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### Course: Equations & Inequalities 229-236>Unit 1

Lesson 1: Exponent properties intro

# Exponent properties with parentheses

Learn two exponent properties: (ab)^c = (a^c)*(b^c) and (a^b)^c = a ^ (b*c). See WHY they work and HOW to use them. In other words, multiplying two numbers, then raising the product to an exponent is the same as raising each number to that exponent and then multiplying. Raising a number to an exponent and then to another exponent equals raising the base to the product of the two exponents. Created by Sal Khan.

## Want to join the conversation?

• a^3 * a^3 = a^6 What happens when you have a^3 + a^3?
(53 votes)
• if you take a^3 and say it is equal to an arbitrary letter b (a^3 = b)
you can simply say that b + b = 2b which leads to 2a^3 since b is equal to a^3
(38 votes)
• can a exponent be a negative fraction or mixed number?
(33 votes)
• yes, exponents can be negative fraction or mixed number
(12 votes)
• Can an exponent be a fraction or a decimal? Or must it be a whole number?
(14 votes)
• Yes, exponents can be in fraction or a decimal. Although decimal exponents are less common and you will see exponents as fractions almost all the time.
(19 votes)
• Can negative numbers have a negative exponent?
(8 votes)
• Yes. For example, (-3)^(-4) = 1/[(-3)^4] = 1/81, and (-4)^(-3) = 1/[(-4)^3] = 1/(-64) = -1/64.
Have a blessed, wonderful day!
(14 votes)
• can you use exponents to show exponents? For example 3 to the 6th power and the 6th power is shown as 6 to the 1st power?
(8 votes)
• yes, you can, 3^6^1 = 3^6 = 729, for a better example, 2^2^3 = 2^8 = 256, but (2^2)^3 = 4^3 = 64
(7 votes)
• 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
(7 votes)
• you can do it the other way as well if need be but I agree
(5 votes)
• can a exponent be a fraction
(4 votes)
• 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.
(9 votes)
• Why does every use "^"? ex. (a^b)
What does it stand for?
(4 votes)
• The "^" is a caret symbol on the keyboard which is a way to denote exponents while typing, which you can't do as correctly and that's why the "^" acts a substitute for the exponent on the higher end. For example: 5^3 = 5x5x5 = 125. So the last number after the "^" is the exponent. Hopefully that's helpful.
(8 votes)
• This video is labelled "Algebra". Im doing the 7th Grade math (FL B.E.S.T.) course so I do have 8th grade math and such. Could someone explain Algebra in some simple words for me to understand?
(5 votes)
• Think of Algebra as a type of math. It's a category, like how you might have "Vocabulary" in Language Arts. It doesn't really have a definition of its own, other than being the category where equations and expressions come into play.
(3 votes)
• Will the property still hold when ((a^x)^y)^z is the same as multiplying a^x*y*z ?
(6 votes)
• Yes. Doing the parentheses the way you show them is critical though. It would not be true without the nested parentheses.
(3 votes)

## 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.