If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Exponent properties with parentheses

Learn two exponent properties: (ab)^c and (a^b)^c. See WHY they work and HOW to use them. Created by Sal Khan.

Want to join the conversation?

  • sneak peak green style avatar for user Cybernetic Organism
    a^3 * a^3 = a^6 What happens when you have a^3 + a^3?
    (52 votes)
    Default Khan Academy avatar avatar for user
  • male robot hal style avatar for user isaiah.armstrong.2018
    can a exponent be a negative fraction or mixed number?
    (30 votes)
    Default Khan Academy avatar avatar for user
  • piceratops ultimate style avatar for user Leonardo Padro
    Can an exponent be a fraction or a decimal? Or must it be a whole number?
    (8 votes)
    Default Khan Academy avatar avatar for user
  • starky ultimate style avatar for user Logan
    Can negative numbers have a negative exponent?
    (5 votes)
    Default Khan Academy avatar avatar for user
  • piceratops ultimate style avatar for user Leonardo Padro
    Isn't it easier just to multiply the exponent on the outside by the one in the parenthesis?

    example:
    (2^3)^2

    2*3=6

    so 2^6
    (6 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user 2711301379
    can a exponent be a fraction
    (3 votes)
    Default Khan Academy avatar avatar for user
    • primosaur seed style avatar for user Ian Pulizzotto
      Great question! Yes, exponents can be fractions! When we take a number to a fractional power, we interpret the numerator as a power and the denominator as a root. For example:

      25^(1/2) = [sqrt(25)]^1 = sqrt(25) = 5.
      16^(3/4) = [4throot(16)]^3 = 2^3 = 8.
      1,000^(4/3) = [cuberoot(1,000)]^4 = 10^4 = 10,000.
      (4 votes)
  • blobby green style avatar for user AudreyonnaM
    Can an exponent be a fraction or a decimal?
    (2 votes)
    Default Khan Academy avatar avatar for user
    • blobby green style avatar for user 9039736
      And now I want to go over some of the other core exponent properties. But they really just fall out of what we already know about exponents. Let's say I have two numbers, a and b. And I'm going to raise it to-- I could do it in the abstract. I could raise it to the c power. But I'll do it a little bit more concrete. Let's raise it to the fourth power. What is that going to be equal to? Well that's going to be equal to-- I could write it like this. Copy and paste this, copy and paste. That's going to be equal to ab times ab times ab times ab times ab. But what is that equal to? Well when you just multiply a bunch of numbers like this it doesn't matter what order you're going to multiply it in. This right over here is going to be equivalent to a times a times a times a times-- We have four b's as well that we're multiplying together. Times b times b times b times b. And what is that equal to? Well this right over here is a to the fourth power. And this right over here is b to the fourth power. And so you see, if you take the product of two numbers and you raise them to some exponent, that's equivalent to taking each of the numbers to that exponent. And then taking their product. And here I just used the example with 4, but you could do this really with any arbitrary-- actually any exponent. This property holds. And you could satisfy yourself by trying different values, and using the same logic right over here. But this is a general property. That-- let me write it this way-- that if I have a to the b, to the c power, that this is going to be equal to a to the c times b to the c power. And we'll use this to throughout actually mathematics, when we try to simplify things or rewrite an expression in a different way. Now let me introduce you to another core idea here. And this is the idea of raising something to some power. And I'll just use example of 3. And then raising that to some power. What could this be simplified as? Well let's think about it. This is the same thing as a to the third-- let me copy and paste that-- as a to the third times a to the third. And what is a to the third times-- So this is equal to a to the third times a to the third. And that's going to be equal to a to the 3 plus 3 power. We have the same base, so we would add and they're being multiplied. They're being raised to these two exponents. So it's going to be the sum of the exponents, which of course is going to be equal to a-- that's a different color a-- it's going to be a to the sixth power. So what just happened over here? Well I took two a to the thirds. And I multiplied them together. So I took these two 3s and added them together. So this essentially right over here, you could view this as 2 times 3. That's how we got the 6. When I raise something to one exponent, and then raised it to another, that's the equivalent to raising the base to the product of those two exponents. I just did it with this example right over here. But I encourage you try other numbers to see how this works. And I could to do this in general. I could say a to the b power. And then-- let me copy and paste that-- and then I'm going to raise that to the c power. Well what is that going to give me? Well I'm essentially going to have to take c of these, so one, two, three. I don't know how large of a number c is, so I'll just do the dot, dot dot. So dot, dot, dot. I have c of these, right over here. So what is that going to be equal to? Well that is going to be equal to a to the-- well for each of these c, I'm going to have a b that I'm going to add together. So let me write this. So I'm going to have a b plus b plus b plus dot, dot, dot plus b. And now I have c of these b's, so I have c b's right over here. Or you could view this as a, this is equal to a to the c times b power. c or a, you could do a to the cb power. So very useful. So if someone were to say what is 35 to the third power, and then that raised to the seventh power? Well this is going to obviously be a huge number. But we could at least simplify the expression. This is going to be equal to 35 to the product of these two exponents. It's going to be 35 to the 3 times 7, or 35 to the 21, or to the 21st power.
      (3 votes)
  • mr pink orange style avatar for user Dylan Chan
    I know this probably isn't in this video, but can somebody explain to me if you have an equation like this : (a+b)^2, why can't you just distribute the power to a and b, so it's a^2+b^2, instead of it being a^2+b^2+ab^2?
    (2 votes)
    Default Khan Academy avatar avatar for user
    • mr pink red style avatar for user David Lee
      So (a+b)^2 is not a^2+b^2+ab^2. It's a^2+2ab+b^2. Well, this is because you are multiply a+b twice. If you actually do (a+b)(a+b), you'll et a^2+ab+ab+b^2 which is a^2+2ab+b^2.
      Hope this helps! If you have any question or need help, please ask! :)
      (3 votes)
  • duskpin tree style avatar for user asimon423
    a^3 * a^3 = a^6 What happens when you have a^3 + a^3?
    (2 votes)
    Default Khan Academy avatar avatar for user
  • piceratops ultimate style avatar for user 𝓜𝓪𝓱𝓮𝓼𝓱 𝓜𝓪𝓱𝓮𝓷𝓭𝓻𝓪𝓴𝓪𝓻
    Will the property still hold when ((a^x)^y)^z is the same as multiplying a^x*y*z ?
    (2 votes)
    Default Khan Academy avatar avatar for user

Video transcript

And now I want to go over some of the other core exponent properties. But they really just fall out of what we already know about exponents. Let's say I have two numbers, a and b. And I'm going to raise it to-- I could do it in the abstract. I could raise it to the c power. But I'll do it a little bit more concrete. Let's raise it to the fourth power. What is that going to be equal to? Well that's going to be equal to-- I could write it like this. Copy and paste this, copy and paste. That's going to be equal to ab times ab times ab times ab times ab. But what is that equal to? Well when you just multiply a bunch of numbers like this it doesn't matter what order you're going to multiply it in. This right over here is going to be equivalent to a times a times a times a times-- We have four b's as well that we're multiplying together. Times b times b times b times b. And what is that equal to? Well this right over here is a to the fourth power. And this right over here is b to the fourth power. And so you see, if you take the product of two numbers and you raise them to some exponent, that's equivalent to taking each of the numbers to that exponent. And then taking their product. And here I just used the example with 4, but you could do this really with any arbitrary-- actually any exponent. This property holds. And you could satisfy yourself by trying different values, and using the same logic right over here. But this is a general property. That-- let me write it this way-- that if I have a to the b, to the c power, that this is going to be equal to a to the c times b to the c power. And we'll use this to throughout actually mathematics, when we try to simplify things or rewrite an expression in a different way. Now let me introduce you to another core idea here. And this is the idea of raising something to some power. And I'll just use example of 3. And then raising that to some power. What could this be simplified as? Well let's think about it. This is the same thing as a to the third-- let me copy and paste that-- as a to the third times a to the third. And what is a to the third times-- So this is equal to a to the third times a to the third. And that's going to be equal to a to the 3 plus 3 power. We have the same base, so we would add and they're being multiplied. They're being raised to these two exponents. So it's going to be the sum of the exponents, which of course is going to be equal to a-- that's a different color a-- it's going to be a to the sixth power. So what just happened over here? Well I took two a to the thirds. And I multiplied them together. So I took these two 3s and added them together. So this essentially right over here, you could view this as 2 times 3. That's how we got the 6. When I raise something to one exponent, and then raised it to another, that's the equivalent to raising the base to the product of those two exponents. I just did it with this example right over here. But I encourage you try other numbers to see how this works. And I could to do this in general. I could say a to the b power. And then-- let me copy and paste that-- and then I'm going to raise that to the c power. Well what is that going to give me? Well I'm essentially going to have to take c of these, so one, two, three. I don't know how large of a number c is, so I'll just do the dot, dot dot. So dot, dot, dot. I have c of these, right over here. So what is that going to be equal to? Well that is going to be equal to a to the-- well for each of these c, I'm going to have a b that I'm going to add together. So let me write this. So I'm going to have a b plus b plus b plus dot, dot, dot plus b. And now I have c of these b's, so I have c b's right over here. Or you could view this as a, this is equal to a to the c times b power. c or a, you could do a to the cb power. So very useful. So if someone were to say what is 35 to the third power, and then that raised to the seventh power? Well this is going to obviously be a huge number. But we could at least simplify the expression. This is going to be equal to 35 to the product of these two exponents. It's going to be 35 to the 3 times 7, or 35 to the 21, or to the 21st power.