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Multiplying & dividing powers (integer exponents)

For any base a and any integer exponents n and m, aⁿ⋅aᵐ=aⁿ⁺ᵐ. For any nonzero base, aⁿ/aᵐ=aⁿ⁻ᵐ. These are worked examples for using these properties with integer exponents.

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  • ohnoes default style avatar for user Anthony Cajamarca
    in how did he get 1/4 3
    (12 votes)
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  • aqualine ultimate style avatar for user Chloe
    If negative exponents such as 10^-5 is equal to 1/10^5, what would fractions with negative exponents such as 1/10^-5 be equal to?
    (10 votes)
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    • leaf green style avatar for user Swapnal
      Apply the same rule you have cited. As you put it (10)^-5 = 1/(10)^5
      The expression in question is 1/(10)^-5.
      Lets see! We can write (10)^-5 as 1/(10)^5 (as you wrote).
      So 1/(10)^-5 can essentially be written as 1/(1/10^5)
      Which is nothing but 10^5 itself( We're basically taking the reciprocal of 1/10^5)

      So 1/(10)^-5 =10^5
      Cheers!

      EDIT: Since perhaps that's a bit long, you can remember it for a general case as:
      1/a^-m = a^m
      where a and m can be any of positive or negative integers(but not zero!)
      Hope that helps!
      (19 votes)
  • starky tree style avatar for user Bruh
    How do you divide exponents by exponents? I kinda really don't understand that part.
    (10 votes)
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    • old spice man blue style avatar for user Lauren Kirkemier
      An easier way to think about this is to treat the multiplication sign as an addition sign and treat the division sign as a subtraction sign. I'll put an example down below! :)
      Therefore, 4_^-3 x 4_^5 is equal to 4_^2.
      You would add -3 + 5, which is equal to 2. Then keep the 4 and put the 2 as the exponent!
      (12 votes)
  • aqualine ultimate style avatar for user Asher frank
    how do you do it when both powers are negative and you are multiplying.
    (5 votes)
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  • blobby green style avatar for user blondela
    For the dividing part, how did you make the exponent of 12^-5 positive and the exponent of x^5 negative?
    (3 votes)
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    • mr pink green style avatar for user David Severin
      The rule for dividing same bases is x^a/x^b=x^(a-b), so with dividing same bases you subtract the exponents. In the case of the 12s, you subtract -7-(-5), so two negatives in a row create a positive answer which is where the +5 comes from. In the x case, the exponent is positive, so applying the rule gives x^(-20-5).
      If you want to use two different laws of exponents, you can use the negative exponent rule, if you move an exponent from numerator to denominator (or from denominator to numerator), you have to change the sign. So 12^-5 in the denominator would be the same as 12^5 in the numerator and x^5 in the denominator would be x^-5 in the numerator. Then you would have to use the rule for multiplying same bases shown as x^a * x^b=x^(a+b). Thus, x^-7*x^5 (as moved above) you still get 12^(-7+5) and x^-20 * x^-5 = x^(-20-5).
      (10 votes)
  • mr pants teal style avatar for user Hanae
    what if you don't have the same base? like if you have 5 to the power of 3 times 6 to the power of 2?
    (2 votes)
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  • scuttlebug green style avatar for user maewadd2025
    Isn't multiplying exponents the same as adding exponents then? Because for multiplying exponents you add the exponents and for adding exponents you add the exponents. What is the difference?
    (2 votes)
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    • stelly blue style avatar for user Kim Seidel
      Your terminology is a little off...
      If you are multiplying a common base, then you add the exponents. For example: x^7 * x^2 = x^(7+2) = x^11
      There is no multiplication of the exponents in this problem. The exponents are beind added. The base values "x" are what is being multiplied.

      Multiplying exponents occurs when you have an expression that involves and exponent and that expression is raised to an exponent. For example: (x^7)^2 = x^(7*2) = x^14

      Hope this helps.
      (8 votes)
  • blobby green style avatar for user 26smarrl
    What is an integer?
    (3 votes)
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  • hopper cool style avatar for user 뽀롱이OreoYellowieBuffyFluffyCheerio
    What's a reciprocal?
    (2 votes)
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  • marcimus pink style avatar for user Raindrop9
    Why is their a one as the numerator for the negative exponent?
    (3 votes)
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Video transcript

- [Narrator] Let's get some practice with our exponent properties, especially when we have integer exponents. So, let's think about what four to the negative three times four to the fifth power is going to be equal to. And I encourage you to pause the video and think about it on your own. Well there's a couple of ways to do this. See look, I'm multiplying two things that have the same base, so this is going to be that base, four. And then I add the exponents. Four to the negative three plus five power which is equal to four to the second power. And that's just a straight forward exponent property, but you can also think about why does that actually make sense. Four to the negative 3 power, that is one over four to the third power, or you could view that as one over four times four times four. And then four to the fifth, that's five fours being multiplied together. So it's times four times four times four times four times four. And so notice, when you multiply this out, you're going to have five fours in the numerator and three fours in the denominator. And so, three of these in the denominator are going to cancel out with three of these in the numerator. And so you're going to be left with five minus three, or negative three plus five fours. So this four times four is the same thing as four squared. Now let's do one with variables. So let's say that you have A to the negative fourth power times A to the, let's say, A squared. What is that going to be? Well once again, you have the same base, in this case it's A, and so since I'm multiplying them, you can just add the exponents. So it's going to be A to the negative four plus two power. Which is equal to A to the negative two power. And once again, it should make sense. This right over here, that is one over A times A times A times A and then this is times A times A, so that cancels with that, that cancels with that, and you're still left with one over A times A, which is the same thing as A to the negative two power. Now, let's do it with some quotients. So, what if I were to ask you, what is 12 to the negative seven divided by 12 to the negative five power? Well, when you're dividing, you subtract exponents if you have the same base. So, this is going to be equal to 12 to the negative seven minus negative five power. You're subtracting the bottom exponent and so, this is going to be equal to 12 to the, subtracting a negative is the same thing as adding the positive, twelve to the negative two power. And once again, we just have to think about, why does this actually make sense? Well, you could actually rewrite this. 12 to the negative seven divided by 12 to the negative five, that's the same thing as 12 to the negative seven times 12 to the fifth power. If we take the reciprocal of this right over here, you would make exponent positive and then you would get exactly what we were doing in those previous examples with products. And so, let's just do one more with variables for good measure. Let's say I have X to the negative twentieth power divided by X to the fifth power. Well once again, we have the same base and we're taking a quotient. So, this is going to be X to the negative 20 minus five cause we have this one right over here in the denominator. So, this is going to be equal to X to the negative twenty-fifth power. And once again, you could view our original expression as X to the negative twentieth and having an X to the fifth in the denominator dividing by X to the fifth is the same thing as multiplying by X to the negative five. So here you just add the exponents and once again you would get X to the negative twenty-fifth power.