Skip to 0 minutes and 4 seconds MICHAEL: So sometimes it’s better to leave our answers in surd form. So let’s see how we can work with surds.

Skip to 0 minutes and 13 seconds PAULA: OK, so by ‘surd form’ what we mean is when we have our final answer, we’ll have some numbers and we’ll also have numbers written with a square root symbol. OK. So probably easier to show you what I mean rather than discuss.

Skip to 0 minutes and 27 seconds For example: if we had, we’ll call this root 36. So we’re quite familiar that 36 is a square number. We know that root 36 gives us 6.

Skip to 0 minutes and 38 seconds MICHAEL: So we’ve got an answer. This isn’t a surd, because we’ve actually found an answer that ends, it’s 6.

Skip to 0 minutes and 43 seconds PAULA: Square numbers in this context are very useful. We could also notice we could break down this 36 into 9 multiplied by 4.

Skip to 0 minutes and 55 seconds MICHAEL: OK, yeah 9 times 4 is 36.

Skip to 0 minutes and 57 seconds PAULA: This whole lot is square rooted. We could also write this as root 9, multiplied by root 4.

Skip to 1 minute and 8 seconds MICHAEL: OK, so is this going to give us the same answer? Let’s have a look.

Skip to 1 minute and 11 seconds PAULA: It should do. We know root 9 is just going to be 3. 3 squared is 9. We multiply this by, we know root 4 is 2, so we multiply by 2. 3 multiplied by 2 gives us 6.

Skip to 1 minute and 27 seconds MICHAEL: So it seems to work. If we can split our original number down into two numbers that multiply together to make it, then we can square root those individually you’ll still get the same answer.

Skip to 1 minute and 37 seconds PAULA: So we could write down all the numbers in the whole world and be here for a while! So as we often do, we generalise using some algebra. So like we had before, we could say we have the square root of two numbers multiplied together…

Skip to 1 minute and 53 seconds MICHAEL: a and b.

Skip to 1 minute and 54 seconds PAULA: a and b… and this could be any two numbers at all. This is exactly the same as we did here, as having our root of the first number…

Skip to 2 minutes and 4 seconds MICHAEL: So the square root of a…

Skip to 2 minutes and 5 seconds PAULA: Multiplied by the root of the second number (b).

Skip to 2 minutes and 9 seconds MICHAEL: OK, so in general, if we’ve got the square root of a multiplied by b, that’s the same as the square root of a, multiplied by the square root of b.

Skip to 2 minutes and 18 seconds PAULA: Perfect. —

Skip to 2 minutes and 19 seconds MICHAEL: So let’s see how this rule can help us to simplify surds.

Skip to 2 minutes and 23 seconds PAULA: OK, so if we start with the number, say, root 50, so we can see straight away it’s not a square number…

Skip to 2 minutes and 32 seconds MICHAEL: No.

Skip to 2 minutes and 33 seconds PAULA: So not as nice as our previous example.

Skip to 2 minutes and 35 seconds MICHAEL: So this is definitely a surd.

Skip to 2 minutes and 37 seconds PAULA: Definitely a surd. So in our calculator we’ll have a long decimal that would continue for ever and ever.

Skip to 2 minutes and 45 seconds MICHAEL: OK.

Skip to 2 minutes and 46 seconds PAULA: What we can do thought is think about this and are there any square numbers that 50 can be divided by?

Skip to 2 minutes and 52 seconds MICHAEL: OK.

Skip to 2 minutes and 53 seconds PAULA: If we’re not sure, I always ask students to list their square numbers. When they do that they might find that 4 goes into 50, but that would give them 12.5 and we don’t want to have a decimal.

Skip to 3 minutes and 5 seconds MICHAEL: So just whole number solutions, that multiply together to get 50 with one or both of them being square numbers.

Skip to 3 minutes and 13 seconds PAULA: Fantastic, OK, so we know that the largest square number that goes into 50 is 25. So I can just rewrite the same number as, rather than root 50, as root 25 multiplied by 2.

Skip to 3 minutes and 28 seconds MICHAEL: That makes sense.

Skip to 3 minutes and 29 seconds PAULA: Now this is really useful because using our general rule from earlier, I’m going to rewrite this again, still the same number just written a new way, as root 25 multiplied by root 2.

Skip to 3 minutes and 46 seconds MICHAEL: So we’re splitting out multiplication up so we can square root them individually.

Skip to 3 minutes and 50 seconds PAULA: Perfect, exactly yes. This is really useful because we know that root 25 is just 5. I’m going to write this root 2 right next to it, because we know that when numbers are written next to each other we just multiply. Also, because we know that root 2 is irrational, it’s just going to continue forever. So this is going to be my answer in simplest surd form.

Skip to 4 minutes and 15 seconds MICHAEL: So what we seemed to have done is taken a large number we’re square rooting and we’ve broken it down so that now we’ve just got the smallest number possible in the square root sign and a number outside.

Skip to 4 minutes and 26 seconds PAULA: Fantastic MICHAEL: So 5 root 2 is root 50 in its simplest surd form.

Skip to 4 minutes and 32 seconds PAULA: If you weren’t sure, you could type in root 50 into your calculator, it would be the same as typing in 5 lots of root 2. But it would be such a long decimal, even your calculator would round it off, so this is the most exact way of writing root 50.

# Exact answers: surd form

Take a square number, say 36 and use your calculator to find the square root of 36. Most calculators will give the answer as 6. If we square root any square number we will be left with a whole number.

However, this is not the case for other numbers. For example \(\sqrt{32} = 5.656854249...\). To express this number in exact form we write it as a multiple of a square root. This is called leaving your answer in surd form.

In this video, Paula and Michael look at how to manipulate surds in order to leave exact answers involving as square root in their simplest form.

## Problem worksheet

Now complete question 6 from this week’s worksheet.

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