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 of the equation we can subtract 4 times the cube root of Y from both sides of this equation so let's subtract subtract 4 we want to subtract 4 times the cube root of Y from both sides of this equation 4 times the cube root of Y from both sides and so your left hand side your left hand side you already have negative 1 times the cube root of Y then 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 a 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're almost we've almost isolated this cube root of Y we just have to divide both sides of the equation by negative 5 so we 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 3rd power and another way this statement right here is the exact same statement this is the exact same statement is saying Y to the one-third 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 3rd power if we take both sides of this equation to the 3rd power that's like taking both sides of this equation to the 3rd power both sides of that to the 3rd power and you can see here why to the one third to the third Y to the one-third and then to the third that's like saying Y to the one-third 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 one in the third power negative one times negative one is one times negative one again is negative one so we get Y is equal to negative one as our solution now let's make sure that it actually works let's go back to our original equation our original equation and I'll put negative one in for our Y's we had the negative of the cube root of this time negative one has to be equal to four times the cube root of negative one plus five let's verify that this is the case the cube root of negative one is negative one right negative one of the third power is negative one so this is equal to the negative of negative one has to be equal to four times the cube root of negative one is negative one plus five the negative of negative one is just positive one so one needs to be equal to 4 times negative one negative four plus five this is true negative four plus five is one so this works out this is our solution