiation is repeated multiplication, which is repeated addition". But the part where it gets "hard", although it requires a lot of effort to understand all of it, is much more rich, rewarding, and unveils a lot about how maths works at a higher level!
      (3 votes)
  • marcimus pink style avatar for user Amber Spence
    when you have very big numbers like:220 is it still possible that you multiply
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  • leafers sapling style avatar for user Trees2006
    () What are powers? I'm really confused...
    (2 votes)
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Video transcript

You already know that we can view multiplication as repeated addition. So, if we had 2 times 3 (2 × 3), we could literally view this as 3 2's being added together. So it could be 2 + 2 + 2. Notice this is [COUNTING: 1, 2] 3 2's. And when you add those 2's together, you get 6. What we're going to introduce you to in this video is the idea of repeated multiplication – a new operation that really can be viewed as repeated multiplication. And that's the operation of taking an 'exponent.' And it sounds very fancy. But we'll see with a few examples that it's not too bad. So now, let's take the idea of 2 to the 3rd power (2^3) – which is how we would say this. (So let me write this down in the appropriate colors.) So 2 to the 3rd power. (2^3.) So you might be tempted to say, "Hey, maybe this is 2 × 3, which would be 6." But remember, I just said this is repeated multiplication. So if I have 2 to the 3rd power, (2^3), this literally means multiplying 3 2's together. So this would be equal to, not 2 + 2 + 2, but 2 × ... (And I’ll use a little dot to signify multiplication.) ... 2 × 2 × 2. Well, what's 2 × 2 × 2? Well that is equal to 8. (2 × 2 × 2 = 8.) So 2 to the 3rd power is equal to 8. (2^3 = 8.) Let's try a few more examples here. What is 3 to the 2nd power (3^2) going to be equal to? And I'll let you think about that for a second. I encourage you to pause the video. So let's think it through. This literally means multiplying 2 3's. So let's multiply 3 – (Let me do that in yellow.) Let's multiply 3 × 3. So this is going to be equal to 9. Let’s do a few more examples. What is, say, 5 to the – let's say – 5 to the 4th power (5^4)? And what you'll see here is this number is going to get large very, very, very fast. So 5 to the 4th power (5^4) is going to be equal to multiplying 4 5's together. So 5^4 = 5 × 5 × 5 × 5. Notice, we have [COUNTING: 1, 2, 3] 4 5's. And we are multiplying them. We are not adding them. This is not 5 × 4. This is not 20. This is 5 × 5 × 5 × 5. So what is this going to be? Well 5 × 5 is 25. (5 × 5 = 25.) 25 × 5 is 125. (25 × 5 = 125.) 125 × 5 is 625. (125 × 5 = 625.)