Evaluating fractional exponents: fractional base
About this video
Sal evaluates fractional exponents with fraction bases. (25/9)^(1/2) is 5/3 because 5/3 times 5/3 is 25/9. For (81/256)^(-1/4), he flips the base to remove the negative, applies the 1/4 power to top and bottom, and uses 4^4 = 256 and 3^4 = 81 to get 4/3.
You'll be able to
- find a fractional root of a fraction by taking the root of the numerator and denominator.
- remove a negative exponent by taking the reciprocal of a fraction base.
- recognize 256 as 4^4 and 81 as 3^4.
Key words
- principal square root
- The non-negative number that, times itself, gives the original number.
- fourth root
- The number that, multiplied by itself four times, gives the original; the same as the 1/4 power.
- reciprocal
- A fraction flipped upside down; the reciprocal of 81/256 is 256/81.
- negative exponent
- Tells you to take the reciprocal of the base and use the positive exponent.
Transcript
Fractional exponents can be a little daunting at first, so it never hurts to do as many examples as possible. So let's do a few. What if we had 25/9, and we wanted to raise it to the 1/2 power? So we're essentially just saying, well, what is the principal square root of 25/9?
So what number times itself is going to be 25/9? Well, we know 5 times 5 is 25, and 3 times 3 is 9. So why don't we just go with 5/3? Because notice, if you have 5/3 times 5/3, that is going to be 25/9. Or another way of saying this, that 5/3 squared is equal to 25/9.
So 25/9 to the 1/2 is going to be equal to 5/3. Now let's escalate things a little bit. Let's take a really hairy one. Let's raise 81/256 to the negative 1/4 power. I encourage you to pause this and try this on your own. So what's going on here? This negative-- the first thing I always like to do is I want to get rid of this negative in the exponent.
So let me just take the reciprocal of this and raise it to the positive. So I could just say that this is equal to 256/81 to the 1/4 power. And so now I can say, well, what number times itself times itself times itself is going to be equal to 256, and what number times itself times itself times itself-- did I say that four times?
Well, what number, if I take four of them and multiply, do I get 81? And one way to think about it, this is going to be the same thing-- and we'll talk about this in more depth later on when we talk about exponent properties. But this is going to be the exact same thing as 256 to the 1/4 over 81 to the 1/4.
You, in fact, saw it over here. This over here was the same thing as the square root of 25 over the square root of 9. Or 25 to the 1/2 over 9 to the 1/2. So we're just doing that over here. So one, we still have to think about what number this is. And this is a little bit of, there's no easy way to do this.
You kind of have to just play around a little bit to come up with it. But 4 might jump out at you if you recognize that 16 times 16 is 256. We know that 4 to the fourth power, or you're about to know this, is 4 times 4 times 4 times 4. And 4 times 4 is 16, times 4 is 64, times 4 is equal to 256.
So 4 to the fourth is 256, or we could say 4 is equal to 256 to the 1/4 power. Fair enough? Now what about 81? Well, 3 might jump out at you. We know that 3 to the fourth power is equal to 3 times 3 times 3 times 3, which is equal to 81. So 3 is equal to 81 to the 1/4.
So this top number, 256 to the 1/4, that's just 4. 81 to the 1/4, that is just 3. So this right over here is going to be equal to 4/3.
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