The square of the cube of 𝑘 is eight, and 𝑘 is positive. Circle the value of 𝑘. The options are the cube root of 64, the square root of eight, square root of two, or the square root of 512.
So first of all, we need to think about how we’re going to represent this. So what we want to represent is the square of the cube of 𝑘 is eight. So we’re gonna work backwards. We’re gonna start with the cube of 𝑘. So we’ve got 𝑘 cubed. So we have 𝑘 cubed all squared because we’ve squared 𝑘 cubed. And this is gonna be equal to eight.
Now if we look at what we could do next. Well, we can’t take the square root of eight because this isn’t really gonna help us. But what we can do is we can swap here around. So we have 𝑘 squared all cubed because this is gonna become helpful. And the reason we can do this is because both 𝑘 cubed all squared and 𝑘 squared all cubed will give us the result of 𝑘 to the power of six.
And we know this because we get it from one of our index laws, which tells us that if we have 𝑥 to the power of 𝑎 all to the power of 𝑏, then this is gonna be equal to 𝑥 to the power of 𝑎𝑏. So we multiply the powers. And this is what we do in both of ours because if we multiply three and two, we’re gonna get six and it doesn’t matter which way around we do it, so three multiplied by two or two multiplied by three. And the reason we want it in this format is because eight is a cube number because we know that two multiplied by two multiplied by two is equal to eight.
So therefore, what we can do is take the cube root of both sides of our equation. And if we do that, what we’re going to get is 𝑘 squared is equal to two. And we don’t have to worry here about a negative result because-because it’s a cube root and we’re looking at a cube number. Then, the negative wouldn’t work because we’d have negative two multiplied by negative two which would give us positive four then multiplied by negative two would give us negative eight. And we don’t have negative eight; we have eight.
Okay, so we’ve got 𝑘 squared is equal to two. So now to find 𝑘, what we’re gonna do is take the square root of both sides of the equation. And when we do that, we’re gonna get 𝑘 is equal to positive or negative root two. And that’s because if we have a positive multiplied by a positive, it gives us a positive. And if we have a negative multiplied by a negative, it will also give us a positive. So in this case, we had negative root two multiplied by negative root two, we’d get two. So there are two possible answers to 𝑘.
However, if we look at the question: we’re told that 𝑘 is positive. So therefore, if we’re told that 𝑘 is positive, we can discard negative root two. So therefore we can say that 𝑘 is equal to root two. So we know that if the square of the cube of 𝑘 is eight and 𝑘 is positive, the value of 𝑘 is root two. And I’ve circled it; it’s the third answer.