Video Transcript
Complete the following: The cube root of 27 equals the square root of blank.
Let’s begin this question by seeing if we can simplify the left-hand side, the cube root of 27. We can recall that if we’re finding something like the cube root of 27, we’re really asking, what value, 𝑦 for example, to the third power would give us 27? When a number is taken to the third power, that means it’s written three times and multiplied. We can answer questions like this using a trial and improvement method.
We have written one times one times one here, but in reality we never need to choose one as a possibility because we know that the answer will always be one. So, what about two times two times two? Well, we could work that out that as eight, but remember any even number multiplied by an even number will always give us an even number. And we’re looking for an odd value of 27 here. So, if we’d thought about it, we wouldn’t even need to calculate the value of two to the third power.
So, what about three times three times three? Well, three multiplied by three will give us nine, and nine multiplied by the remaining three gives us a value of 27. And that’s the value that we were looking for. Now we know that three multiplied by three multiplied by three gives us 27, we know that three to the third power is 27. And so, the inverse operation, that is, the cube root of 27, gives us three.
Now that we’ve found that the value is three, does that mean that our missing answer must be three? Well, not quite. The value that we’re looking for is the square root of something gives us three. In order to find the answer to the square root of what gives us three, we must perform the inverse operation to finding the square root. And that’s squaring. So, our value that we’re missing must be three squared. Three squared is three multiplied by three. And that’s nine. And so, our answer is the value of nine. We can double-check each side of the equation. The cube root of 27 we found was the value of three, and the square root of nine is also the value of three.