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We're asked to solve for y.

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So we're told that the negative of the cube root of y

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is equal to 4 times the cube root of y plus 5.

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So in all of these it's helpful to just be able to

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isolate the cube root, isolate the radical in the equation,

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and then solve from there.

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So let's see if we can isolate the radical.

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So the simplest thing to do, if we want all of the radical

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onto the left-hand side equation, we can subtract 4

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times the cube root of y from both sides of this equation.

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So let's subtract 4.

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We want to subtract 4 times the cube root of y from both

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sides of this equation.

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And so your left-hand side, you already have negative 1

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times the cube root of y, and you're going to subtract 4

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more of the cube root of y.

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So you're going to have negative 5 times the

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cube root of y.

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That's your left-hand side.

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Now the right-hand side-- these two guys-- cancel out.

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That was the whole point behind subtracting this value.

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So that cancels out and you're just left with a 5 there.

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You're just left with this 5 right over there.

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Now, we've almost isolated this cube root of y.

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We just have to divide both sides of the equation by

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negative 5.

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So you just divide both sides of this

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equation by negative 5.

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And these cancel out.

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That was the whole point.

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And we are left with the cube root of y is equal to-- 5

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divided by negative 5 is negative 1.

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Now, the cube root of y is equal to negative 1.

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Well the easiest way to solve this is, let's take both sides

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of this equation to the third power.

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This statement right here is the exact same statement as

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saying y to the 1/3 is equal to negative 1.

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These are just two different ways of

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writing the same thing.

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This is equivalent to taking the 1/3 power.

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So if we take both sides of this equation to the third

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power, that's like taking both sides of this equation to the

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third power.

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And you can see here, y to the 1/3 to the third-- y to the

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1/3 and then to the third, that's like saying y to the

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1/3 times 3 power.

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Or y to the first power.

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That's the whole point of it.

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If you take the cube root of y to the third power, that's

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just going to be y.

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So the left-hand side becomes y.

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And then the right-hand side, what's negative 1

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to the third power?

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Negative 1 times negative 1 is 1.

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Times negative 1 again is negative 1.

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So we get y is equal to negative 1 as our solution.

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Now let's make sure that it actually works.

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Let's go back to our original equation.

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And I'll put negative 1 in for our y's.

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We had the negative of the cube root of-- this time,

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negative 1-- has to be equal to 4 times the cube root of

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negative 1 plus 5.

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Let's verify that this is the case.

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The cube root of negative 1 is negative 1.

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Negative 1 to the third power is negative 1.

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So this is equal to the negative of negative 1 has to

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be equal to 4 times-- the cube root of negative 1 is

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negative 1 plus 5.

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The negative of negative 1 is just positive 1.

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So 1 needs to be equal to-- 4 times negative 1,

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negative 4, plus 5.

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This is true.

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Negative 4 plus 5 is 1.

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So this works out.

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This is our solution.

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