Let’s take a look at simplifying radicals. Now depending on where you live, you may use words like surds or irrational numbers, but they all mean the same thing. Now what we’re gonna concentrate on is this idea of minimising the radical. So if we’ve got something in this format here, like the square root of twenty, and it’s considered polite amongst mathematical people to minimise the number that is inside that square root sign. So we look for factors which are square numbers and then we factorize those out like we have here and simplify them. And so that the number that’s left inside the square root or the radical is as small as it can possibly be. That is called “simplifying radicals.” So let’s take a look at a few examples.
Right then, our first example is to simplify the square root of eight. So what we need to do is look at eight and see if we can find any factors which are square numbers. In fact what we really like to do is look for the largest number, which is a square number and is also a factor of eight, that we can find. And we can see that four and two are factors of eight. And four is a square number, so we’re gonna have to take the square root of four and get an integer. And then we’re gonna write it slightly differently. So the square root of eight is the same as the square root of four times two. And we can separate that out into two separate terms: so the square root of four and the square root of two, and we’re multiplying those two together. So this bit here and this bit here are all equivalent. So the square root of four is two. So that gives us two times root two, so that’s this answer here.
So the next question is to simplify the square root of fifty. Well there’re quite a few different factors of fifty. So we’ve got five; we’ve got ten, but they’re not square numbers. So what we’re looking for remember is the largest factor of fifty, which is a square number. So I tend to just go through and say what’s fifty divided by two, what’s fifty divided by three, what’s fifty divided by four and keep doing that until my answer is a square number. And in fact fifty divided by two is twenty-five, and twenty-five is a square number. So I can rewrite the square root of fifty as the square root of twenty-five times two. And that’s the same as the square root of twenty-five times the square root of two. And of course the square root of twenty-five is five. So that’s equivalent to five root two.
And the next one, simplify the square root of twenty-eight. Okay, so again we’re looking for factors of twenty-eight that are square numbers. So twenty-eight divided by two is fourteen; two and fourteen, they are not square numbers. Twenty-eight divided by four is seven, so- well four and seven, four is a square number. Let’s try does it divide by three? No. And then we’re dividing by four; oh we’ve got back to four again. So we’ve run out of factors. So we know that the largest square factor is four. So we can write that out as the square root of four times seven, which is equivalent to the square root of four times the square root of seven. And of course the square root of four is two. So that gives us our answer of two times root seven or just two root seven.
Okay, so the last quick example that we’re gonna look at is this one: the square root of thirty-two. So let’s think of factors which are square numbers. So if I try dividing by two, then by three, then by four and see which of the other factors comes out as a square number, thirty-two divided by two is sixteen. And of course sixteen is a square number. So root thirty-two is the same as root sixteen times two. And as we saw before, that’s the same as the square root of sixteen times the square root of two. And because the square root of sixteen is four; that’s equivalent to four times root two or just four root two.
Now it’s just worth doing this one again in a slightly different way just to show that some of the problems that you can encounter if you don’t find the largest factor of that number, which is a square number. So for example, thirty-two also has factors four and eight, so four times eight. Four is a square number. So I’ve written that out as the square root of four times eight, which of course is the square root of four times the square root of eight, which gives us two root eight. So we think we’ve got two different answers for the same question. But the problem is that this second one that we’ve got here isn’t fully simplified because as we saw up here the square root of eight can be written as two root two; so that is in its simplest form. Because we didn’t find the largest factor of thirty-two, which is a square number, we have simplified it a bit, but we haven’t fully simplified that expression. So because root eight is the same as two root two, this expression here means two times two root two, which is obviously four root two. So by carrying on and spotting the fact that I haven’t fully simplified it, I can still get to the same correct answer. But life is just so much easier if you found the largest square factor of the number in the first place.
Okay then, let’s have a look at this question. Solve 𝑥 squared equals two hundred, leaving your answer in surd format in its simplest form. So it says surd format here. That might say in radical format or it might say expressing it as a multiple of irrational number; you could encounter that in either of those forms. So let’s just write that expression out: 𝑥 squared equals two hundred. Well if we’re solving that, we want to know what 𝑥 equals. So what we have to do to 𝑥 squared to turn it into 𝑥? Well, we’ve got to take the square root of both sides of the equation. So the square root of 𝑥 squared is 𝑥 and that’s equal to the square root of two hundred. But it’s not quite as simple as that because it could be positive root two hundred or it could be negative root two hundred because negatives times negatives make positives.
So there we are, 𝑥 is equal to plus or minus root two hundred. Well we’ve solved it, and but we still need to put it into its simplest form. So we’ve got to try to look for factors of two hundred, which are square numbers and we want the biggest one of those that we can. So I’m looking for the largest square factor that I can find of two hundred and then times something else. So two hundred, so we’re gonna divide by two, divide by three, divide by four until we find the other factor which is a square number. So two hundred divided by two is a hundred. Ah, that is a square number. So it’s a hundred times two. And we can split the hundred and the two out remember, so that’s plus or minus the square root of a hundred times the square root of two. And because the square root of a hundred is ten. That becomes ten times root two or ten root two. So our final answer here is plus or minus ten root two.
So next example, a rectangle has sides of five plus root seven centimetres and five take away root seven centimetres. Find the perimeter and the area of the rectangle, giving your answers in their simplest form. So first of all, I strongly recommend drawing a diagram, always just helps you to collect your thoughts and understand what you’ve got to do. So essentially we’ve got one side which’s got a length of five plus root seven. I’m gonna put that in brackets just to make some of our calculations a bit more clear. And the other side is five minus root seven. So to work out the perimeter, I’m just gonna add up the length of all the sides, so that one plus that one plus that one plus that one. And to work out the area, you just do length times width of the rectangle.
So to work out the perimeter, we’ve got five minus root seven over here and five minus root seven over here. So we’re gonna add two of those together, so that’s these two here. So two lots of five minus root seven and we’ve got five plus root seven up here. And we’ve got to add that to another five plus root seven here, so we’ve got two of those that we’re adding in here. So just it’s kind of multiplying the brackets out like this. These two lots of five and two lots of root seven and then two lots of five and two lots of negative root seven. So the first bracket, two lots of five is ten and two lots of root seven is two root seven. And for the second bracket, we’ve got two lots of root five- oh sorry! two lots of five which is another ten, and we’ve got two lots of negative root seven, which is negative two root seven. So I’ve got ten and I’m adding another ten to that, which gives us twenty. And then I’ve got two root seven. And then I’m taking away the same amount — another two root seven. So those two are just gonna come and cancel each other out. So the total length there is twenty. Remember to add in the length- the units which are centimeters. So the answer is the perimeter is twenty centimetres.
And as we set to work out the area, I’m gonna multiply the length by the width. So that’s five plus root seven times five minus root seven. So to multiply out those brackets, I’m gonna use my FOIL — First, Outer, Inner, Last — method. So five times five is twenty-five; five times negative root seven is negative five root seven. Then I’ve got five times root seven again, but this is positive times positive in this case, that’s making a positive five root seven. And then root seven times root seven, and we’ve got a positive times a negative makes a negative.
So the expression is twenty-five take away five root seven add five root seven. Well they’re gonna cancel each other out. For if I’ve got a negative five root seven and I add five root seven, I’m gonna- I’m adding something to the negative itself; I’m gonna get up to zero and then I’ve got minus root seven times root seven.
Now the definition of a square root is what is it that when you multiply it the root of it by itself you get that number. So root seven times root seven is gonna be seven. Think about it, if you had the square root of four times the square root of four, well the square root of four is two. So two times two is four. If I had the square root of sixteen times the square root of sixteen, square root of sixteen is four, so that’s four times four would give us a sixteen. So the square root of seven times the square root of seven is just seven. So this means we’ve got twenty-five take away seven, and twenty-five take away seven is eighteen. Remember the unit for area is centimetres squared in this case because the measurements were in centimetres. So our answer is eighteen centimetres squared.
Right, let’s move on to our final example then in this section. Simplify fully one plus root two times four minus root two, giving your answer in the form a plus b root two, where 𝑎 and 𝑏 are integers. So what they’re telling you to do basically is multiply these brackets together. And we’ve just seen an example of that using the FOIL method, but they’re asking for the answer in a very specific format. And they do intend to throw these kinds of little twists into trying to throw you off the sense sometimes. 𝑎 plus 𝑏 root two, so this just means an integer — so a whole number — plus some whole number times the square root of two. So they’re not asking you for the value of 𝑎 or 𝑏 particularly, they’re just asking you to represent your answer in that kind of layout. Okay, let’s write out the question and stop multiplying out the brackets.
So we’re gonna do one times four, which gives us four. We’re then gonna do one lot of negative root two, which is just negative root two. Then root two times four or four lots of root two, they’re both positive number; so it’s gonna be a positive answer. And then we’ve got positive root two times negative root two. So positive times negative is gonna give us a negative answer and then we’ve got root two times root two. So I just really need that for the moment, but we do know that root two times root two as we just saw is just two. So we’ve got four take away two in terms of just normal rational numbers, and four take away two is two. And then we’re starting off with negative root two. Well that really means there’s just one of them. So it’s really negative one root two, one lot of root two. And then we’re adding another four lots of root two. So if we start off a negative one on the number line and add four, we’re gonna go one, two, three, four steps up to positive three.
So here’s our answer then, two plus three root two. And this tallies with the format that we’ve been asked to give the answer in. So in this case, 𝑎 would be two and 𝑏 is the multiplier of the root two here would be equal to three. Just check the signs carefully. So we were asked for 𝑎 plus 𝑏 root two. Well we’ve got two plus three root two. So we know that 𝑎 will be two and 𝑏 will be three. As I said before, they didn’t actually ask for the value of 𝑎 and 𝑏. But in some questions, they do; so at least you now know how to answer those sorts of questions as well. So hopefully that’s helped you to see how to simplify radicals or irrational numbers or surds in a few basic examples.