Find the value of the cube root of 64 over 27 minus the cube root of negative one over eight.
To enable us to solve this problem, we’ve got a rule. And that rule tells us that if we’ve got the cube root of 𝑎 over 𝑏, it’s equal to the cube root of 𝑎 divided by the cube root of 𝑏.
So therefore, using this, we can actually rewrite our expression. So when we rewrite it, what we’ve got is the cube root of 64 over the cube root of 27 minus then we’ve got negative the cube root of one over the cube root of eight.
So if we take a look at what we’ve got in the expression, we can see that each of the values we’ve got — so 64, 27, one, and eight — are all cube numbers. So we’re going to be able to actually simplify this using the fact that they are cube numbers and we have cube roots.
So therefore, our first term is gonna be four over three and that’s because the cube root of 64 is four and that’s because four multiplied by four multiplied by four is 64. So as we said, we’ve got that. And then on the denominator, we’re gonna have three because the cube root of 27 is three. And then, this is gonna be minus negative a half and again that’s because the cube root of one is one and the cube root of eight is two.
So as we got four-thirds or four over three minus negative a half, we’re actually gonna add them because if you subtract a negative, that means we’re actually turning it into an add. But to enable us to do that, what we need to have is a common denominator. Well, the common denominator we’re gonna use is six. And that’s because that’s the lowest common multiple of both three and two.
So therefore, if I turn four-thirds or four over three into a fraction with a common denominator of six, we’re gonna get eight over six or eight-sixths. And we got that because you have to multiply three by two to get six on the denominator. So therefore, you have to do the same to the numerator. So four multiplied by two gives us eight.
And similarly, we have to do the same with the other fraction. So we’ve got a half turns into three over six or three-sixths. And that’s because we multiplied the denominator by three to get from two to six. So we have to do the same to the numerator. So now we have the same denominator, all we do is we add the numerators.
So therefore, we can say that the value of the cube root of 64 over 27 minus the cube root of negative one over eight is equal to eleven sixths or 11 over six.