Video Transcript
Given that 𝑥 equals the sixth root of 64 plus the square root of 81 plus the third root of negative 27 subtract the fourth root of 16, find 𝑥.
In order to find the value of 𝑥, we’ll need to simplify each of the terms on the right-hand side. There’s one in here that we should be familiar with, the square root of 81. When we’re finding the square root of a number, we’re really asking, what number multiplied by itself would give us this value? Well, we knew that nine squared is equal to 81. Therefore, the square root of 81 is equal to nine.
Let’s have a look at seeing if we can figure out the sixth root of 64. In order to work this out, we can think of it in the inverse way. That is, we’re saying what value of 𝑎 to the sixth power would give us 64. 𝑎 to the sixth power is equivalent to 𝑎 six times multiplied together. Let’s try a small value. For example, we can take the value of 𝑎 to be equal to two. We know that two multiplied by two gives us four. And we realize that we have four multiplied by four multiplied by four. Four fours are 16. And 16 times four will give us 64. And that’s the value that we were looking for. Now, we know that the value to the power of six is two. Two to the power of six is 64. Therefore, we can say that the sixth root of 64 is two.
Next, let’s have a look at the third term on the right-hand side, the cube root of negative 27. It’s easy to be a little confused here. We might have been told that we can’t find the square root of a negative number, but it’s a little bit different with finding the cube root. This time, we’re asking, what value of 𝑏 cubed, or taken to the third power, would give us negative 27? Well, we know that 𝑏 must be equal to a negative number since if we multiply a negative by a negative, we get a positive value and then if we multiply a positive value by a negative value, we get a negative value.
So, if we wished, we could try a value of 𝑏 equal to negative two. So, we calculate negative two multiplied by negative two multiplied by negative two. The first two lots of negative two multiplied would give us four. And four multiplied by negative two would give us negative eight. But we’re looking for a value of negative 27. So, this value of 𝑏 as negative two would not work.
We can try a value of negative three next. We know that negative three multiplied by negative three will give us nine, and then multiplied by a further negative three would give us the value of negative 27. And that’s the value that we were looking for. This means that the value of 𝑏, which is cubed, must be negative three. So, the cube root of negative 27 is negative three.
We have one final term left to investigate, the fourth root of 16. This time we’re asking, what value of 𝑐 to the fourth power would give us 16? Let’s begin by trying a value of two. Two times two is four, four times two is eight, and eight times two is 16. So, it looks like we’ve found the value of 𝑐. Since two to the fourth power is 16, then the fourth root of 16 is two.
So, let’s put all of these simplified terms together to find the value of 𝑥. So, we have 𝑥 is equal to the sixth root of 64, which we found is two, plus the square root of 81, which is nine, plus the cube root of negative 27, which is negative three, subtract the fourth root of 16, which is two. When we add a value like negative three, this is equivalent to subtracting three. We must remember to apply the order of operations when we’re adding and subtracting. So, we work from left to right. Two plus nine is 11, subtract three is eight, subtract two is six. And so, our answer is that 𝑥 is equal to six.
