### Video Transcript

The radius π of a sphere is given by the formula π equals three π over four π to the power of a third, where π is the sphereβs volume. Determine the difference in radius between a sphere with volume 36π and a sphere with 2304π.

In order to solve this problem, weβre gonna need to find the radius of the sphere with the volume 36π and the radius of the sphere with volume 2304π. And Iβm gonna start with the sphere that has the volume 36π. So weβre gonna use the equation π equals three π over four π to the power of third.

Okay, so letβs substitute our value 36π in for π. So weβre gonna get π is equal to three multiplied by 36π over four π all to the power of third. Okay, so now letβs simplify. Well, first of all, if we divide 36 by four, we get nine. And then if we actually divide 36π by π, the πs cancel out. So weβre just left with three multiplied by nine all to the power of a third.

And now, what we can actually do is we can use one of our exponent rules that says that if we have π₯ to the power of one over π, this is equal to the πth root of π₯. So therefore, weβve got three multiplied by nine. So thatβs gonna be 27 and then itβs gonna be 27 to the power of third, which is the cube root of 27, which is equal to three. So great, we found the radius of our first sphere. And the radius of our first sphere is three.

Now, letβs move on to the sphere with the volume 2304π. This time weβre actually gonna substitute π equals 2304π into our formula. So weβre gonna get π is equal to three multiplied by 2304π over four π all to the power of a third.

Again, weβre gonna simplify. And first of all, weβre gonna divide 2304 by four. Weβre just gonna do that using this method here. So we have four is into two donβt go. So itβs zero. And then, we carry the two. And then, we see four is into 23 go five times remainder three. Then four is into 30 goes seven remainder two. Then, four is into 24 goes six times. So we have 576. And also once again, our πs cancel out because π divided by π is just one. So weβre left with π is equal to three multiplied by 576 to the power of third.

So again, we use our expanded rule that tells us that π₯ to the power of one over π equals the πth root of π₯. So we get the cube root of 1728. So therefore, we get π is equal to 12.

So now, we move on to the final part of the problem that says βdetermine the difference in radius between a sphere with volume 36π and a sphere with volume 2304π.β So to determine the difference, weβre gonna need 12 because thatβs the radius of our sphere with volume 2304π minus three because thatβs the radius of our sphere with volume 36π.

So therefore, we can say that given the formula π equals three π over four π to the power of third, we can say that the difference in radius between a sphere with volume 36π and a sphere with 2304π is nine.