Algebra

Fractional exponents with numerators other than 1 | Algebra I | Khan Academy

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We have already seen what it means 64 to the degree of 1/3. We saw that this was exactly the same thing, such as finding the third root of 64. And since we know that 4 by 4, by 4, or 4 to the third power is equal to 64, if we look for the third root of 64, we will look for a number where that number by that number, by the same number, it will be equal to 64. We know that the number is 4, so this thing here is going to be 4. We will think now for slightly more complex fractions. What we see here has 1 in the numerator. Now we will see something different. What I want to do is think about how much it is 64 to degree 2/3. And here I will use a property of degrees, which we will learn later. But this property of degrees is the idea that – let’s take a prime number – if I raise something to the third degree and then raised this, for example, to the fourth degree, it will be the same thing as lifting 2 to grade 3 by 4 or 2 to grade 12, which can also to write as raising it to the fourth degree and after of the third degree. All this says that if I raise something to a degree and then raised this whole thing to another level, it’s the same thing as multiplying the two exponents. It’s the same thing as 2 on the 12th. We can use this property here, to say that 2/3 is the same thing as 1/3 by 2. So we can go the other way. We can say – Hey, look, this it will be the same thing as 64 to the power of 1/3 and then this thing squared. Notice that I’m raising something and then I raise that to a degree. If I have to multiply these two things, I will get 64 to the power of 2/3. Why did I do this? I already know what 64 is to the 1/3 power. We just calculated it. This is equal to 4. We can say that is equal to … and I will write it with this same yellow color – this is equal to 4 squared, which is equal to 16. So 64 to the power of 2/3 is equal to 16. The way I look at it is to find it first root third of 64, which is 4, and then I square it. And that will take me to 16. Now I will give you an even more complicated task. And I recommend you try this one yourself, before I do it. So, we will deal with 8 on 27. And we will raise this to … I will try to mean it in color … to the degree -2/3, of degree -2/3. I recommend you stop and try this for yourself. The first thing I will do, regardless that I see a negative degree is to say: How can I get rid of this negative degree? And I just remember the negative degree it just says to take the reciprocal on the basis to a positive degree. So that’s going to be equal to … the reciprocal of this is 27 … I use a different color. I will use this light pale purple color. So that will be equal to 27 on 8. I just take the reciprocal of that here. This is equal to 27/8 to the power plus 2/3. On a positive level 2/3. Notice that all I did was get rid of the downside and I took the reciprocal on the basis here. 8/27 is the basis, -2/3 is the degree. How can I handle this? We have already seen that if I have a numerator of some degree on a denominator of some degree – and this is another very important property of degrees – it’s the same as lifting 27 to level 2/3 (to level 2 on 3) on 8 to degree 2/3. 8 to degree 2/3. This is another very powerful property of degrees. Notice that if I have something divided into something, and raised the whole thing to a degree, I can essentially raise the numerator to this degree and raise the denominator to this degree. Let me think what that is. Just like we saw before, this it will be the same thing it’s the same thing as 27 to the 1/3 power and then squared it, because 1/3 by 2 is 2/3. So I’m going to raise 27 to 1/3 and then we square it, no matter what it is. For all this color coding you will need to change a lot of colors. This will be over 8 to the 1/3 power. 8 to the degree of 1/3. And then it will be raised to the second degree. We do the same thing in the denominator – we raise 8 by 1/3 and then we square it. So how much will that be? 27 of degree 1/3 is the third root of 27. It’s a number – that number on that same number, by this same number and will be equal to 27. Okay, maybe you already guessed that 3 on the third is equal to 27 or that 27 to 1/3 is equal to 3. So in the numerator we will end with 3 squared. And then in the denominator we will have … what is 8 to the power of 1/3? This is 2 by 2 by 2, which is 8. So that’s 8 to 1/3, which is 2. And then you will … Let me do this with the same orange color. 8 to 1/3 is 2 and then we’ll square that. Everything will be simplified to 3 squared on 2 squared, which is just equal to 9/4. And if you just break it step by step, it’s not really scary at all.

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