Skip to 0 minutes and 14 secondsNow we want to consider other powers of x-- not just integer values, like y equals x squared or y equals x cubed, but other fractional powers, like y equals x to the 1/2 or y equals x to the 3/4. We'll see that these have interesting applications to biology and other aspects of the modern world. So first, we have to review a few important things about powers generally and review some index laws. So let me remind you that if you have, say, 2 cubed and you multiply by 2 to the fourth, then you get 2 to the 7th. There's three 2s multiplying together there, four 2s multiplying together there. If you multiply them all together, you have seven 2s multiplying altogether.

Skip to 1 minute and 2 secondsIf you have 2 cubed and you, say, raise it to the fifth power, then you are multiplying 3 by 5. There's altogether 15 2s that I multiplied together, so you get 2 to the 15. And let me remind you that, say, 2 to the minus 1 is the same as 1/2. So for example, if we wanted 2 cubed to the minus 1, that's the same as 2 to the minus 3, which is really the same as 1 over 2 cubed.

Skip to 1 minute and 35 secondsSo in particular, if we want to introduce something like x to the 1/2, then x to the 1/2 will have the property that if we square it, we'll get x to the 1/2 times 2, which is x to the 1 or just x. That's telling us that x to the 1/2 is really the same as what we usually call the square root of x. So here's the graph of y equals x to the 1/2 or y equals the square root of x. And it's a familiar shape. It's really the same shape as y equals x squared. We can see that because this equation can be rewritten if we square things as y squared equals x.

Skip to 2 minutes and 14 secondsSo it has the same kind of relation as this one, except that it's reflected in the line y equals x. If we reflected this entire curve, we would get a parabola with the usual two arms. But the agreement is that the square root of x will only have this one upper branch, so it's a bona fide function. Now, y equals x to the 1/2 has quite a different character than these other functions, because it can only be evaluated, generally, approximately. So we can't calculate values by hand like we can for these algebraic polynomial relations. But nevertheless, still, even if it's an approximate function, it's still very useful for applications.

Skip to 2 minutes and 58 secondsAnd there's nothing to prevent us from considering other powers of x. For example, we might consider y equals x to the 3/4.

Skip to 3 minutes and 8 secondsThat would be a function that's between y equals x to the 1/2 and y equals x, some kind of shape that looks something like that. We'll see that, in fact, that's actually a very important function in biology. Now that we have control over more general exponents, we can introduce the idea of power law relations, where we have two quantities, x and y, and y is a fixed multiple ‘a’ of a power of x-- say x to the k. So this includes the case x equals 1, x equals 2, x equals minus 1, x equals 3.

Skip to 3 minutes and 52 secondsBut we're interested in the possibility of having other values of k-- in particular intermediate values of k, say between 1 and 2, or between 2 and 3.

Skip to 4 minutes and 4 secondsWe'll show you two interesting examples, one having to do with gas stations, another one having to do with a very famous law of Isaac Newton. So it's been observed that if you have a city and you count the number of gas stations and the population of that city grows, well, then of course, the gas stations increase as well. And you might naively think that if you double the population of a city, that the number of gas stations ought to also double. That would be a linear kind of relationship. But it turns out that the actual relationship is not quite like that. It's more a power law where the exponent is less than 1.

Skip to 4 minutes and 46 secondsSo if n represents the number of gas stations in a city and p represents the population, then empirically what was found is that the number of gas stations is roughly a fixed multiple of p to the 0.77. This is one of these curves, like a square root. Pretty close to ‘a’ times p to the 3/4, because 3/4 is pretty close to 0.77. So there's some kind of efficiency that's gained by having a larger population. You don't quite need as many new gas station as you might naively think. This is important for economies of scale, in terms of if you're deciding whether you need to open a new gas station in a certain town because it's growing bigger.

Skip to 5 minutes and 32 secondsYou might want to be aware of this kind of law that's going to govern how many naturally are going to be there. A very historically important example of a power law is Newton's law of gravitation, which is really an inverse square law. Very famous. And this is really part of Newton's explanation of why Kepler's laws work, why planets revolve around the sun in elliptical fashion, and why the speed slows down when the thing is far away and speeds up when it's closer. So the fundamental law is this one here. It's the law of gravitation, which tells us how much attractive force there is between this orbiting planet and the sun.

Skip to 6 minutes and 30 secondsThe law is that the size of the force, which is always directed from the planet towards the sun, is some proportional constant ‘a’ over r squared, where r is the distance from the sun to the planet. So when the distance is big, like it is here, then the 1 over r squared term, that's altogether relatively small, and so the planet feels a relatively small force towards the sun. Over here where the r is much smaller, then 1 over r squared will be proportionately bigger than it is there, and then the size of the force will actually be quite a lot bigger than it was over there.

Skip to 7 minutes and 14 secondsAnd this is really the key point that allowed Newton to actually calculate exactly what the orbits were mathematically, using calculus and his other famous law, that f equals ma. Once we know what the force is, then we can calculate what the acceleration is. And then using a little bit of calculus, we can go from knowing the acceleration to knowing the velocity, and then from knowing the velocity to knowing the position. It's really one of the great tour de forces. And it really centres very much on this inverse square law, which is really at the heart of this phenomenon. And actually, it's very important in other aspects of physics as well.

# How many gas stations has your city?

A power law relation has the form \(\normalsize{y=ax^k}\) for some number \(\normalsize{k}\), not necessarily an integer. In this video we introduce this more general kind of relation, explain how the usual index laws apply, and look at two interesting examples, relating to the numbers of gas stations in a city, and Newton’s famous inverse square law of gravitation.

## The square root function and other fractional powers

Real life doesn’t always model itself in a polynomial way; there are also other kinds of useful mathematical relations. In fact the inverse relationships like \(\normalsize{y=\frac{1}{x}}\) are not polynomial. But they are not that far removed either, as this is just the case of \(\normalsize{y=x^n}\) with \(\normalsize{n=-1}\), a negative exponent. Another example of a different kind of power law is

\[\Large{y=x^\frac{1}{2}=\sqrt{x}}.\]This has larger values than \(\normalsize{x}\) in the range \(\normalsize{(0,1)}\) but then grows more slowly for \(\normalsize{x \gt 1}\). Here is the graph:

We also see the fractional power \(\normalsize{y=x^\frac{3}{4}}\), which is intermediate between \(\normalsize{y=x^\frac{1}{2}}\) and \(\normalsize{y=x}\), and occurs in real-life applications, as we shall see.

## Identifying powers of \(\normalsize{x}\)

\[\Large{ k=-1,0,\frac{1}{2},1,\frac{3}{2},2,3}.\]

Q1(M): The following diagram has various powers \(\normalsize{y=x^k}\), including exponentsThere is also one more that is not on the list. Can you guess what this other one is?

## Connection with the quadratic relation

If \(\normalsize{y=x^2}\) then \(\normalsize{x=\sqrt{y}}\) or \(\normalsize x = -\sqrt{y}\). That means that if we invert a quadratic relation, we get a square root relation. Depending on the context, we sometimes omit the negative square root from consideration. For example we know that for a square,

\[\Large{\operatorname{area}=\operatorname{length}^2}.\]So conversely

\[\Large{\operatorname{length}=\operatorname{area}^{\frac{1}{2}}}.\]Geometrically we are just interchanging the axes!

While calculating the square of a number is straightforward, calculating a square root is, in general, an infinite procedure that can never be completed, so we usually rely on a calculator to give us an approximate value.

Q2(M): If we had a spherical blimp filled with helium, what power of the radius \(\normalsize r\) would we expect the ratio of the buoyant force \(\normalsize B\) to the weight of the blimp \(\normalsize W\) to be proportional to?

## Gas stations and Newton’s Inverse square law

We mention two examples of power law relations in this video. The first is the relationship between the size of a city and the number of gas stations in it — which has been empirically observed in a number of cities. As a city grows, the number of gas stations does not grow linearly; rather, there is a power law relation with an exponent rather closer to \(\normalsize{\frac{3}{4}}\). The second example is one of the most famous laws in physics: Newton’s *law of gravitation*, which is a classic example of an inverse square law relation.

## Answers

A1.The unaccounted for graph is the light green one, somewhere between the red one of \(\normalsize y=x^2\) and the dark green one of \(\normalsize y=x^3\). So it might have equation \(\normalsize y=x^{5/2}\).

A2.The weight of the blimp (or the force due to gravity) is due to thesurface areaof the canvas, and so \(\normalsize W\) is proportional to \(\normalsize r^2\). The buoyant force is proportional to thevolumeof air the blimp displaces, and so \(\normalsize B\) is proportional to \(\normalsize r^3\). The ratio of these two forces \(\normalsize \frac{B}{W}\) is thus proportional to \(\normalsize \frac{r^3}{r^2}=r.\) How would you interpret this for small and large values of \(\normalsize r\)?

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