Solving Equations with Squares and Cubes - Concept
Many of the formulas for calculating volumes (prism volume, volume of cylinders, volume of cones formulas and volume of pyramids) require solving equations with square roots or cube roots. To solve these equations, we use many of the same principals that we learned when solving equations with square roots in Algebra such as isolating the unknown variable and simplifying.
In Geometry you're going to solve with square roots and with cube roots. So we'll look at 3 quick problems here.
The first one you have an equation with one variable r and r is being squared. So you're going to have to do a couple of steps here. First step is you're going to eliminate what's multiplying r squared and that is 4 pi. So I'm going to divide both sides by 4 pi. So it should look familiar, it's something that you did last year in Algebra. 4 pi divided by 4 pi is 1. Anything divided by itself is 1. The only thing left on the right side is r squared. On the left side here we have pi divided by pi which is 1, and we have 80 divided by 4, which is 20. So to solve this for r, I need to undo squaring which is squaring square rooting. So the square root of 20 the way I like to simplify that is to think of it as two square roots being multiplied together. And I can say that 20 is 10 times 2, but I don't know either of those square roots as a whole number but I can write it out as square root of 4 times square root of 5. Square root of 4 is 2. So we are going to say that r is equal to 2 times the square root of 5.
Let's look at two more. Here's a next one. x cubed is equal to 27. Well to undo cubing something, I'm going to take not the square root but the cube root. So the cube root of x cubed is going to be x. I have to do the same operation on the other side and the cube root of 27 is going to be 3. Now if we go back to our first problem, something that we'll notice is that I could have said that this was positive or negative 2 times the square root of 5. Since we're in Geometry and we're almost always talking about distances, we're going to almost always take the positive root. Because, 2 times the square root of 5 times itself is 20 and if I took the negative of that multiplied by itself we'd end up with 20 as well. So there it could be two answers. If we go back to this cube root however, if I said x could be -3, let's just look at this real briefly. -3 cubed. -3 times -3 is 9. So I'll have 9 times -3 which is -27. So notice that with the cube root you're only going to end up with this one answer. The negative is not going to be one of your answers.
The last one that we're going to look at is something that you'll be solving when you're talking about the volume of the sphere. To isolate r here, first we're going to take the reciprocal of four thirds. So I'm going to multiply both sides by three fourths. So it's three fourths times 823. I'm going to type that into my calculator. Three fourths times 823 is 617.25. So what we've done is we've isolated pi times r cubed. I can't take the cube root just yet so what I'm going to do is I'm going to divide both sides by pi. So now I'm going to get another decimal, I'm going to divide this by pi and I get 196.5 we'll round. 196.5 is equal to r cubed.
Now that the only thing that we have here is an r cubed, we can take the cube root and isolate r. So we take the cube root of both sides and I'm going to say that the cube root of our cube is r, and in my calculator, what I'm going to type, if your teacher hasn't shown you, the way you type this in is you're going to type in 196.5 and then to tell it to raise it to a fraction. Because a fractional root, or excuse me a fractional expone is actually taking a root. So here you're going to raise it to the one third power. So in my calculator I'm going to type in 196.5 and we'll raise it to, now remember to have these parentheses here, otherwise your calculator will just raise it to the first and then divide everything by 3. So I'm going to say one third and I get 5.8. So I'm going to write that little bit over here. r = 5.8 and we don't know what our units are so we'll just leave it like that.
So remember that when you're trying to solve problems with surface area, any time something is squared, any time something is cubed, you're going to be taking the square root or the cube root to isolate your variables.