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Video transcript
Now that we know that we can treat units algebraically, let's see if we can use that knowledge to figure out what the units of a variable might be from some formula. So let's say that I am given this formula, that capital K is equal to lowercase b over s squared. And let's say that we are also told that b is going to be in centimeters, and s is going to be in grams per centimeter. So that's the units of s is grams per centimeter. B is in centimeters. What is the units of K going to be? And I encourage you to pause this video and think about this. Well, let's try it out with just some value and some values for b and s, and see what happens. Let's say that b is equal to 1 centimeter, and I'm just using 1, because it makes the math really easy. And let's say that s is 1 gram per centimeter. So, we know that K is going to be equal to 1 centimeter-- that's b-- over 1 gram per centimeter squared. Now, we've already talked about the idea that we should treat our units like algebraic quantities, so this should be equal to 1 centimeter over-- now, you could treat this as 1 squared times-- let me rewrite this. You could rewrite this as 1 squared times grams per centimeter squared, and this just comes out of the idea that if I have a times b to some power, say, the second power, that's equal to a squared b squared. And we also know that if I have a/b to some power, say, squared, that's the same thing as a squared over b squared. So I could rewrite this as this is going to be equal to 1 centimeter over-- 1 squared is just 1-- 1 gram squared per centimeter squared. Now, what is this going to be equal to? Let me write it over here. Well, that's going to be 1 centimeter in our numerator. Let me do it in that green color, just to keep track of things. 1 centimeter from our numerator, instead of dividing it by 1 gram squared centimeter squared, I can multiply it times the reciprocal of this. So times 1/1, which is just going to be 1, but let me just write it that way just so that you get that I'm taking the reciprocal of 1. So 1/1 times the reciprocal of grams squared per centimeter squared. Well, that's just going to be centimeters squared per grams squared. So what's this going to be equal to? Well, I have 1 times 1, so if I take the numeric parts, 1 times 1, it's just going to be 1. And now remember, we can treat the units algebraically. I have a centimeters times centimeters squared. Well, that's going to get me to centimeters cubed, centimeters to the third power. And this isn't c times n to the third power. This is centimeters to the third power-- or you could write it like this if you wanted to be clear about it-- per-- and over here we still have grams squared-- per grams squared. So the units, I just picked the numbers 1 arbitrarily, so the 1 isn't what matters here. I just picked those numbers. What matters here is what are the units. What are the units for this formula when b is in centimeters and s is in grams per centimeters? We see that K is going to be expressed in terms of cubic centimeters per gram squared. And I can't tell you offhand kind of an intuitive physical representation of that, but you at least know the units. You have cubic centimeters, which you can view as a volume-- divided by, I guess, a mass squared. And a mass squared is a little bit less intuitive for me, but at least we know what the units of K would be in this formula given the units of b and s.