Solving cube-root equations

We're asked to solve for y. So we're told that the negative of the cube root of y is equal to 4 times the cube root of y plus 5. So in all of these it's helpful to just be able to isolate the cube root, isolate the radical in the equation, and then solve from there. So let's see if we can isolate the radical. So the simplest thing to do, if we want all of the radical onto the left-hand side equation, we can subtract 4 times the cube root of y from both sides of this equation. So let's subtract 4. We want to subtract 4 times the cube root of y from both sides of this equation. And so your left-hand side, you already have negative 1 times the cube root of y, and you're going to subtract 4 more of the cube root of y. So you're going to have negative 5 times the cube root of y. That's your left-hand side. Now the right-hand side-- these two guys-- cancel out. That was the whole point behind subtracting this value. So that cancels out and you're just left with a 5 there. You're just left with this 5 right over there. Now, we've almost isolated this cube root of y. We just have to divide both sides of the equation by negative 5. So you just divide both sides of this equation by negative 5. And these cancel out. That was the whole point. And we are left with the cube root of y is equal to-- 5 divided by negative 5 is negative 1. Now, the cube root of y is equal to negative 1. Well the easiest way to solve this is, let's take both sides of this equation to the third power. This statement right here is the exact same statement as saying y to the 1/3 is equal to negative 1. These are just two different ways of writing the same thing. This is equivalent to taking the 1/3 power. So if we take both sides of this equation to the third power, that's like taking both sides of this equation to the third power. And you can see here, y to the 1/3 to the third-- y to the 1/3 and then to the third, that's like saying y to the 1/3 times 3 power. Or y to the first power. That's the whole point of it. If you take the cube root of y to the third power, that's just going to be y. So the left-hand side becomes y. And then the right-hand side, what's negative 1 to the third power? Negative 1 times negative 1 is 1. Times negative 1 again is negative 1. So we get y is equal to negative 1 as our solution. Now let's make sure that it actually works. Let's go back to our original equation. And I'll put negative 1 in for our y's. We had the negative of the cube root of-- this time, negative 1-- has to be equal to 4 times the cube root of negative 1 plus 5. Let's verify that this is the case. The cube root of negative 1 is negative 1. Negative 1 to the third power is negative 1. So this is equal to the negative of negative 1 has to be equal to 4 times-- the cube root of negative 1 is negative 1 plus 5. The negative of negative 1 is just positive 1. So 1 needs to be equal to-- 4 times negative 1, negative 4, plus 5. This is true. Negative 4 plus 5 is 1. So this works out. This is our solution.