Video: Adding and Subtracting Radical Expressions (Surds)

Radical expressions, such as the square root of 5, are sometimes called surds. We look at how to simplify expressions involving surd terms by combining like surds and factoring out square factors of surds to get like surds.

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Video Transcript

In this video, we’re gonna look at some expressions which add or subtract radical, or surd, terms. We’ll be looking at expressions where these terms can be gathered as like terms so that the expressions can be simplified. Radical or surd, terms that don’t simplify can be combined or collected in algebraic expressions in much the same way that you would gather variable terms like three 𝑥 and five 𝑥, or two 𝑦 and seven 𝑦, and so on.

Looking at this example, simplify the square root of seven plus the square root of seven.

Now, let’s imagine that we let 𝑥 equal root seven. Then, we could express root seven plus root seven in a different way. It would be 𝑥 plus 𝑥. Now, if you saw 𝑥 plus 𝑥, you’d quite happily gather those like terms. You’d add one 𝑥 to another 𝑥 and you’d have two 𝑥s. And since we just said up here that 𝑥 was equal to the square root of seven, two 𝑥 means two times the square root of seven, which we write like this, two root seven.

Now, it’s important to remember that a big two in front of that means that it’s two times the square root of seven. And you have to be careful not to confuse that with this expression, which is the small two in that square root sign, which means the square root of seven. Here’s another example.

Simplify the cube root of three plus two times the cube root of three plus three times the cube root of three.

And that first term, just the cube root of three, that means we’ve got one of them. So, we could say it’s one times the cube root of three. So, we’ve got one of the cube root of threes. We’ve got two more of the cube root of threes. And then, we’ve got three more on top of that, these cube root of threes. So, in total, one plus two is three, plus three is six. We’ve got six of them, six times the cube root of three. So, that’s our answer.

Now, we’ve got to simplify the square root of eight plus three times the square root of two minus four times the square root of two.

Now, these second two terms here are clearly like terms. We’ve got three lots of the square root of two and then we’re taking away four lots of the square root of two. So, if we got three of them, and we take away four, we’re left with negative one of them, or we just write negative root two. So, this becomes root eight minus root two. But wait, eight has a square factor. Four is a square number and it’s a factor of eight, so root eight can be written as the square root of four times two. And that can be written as the square root of four times the square root of two.

Now, the square root of four is two. So, the square root of four times the square root of two is two times root two, or as we’d normally write it, just two root two. So, root eight minus root two can be rewritten as two root two minus one root two. And two root two minus one root two is just one root two. Although, obviously, we wouldn’t bother writing the one in front of it, so it’s just the square root of two.

Now, we’ve got a slightly more complicated expression involving root 11 and also just some normal numbers not involving radicals or surds. So, six and negative three are the normal numbers. And four root 11 and two root 11 are like terms because they both involve the square root of 11. They’re radicals or surds. So, we’ve collected the like terms and now we can combine them. Six take away three is three. And four root 11s plus another two root 11s gives me six root 11s. So, that’s our answer. So, here’s another example, this time with parentheses.

Simplify two plus six root five plus nine plus eight root five.

Now, here, the parentheses aren’t really having an effect. They’re telling you to do the calculations in a particular order, but the operations are all additions. And because of the associativity of addition, it will make no difference if you do them in a different order. So, we’ll just remove the parentheses for now and then we’ll collect the like terms. Well, two and nine are the rational numbers, and six root five and eight root five are the radical or surd terms.

So, now we’ve collected the like terms, we can combine them. And two and nine make 11. And then, six root fives plus another eight root fives gives us 14 root fives. So, that’s the simplified version of our original expression. So, let’s look at our last example then.

Simplify root seven minus two minus five minus three root seven.

Now, here, the parentheses are important. The first set are not really having an effect because set root seven minus two is effectively already evaluated, so you can’t simplify that any further. But the second set of parentheses are very important. So, we can remove the first set of parentheses. But that negative sign, we’re taking away five, and we’re taking away negative three root seven. So, that looks like this. Now, if we’re taking away negative three root seven, that’s the same as adding three root seven.

So, we’re now at a situation where we can identify the like terms. So, these are the ones with the radicals, the root sevens, or the surds. And these are just the normal rational numbers. So, we’ve got one root seven plus another three root sevens giving us four root sevens. And we’ve got negative two take away another five, which is negative seven. So, that’s the simplified version of our original expression.

So, to summarise what we’ve done, you can treat radicals or surds as if they were algebraic terms. So, for example, if we had three root seven plus five root seven, we could let 𝑥 equal root seven. And we can think of that then as being three times 𝑥 plus five times 𝑥. So, if we got three of them and five of them, that makes eight of them. And then, we can substitute our root seven back in for 𝑥, giving us eight root seven.

And you need to think carefully about parentheses. As we’ve seen sometimes, they’re not really having an effect, and you can just remove them. And other times, they’re having a big effect, and you have to be very very careful. And look out for signs especially.

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