A cube-shaped container has a capacity of 512 𝑥 to the ninth 𝑦 to the twenty-seventh. Determine the length of one of its sides.
Since we’re talking about the capacity of a cube, we are referencing the volume of a cube. The formula for the volume of a cube is 𝑉 equals 𝑆 cube, where 𝑉 is the volume and 𝑆 is a side length. Now since we’re actually trying to solve for a side length, let’s go ahead and rewrite this.
Let’s go ahead and put 𝑆 on the left-hand side of the equation, since we will eventually be solving for 𝑆, the length of one of its sides. Now we actually already know the volume of this cube-shaped container. They tell us it’s 512 𝑥 to the ninth 𝑦 to the twenty-seventh. So let’s plug that in for 𝑉.
Now we need to solve for one of its sides, so we need to isolate 𝑆, which means to get it by itself. Since 𝑆 is cubed, we need to do the opposite of that; we need to cube-root it. So we must cube-root both sides of the equation.
For 𝑆, the cube root gets rid of the cubed, so 𝑆 is by itself. And we can separate on the right-hand side this into the cube root of 512 times the cube root of 𝑥 to the ninth times the cube root of 𝑦 to the twenty-seventh.
The reason for doing this is because each of these are perfect cubes. So if we can separate these into perfect cubes, it will simplify much easier. 512 is the same as eight times eight times eight. We got it to be something where you multiply with like bases.
Now we do the same for 𝑥 to the ninth. We have the like base of 𝑥 and it’s 𝑥 cubed times 𝑥 cubed times 𝑥 cubed, because three plus three plus three is nine; you add your exponents when you multiply. And the same for 𝑦 to the twenty-seventh, nine plus nine plus nine is 27.
The reason for doing this is because when you have something multiplied to itself three times, that is what cubing something actually is, so eight times eight times eight is eight cubed; 𝑥 cubed times 𝑥 cubed times 𝑥 cubed is 𝑥 cubed cubed; 𝑦 to the ninth times 𝑦 to the ninth times 𝑦 to the ninth is 𝑦 to the ninth cubed.
And why is this useful? Well if you take the cube root of something cubed, it cancels each other out.
So our side length will be eight times 𝑥 cubed times 𝑦 to the ninth, so the length of one of its sides would be eight 𝑥 cubed 𝑦 to the ninth.