### Video Transcript

The radius ๐ of a sphere is given by the formula ๐ equals three ๐ over four ๐ to the power of a third, where ๐ is the sphereโs volume. Determine the difference in radius between a sphere with volume 36๐ and a sphere with 2304๐.

In order to solve this problem, weโre gonna need to find the radius of the sphere with the volume 36๐ and the radius of the sphere with volume 2304๐. And Iโm gonna start with the sphere that has the volume 36๐. So weโre gonna use the equation ๐ equals three ๐ over four ๐ to the power of third.

Okay, so letโs substitute our value 36๐ in for ๐. So weโre gonna get ๐ is equal to three multiplied by 36๐ over four ๐ all to the power of third. Okay, so now letโs simplify. Well, first of all, if we divide 36 by four, we get nine. And then if we actually divide 36๐ by ๐, the ๐s cancel out. So weโre 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 ๐ฅ to the power of one over ๐, this is equal to the ๐th root of ๐ฅ. So therefore, weโve got three multiplied by nine. So thatโs gonna be 27 and then itโs 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, letโs move on to the sphere with the volume 2304๐. This time weโre actually gonna substitute ๐ equals 2304๐ into our formula. So weโre gonna get ๐ is equal to three multiplied by 2304๐ over four ๐ all to the power of a third.

Again, weโre gonna simplify. And first of all, weโre gonna divide 2304 by four. Weโre just gonna do that using this method here. So we have four is into two donโt go. So itโs 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 ๐s cancel out because ๐ divided by ๐ is just one. So weโre left with ๐ is equal to three multiplied by 576 to the power of third.

So again, we use our expanded rule that tells us that ๐ฅ to the power of one over ๐ equals the ๐th root of ๐ฅ. So we get the cube root of 1728. So therefore, we get ๐ is equal to 12.

So now, we move on to the final part of the problem that says โdetermine the difference in radius between a sphere with volume 36๐ and a sphere with volume 2304๐.โ So to determine the difference, weโre gonna need 12 because thatโs the radius of our sphere with volume 2304๐ minus three because thatโs the radius of our sphere with volume 36๐.

So therefore, we can say that given the formula ๐ equals three ๐ over four ๐ to the power of third, we can say that the difference in radius between a sphere with volume 36๐ and a sphere with 2304๐ is nine.