Lesson Video: nth Roots: Integers Mathematics

In this video, we will learn how to work out 𝑛th roots of integers, where 𝑛 is a positive integer greater than or equal to 2.

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Video Transcript

In this video about 𝑛th roots, we’ll learn how to take the 𝑛th root of a number where 𝑛 is a positive integer. For example, we’ll learn how to find the sixth root of 64 or the fourth root of 81. Finding the 𝑛th root of a number is the inverse operation to finding the 𝑛th power. So, the first thing we’ll do is recap some common powers and exponents.

Let’s begin with taking a number to the power of one. For example, two to the power of one just means one two by itself. Two squared means two of the number two multiplied together. And that’s two times two, which is four. Two to the third power means three twos multiplied together. Be careful here as a really common mistake is to say that two to the third power is six, but that would be three twos added together. We can think of this instead in terms of multiplying the first two twos together, which would give us four, and multiplied by the remaining two would give us eight. Two to the fourth power is written as four twos multiplied together. And we know that two cubed is equal to eight. And so, eight multiplied by the remaining two would give us 16.

How does this fit in then in terms with taking roots of numbers? Let’s think in terms of the inverse operation of squaring, which would be taking the square root. Here, we’ve got the square root of four. We can remember that when we write a square root, we don’t need to write this smaller two if we mean just a square root. The square root of four means what number do we multiply by twice to give us a value of four. The answer, of course, is two since two multiplied by two gives us four.

Next, the cube root of eight means what number do we multiply three times to give us a value of eight. And that would be two since two times two times two is equal to eight. Finally, the fourth root of 16 would also be two since two times two times two times two gives us 16.

We can apply the same principles when we’re looking at powers or exponents of three. Three squared is three times three, which is equal to nine. Three to the third power is three times three times three, which is 27. And three to the fourth power would give us 81. We can then find the different roots of nine, 27, and 81 as three.

In general, we can think that the 𝑛th root, 𝑟, of a number 𝑥 is the value which is multiplied 𝑛 times to give 𝑥. We can see an example of this in the fourth root of 81 is equal to three because three to the fourth power is equal to 81.

Before we get into some questions, here’s a helpful tip. It’s very good to be familiar with the first few powers of two, three, four, and even five. That way we’ll have an even quicker recollection of the different roots, particularly in exam questions. So, let’s take a look at the first question.

Evaluate the square root of four.

We can start by recalling that when we have a root symbol and no smaller digit, we’re really taking the square root of a value. When we’re finding the square root of four, we’re really asking, what number multiplied by itself would give us four? We could think of this with any letter. Here, I’ve used 𝑦. So, 𝑦 times 𝑦 equals four. So, 𝑦 must be equal to two since two multiplied by two will give us four. So, we can give our answer that the square root of four is two.

Let’s have a look at this next question.

Evaluate the fourth root of 81.

Before we begin answering this question, it’s important to notice the distinction between this root sign with a small four or a large four. What we have here in the question is the fourth root of 81. This second value, however, would indicate four multiplied by the square root of 81. We must remember that if we’re handwriting these to make sure that when we’re writing the fourth root or something like it to make the value in the root smaller. Four multiplied by the square root of 81 is not what we want to calculate here.

In order to calculate the fourth root of 81, we’re really asking, what value of 𝑦, for example, to the fourth power would give us a value of 81? When we’re finding a number to the fourth power, that means we have four of that number multiplied together. We can often solve these through a series of trial and improvement. If we took the value of 𝑦 equals one, we know that one times one times one times one would only give us one and not 81. So, let’s try a value of two. Two times two would give us four. Four times two would give us eight. And eight times two would give us 16. But 16 isn’t the value that we’re looking for.

Next, we can try the value of three. To work out three times three times three times three, we can do it in several different methods. We could work out the first three times three, which is nine, and then we could multiply it by three and then multiply it by another three. Or we could notice that there’s another three times three in there, which gives us nine. And nine times nine gives us a value of 81. This means that we’ve found the value that we were looking for. Since we now know that three to the fourth power is 81, then that means that the fourth root of 81 is three. And so, three is our answer.

Let’s take a look at another question.

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.

In the next question, we’ll see a more complex question involving a calculation with different roots.

Given that 𝑥 equals the fifth root of 32 subtract the fourth root of 625 plus the square root of 64, find 𝑥.

In this question, in order to find the value of 𝑥, we’ll need to work out the value of all of these different component parts on the right-hand side of the equation. Let’s take each of these terms in turn and see if we can calculate their value.

In order to find the fifth root of 32, what we’re really trying to find is what value of 𝑎 written five times and multiplied would give us a value of 32. We can say that this value of 𝑎 is not going to be one because one multiplied by one however many times would still give us a value of one. If we try a value of two, well, two multiplied by two gives us four, and another two multiplied by two gives us four, and four times four is 16, and 16 multiplied by two does indeed give us 32. This means that two to the power of five is 32. And therefore, the fifth root of 32 is two.

Next, let’s have a look at solving the fourth root of 625. This time, we’re looking for a value, let’s call it 𝑏, written four times and multiplied that would give us 625. We know that 𝑏 isn’t going to be one. And we might even realize that 𝑏 can’t be two either because 625 is an odd number. Two multiplied by two as many times would always give us an even number. So, what about a value of three? Well, three times three is nine. And we have another three times three, which is nine. And nine times nine gives us a value of 81. So, we know that our value is not three.

We could choose the value of four next, but we know that once again four is an even number and the value of 625 that we’re looking for is an odd number. In order to try a value of five, we can see that five times five gives us 25. Multiplying by another five would give us 125. And multiplying by the remaining five would give a value of 625. And that’s the value that we’re looking for. So now, we know that the fourth root of 625 is five.

Finally, let’s look at the square root of 64. We should remember that if we’re trying to find the square root of 64, we’re asking, what value multiplied by itself would give us 64? And that would be eight since eight multiplied by eight is 64. Now that we’ve found each term on the right-hand side in a simplified form, we can find the value of 𝑥.

So, we calculate the fifth root of 32, which is two, subtract the fourth root of 625, which is five, plus the square root of 64, which is eight. We’ll need to apply the order of operations here, which tells us that when we have addition and subtraction, then we perform those in order from left to right. Two subtract five gives us three. And then, adding eight will give us the value of five. Therefore, the answer to the question is that 𝑥 equals five.

Let’s have a look at one final question.

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.

We can now summarize what we’ve learnt in this video. The 𝑛th root, 𝑟, of a number 𝑥 is the value which is multiplied 𝑛 times to give 𝑥. For example, the fourth root of 81 is three since three is multiplied four times to give 81. We must be careful to distinguish between three is multiplied by four, which would be 12, and three multiplied four times, which would be 81.

We can have real number value 𝑛th roots of negative numbers if the 𝑛-value is odd. As we saw in one of the questions, we worked out that the cube root of negative 27 is equal to negative three. And finally, it’s worth being familiar with some of the first powers of two, three, and four. That way, working out the 𝑛th roots of these different values will be so much easier.

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