Exponent properties with quotients
Learn how to simplify expressions like (5^6)/(5^2). Also learn how 1/(a^b) is the same as a^-b. Towards the end of the video, we practice simplifying more complex expressions like (25 * x * y^6)/(20 * y^5 * x^2). Created by Sal Khan and CK-12 Foundation.
Want to join the conversation?
- at4:05why did sal have one in the numerator? we got nothing(19 votes)
- Why does
ab^9? In the other videos Sal showed that
(a^x)^y = a^x*y, so why does it not apply here?(4 votes)
- Yes, the rule you described does apply. However, the answer is not just ab^9 because the a is inside the parentheses and so the exponent of 3 outside the parentheses also applies to the a as well as to the b^3.
(In other words, there's another rule that also applies: (ab)^x = a^x b^x.)
Therefore, (ab^3)^3 = a^3 * (b^3)^3 = a^3 * b^(3*3) = a^3 b^9.
Have a blessed, wonderful day!(18 votes)
- The question is 42w^2 divided by 35w^4. Does that mean they both have the same bases?(6 votes)
- No - The exponents are on only the W's.
For the numbers, you would remove any common factor to reduce that portion of the fraction. For W's, you subtract the exponents.
Hope this helps.(6 votes)
- Why is a number raised to a negative power the same as "1" divided by that same expression? I've been struggling to understand this fully for a while now, and any help would be greatly appreciated.(5 votes)
- So, you know that 3 to the first is 3, right? And 3 squared is 9, 3 cubed is 27, etc...
The pattern is that you divide by 3 when you go down, which makes sense, right?
Then 3 to the 0th is 1. So what is 3 to the negative first? It's simply 1/3. If this still doesn't make sense, then think of the fact that when you multiply, your exponent increases by an amount, and when you divide, your exponent decreases.
Hope you found this helpful.(7 votes)
- how do you do ((9^4)(7^5))^-11? Because there is no exponent that is the same in 9 and 7 to simplify to(3 votes)
- If you distribute the -11 to both of the equations, like so:
Then we multiply the exponents (because an exponent raised to a power is just multiplying the two together):
And then we put them in the denominator (because they have a negative exponent):
(Which can then be simplified to something like... 3.41143722⋅10^−89)(10 votes)
- How do you restore all this information?Some of it easy to understand and some of it is not .(4 votes)
- I may be able to answer your question if a know what you mean by "restore."(6 votes)
- My head is literally singing into the unknown math problem! But, I am a bit confused about7:44.(5 votes)
- It's because of the communitive property, meaning the order of which you multiply doesn't matter, as long as you are only multiplying like terms. Hope this helps!(4 votes)
- What if the denominator and the numerator are different numbers?(5 votes)
- Leave it as a fraction, and simplify the numbers if possible.(4 votes)
- What do we do if the denominator is different.(6 votes)
- Leave it as a fraction, and simplify the numbers if possible.(2 votes)
- What is the sum:21.6x10^4+5.2x10^7 ?(3 votes)
Let's do some exponent examples that involve division. Let's say I were to ask you what 5 to the sixth power divided by 5 to the second power is? Well, we can just go to the basic definition of what an exponent represents and say 5 to the sixth power, that's going to be 5 times 5 times 5 times 5 times 5-- one more 5-- times 5. 5 times itself six times. And 5 squared, that's just 5 times itself two times, so it's just going to be 5 times 5. Well, we know how to simplify a fraction or a rational expression like this. We can divide the numerator and the denominator by one 5, and then these will cancel out, and then we can do it by another 5, or this 5 and this 5 will cancel out. And what are we going to be left with? 5 times 5 times 5 times 5 over 1, or you could say that this is just 5 to the fourth power. Now, notice what happens. Essentially we started with six in the numerator, six 5's multiplied by themselves in the numerator, and then we subtracted out. We were able to cancel out the 2 in the denominator. So this really was equal to 5 to the sixth power minus 2. So we were able to subtract the exponent in the denominator from the exponent in the numerator. Let's remember how this relates to multiplication. If I had 5 to the-- let me do this in a different color. 5 to the sixth times 5 to the second power, we saw in the last video that this is equal to 5 to the 6 plus-- I'm trying to make it color coded for you-- 6 plus 2 power. Now, we see a new property. And in the next video, we're going see that these aren't really different properties. They're really kind of same sides of the same coin when we learn about negative exponents. But now in this video, we just saw that 5 to the sixth power divided by 5 to the second power-- let me do it in a different color-- is going to be equal to 5 to the-- it's time consuming to make it color coded for you-- 6 minus 2 power or 5 to the fourth power. Here it's going to be 5 to the eighth. So when you multiply exponents with the same base, you add the exponents. When you divide with the same base, you subtract the denominator exponent from the numerator exponent. Let's do a bunch more of these examples right here. What is 6 to the seventh power divided by 6 to the third power? Well, once again, we can just use this property. This going to be 6 to the 7 minus 3 power, which is equal to 6 to the fourth power. And you can multiply it out this way like we did in the first problem and verify that it indeed will be 6 to the fourth power. Now let's try something interesting. This will be a good segue into the next video. Let's say we have 3 to the fourth power divided by 3 to the tenth power. Well, if we just go from basic principles, this would be 3 times 3 times 3 times 3, all of that over 3 times 3-- we're going to have ten of these-- 3 times 3 times 3 times 3 times 3 times 3. How many is that? One, two, three, four, five, six, seven, eight, nine, ten. Well, if we do what we did in the last video, this 3 cancels with that 3. Those 3's cancel. Those 3's cancel. Those 3's cancel. And we're left with 1 over-- one, two, three, four, five, six 3's. So 1 over 3 to the sixth power, right? We have 1 over all of these 3's down here. But that property that I just told you, would have told you that this should also be equal to 3 to the 4 minus 10 power. Well. What's 4 minus 10? Well, you're going to get a negative number. This is 3 to the negative sixth power. So using the property we just saw, you'd get 3 to the negative sixth power. Just multiplying them out, you get 1 over 3 to the sixth power. And the fun part about all of this is these are the same quantity. So now you're learning a little bit about what it means to take a negative exponent. 3 to the negative sixth power is equal to 1 over 3 to the sixth power. And I'm going do many, many more examples of this in the next video. But if you take anything to the negative power, so a to the negative b power is equal to 1 over a to the b. That's one thing that we just established just now. And earlier in this video, we saw that if I have a to the b over a to the c, that this is equal to a to the b minus c. That's the other property we've been using. Now, using what we've just learned and what we learned in the last video, let's do some more complicated problems. Let's say I have a to the third, b to the fourth power over a squared b, and all of that to the third power. Well, we can use the property we just learned to simplify the inside. This is going to be equal to-- a to the third divided by a squared. That's a to the 3 minus 2 power, right? So this would simplify to just an a. And you could imagine, this is a times a times a divided by a times a. You'll just have an a on top. And then the b, b to the fourth divided by b, well, that's just going to be b to the third, right? This is b to the first power. 4 minus 1 is 3, and then all of that in parentheses to the third power. We don't want to forget about this third power out here. This third power is this one. Let me color code it. That third power is that one right there, and then this a in orange is that a right there. I think we understand what maps to what. And now we can use the property that when we multiply something and take it to the third power, this is equal to a to the third power times b to the third to the third power. And then this is going to be equal to a to the third power. times b to the 3 times 3 power, times b to the ninth. And we would have simplified this about as far as you can go. Let's do one more of these. I think they're good practice and super-valuable experience later on. Let's say I have 25xy to the sixth over 20y to the fifth x squared. So once again, we can rearrange the numerators and the denominators. So this you could rewrite as 25 over 20 times x over x squared, right? We could have made this bottom 20x squared y to the fifth-- it doesn't matter the order we do it in-- times y to the sixth over y to the fifth. And let's use our newly learned exponent properties in actually just simplify fractions. 25 over 20, if you divide them both by 5, this is equal to 5 over 4. x divided by x squared-- well, there's two ways you could think about it. That you could view as x to the negative 1. You have a first power here. 1 minus 2 is negative 1. So this right here is equal to x to the negative 1 power. Or it could also be equal to 1 over x. These are equivalent. So let's say that this is equal into 1 over x, just like that. And it would be. x over x times x. One of those sets of x's would cancel out and you're just left with 1 over x. And then finally, y to the sixth over y to the fifth, that's y to the 6 minus 5 power, which is just y to the first power, or just y, so times y. So if you want to write it all out as just one combined rational expression, you have 5 times 1 times y, which would be 5y, all of that over 4 times x, right? This is y over 1, so 4 times x times 1, all of that over 4x, and we have successfully simplified it.