Video Transcript
In this video, we’re gonna look at
one of the ways of simplifying radicals. You may know this topic as simplify
surds. We’re gonna be looking at how to
manipulate the expression so that the radicand, that’s the number under the radical,
or the root sign, is as small as possible. Then, we’ll go on to solve some
typical questions which require you to use this skill.
First, let’s remind ourselves of
some terminology. This symbol here, looks like a big
tick, is called the radical, or some people call it the root sign. This number here is called the
index, and that tells you which root you’re looking for. So, I’ll be looking for the square
root, the cube root, the fourth root, or whatever. And the number that you’re taking
the radical, or root, of is called the radicand. Now, when the index is two, we often
don’t bother writing it. So, instead of writing square root
of five like this, we tend to write it like this.
Now, let’s just quickly think about
why we keep numbers in this format, why we work with expressions like this. Well, for example, we could have a
right triangle like this with one side has got a length of three, one side has got a
length of five, and we want to know the length of the other side. Well, we can use the Pythagorean
theorem. And that states for in a right
triangle, the square of the hypotenuse is equal to the sum of the squares of the
other sides. So, in our triangle, 𝑥 squared is
equal to five squared plus three squared, which is 25 plus nine, which is 34.
And taking square root of both
sides, we’ve got 𝑥 is equal to the square root of 34. Now, that is an exact answer. And if you evaluated the square
root of 34 on your calculator, you’d get 5.830951895, and so on, going on forever in
fact. Well, we could round that to one
decimal place or two decimal places. Now, that rounded number is
obviously not 100 percent accurate.
But if you were building a wooden
frame to those dimensions for example, there’s a limit to how accurately you could
cut the wood. So, for practical purposes, one or
two decimal places will probably be a very convenient way to present the answer to a
carpenter who’s making the frame. But in the pure world of
mathematics, we’d rather work with our completely accurate answer the square root of
34.
Now, a set of conventions has
evolved around radicals, about how we present them. And one of these is to try to keep
the number under the root, the radicand, as small as possible. So, let’s take a look at some
situations where we can simplify the radicals using this convention.
Simplify the square root of 50.
Well, let’s think about the factors
of 50. One times 50 are 50. Two times 25 are 50. Three doesn’t go, four doesn’t go,
five times 10. And that’s it; there are all the
factors. And looking at all of the factors,
we can see that 25 is actually a square number. So, let’s rewrite the square root
of 50 as the square root of 25 times two. And the square root of 25 times two
can be rewritten as the square root of 25 times the square root of two. And as we said, 25 is a square
number, so the square root of 25 is exactly five.
So, that becomes five times the
square root of two. And of course, we don’t need to put
the multiplication sign between the two of them. We can just express it as five root
two. So, according to that convention,
five root two is considered to be simpler than root 50. Because the radicand, the contents
of that square root sign there, are smaller than 50. So, five root two is simpler than
root 50.
Now, let’s simplify the square root
of 32.
Well, 32 has got six factors, one
and 32, two and 16, and four and eight. And if we look carefully, we can
see that two of those are square numbers, 16 and four. So, we’re gonna take the largest,
16. And that means we can rewrite root
32 as root 16 times two. And then, we could separate out the
16 and the two. So, we get root 16 times root
two. And 16 being a square number means
that the square root is exactly four. So, that becomes four times root
two, or as we’d normally write it four root two.
And we can see that the radicand is
smaller than it was before, so we’ve simplified that expression. Now, if we’d have chosen the other
square factor of 32, four, we could’ve rewritten root 32 as root four times
eight. Which is root four times root
eight, which is two root eight. And again, that radicand is smaller
than it was before; eight is smaller than 32. So, we have simplified the
expression root 32. But that eight has a square factor,
four times two is eight. And four, as we said before, is a
square number.
So, although we’ve simplified the
radical, root 32, we haven’t fully simplified it because what we’ve got left in this
radical here could still be simplified further. So, it’s equal to two times root
four times two. And we can split up the four and
the two into their own roots. And then, root four is two. So, that becomes two times two
times root two, which is equal to four root two, which is the answer that we got
before.
So, if you want to fully simplify a
radical, make sure that you get into the habit of always checking the radicand in
your answer to see if it’s got any square factors. If it has got square factors, then
you can simplify it down further. Okay, I want you to pause the video
and to have a go at this yourself. So, I’ll just wait three seconds
and then I’ll go through the answer.
Well, we’ve got to fully simplify
the square root of 27.
So, first of all, let’s find out
the factors of 27. And they are one and 27, and three
and nine. Well, nine is a square factor, so
the square root of 27 is root nine times three. And of course, that splits up into
root nine times root three. The square root of nine is
three. So, our answer is three root
three. And just checking that radicand
three, the factors of three are only one and three. None of those are square numbers,
so we can’t simplify any further.
Now, we’ve got to fully simplify
six root 20, so six times root 20.
Let’s write down the factors of
20. And they are one and 20, two and
10, four and five. Well, the biggest square factor
there is four. In fact, the only square factor is
four. So, we can rewrite six root 20 as
six times the square root of four times five. And that’s the same as six times
the square root of four times the square root of five. And the square root of four, of
course, is two. So, that becomes six times two
times the square root of five. And six times two is 12, so the
answer is 12 root five. And again, looking at the radicand
five, that’s got no square factors, So, we’ve simplified this as fully as we
can. Lastly then, let’s look at this
question.
Fully simplify the square root of
two plus the square root of eight plus the square root of 18 plus the square root of
32.
So, we’re just gonna go through
each of these terms individually and see if we can simplify them down. Well, root two, that radicand
doesn’t have any square factors. So, that’s as simple as it can
go. So, let’s consider root eight. And eight has got the factors one
and eight, and two and four. And four is a square number. So, we can rewrite root eight as
the square root of four times two, which is the square root of four times the square
root of two. And the square root of four is two,
so root eight is the same as two root two. And root two, that doesn’t have any
square factors, so that’s fully simplified.
Next, let’s consider root 18. Well, 18 has got six factors, one
and 18, two and nine, and three and six. And nine is the largest square
factor, that is, the only square factor. So, we’re gonna be able to simplify
this one. Root 18 is equal to the root of
nine times two, which can be split into the root of nine times the root of two. And the square root of nine is
three. So, root 18 becomes three root
two. Again, that won’t simplify any
further.
Lastly then, let’s think about the
factors of 32. Well, 32 also got six factors, one
and 32, two and 16, and four and eight. So, the largest square factor there
is 16. So, root 32 is the root of 16 times
two. So, that simplifies to four root
two. Now, we can start trying to
simplify our expression. So, root two plus root eight plus
root 18 plus root 32 is root two plus two root two plus three root two plus four
root two. And root two just means one lot of
root two. So, that means we’ve got one of
these root twos plus another two plus another three plus another four. That makes 10 in total. So, our answer is 10 root two.
Now, that’s quite a lot of working
out for a pretty short little question, so let’s talk about a couple of little
strategies that mainly you can do a lot of this in your head rather than have to
write quite so much working out down. Now, the first strategy is a bit of
exam technique. If you’ve been given this sort of
question in a test or an exam, you’ve been told that the first term is root two. Now, the chances are that these
others are gonna be multiples of root two as well. And that should guide you for
starting points for trying to work out the simplified versions of each of these
other terms.
Now, if these are all gonna be
multiples of root two, then we’re gonna be able to factor those things down into
something times two. And hopefully, they’re going to be
square numbers. So, let’s just try that. Half of eight is four. And yeah, four is a square
number. Half of 18 is nine, and nine is a
square number. And half of 32 is 16, so 16 is a
square number. So, we didn’t have to work out all
the factors of those numbers. The question gave us a clue for a
good starting point to start making guesses as to what those numbers might be.
And you can apply some techniques
even if you’re not given clues in the question. For example, if you were asked to
fully simplify the square root of 80, you could try dividing by two then three then
four then five. And as soon as you get a square
factor, you know you got the largest square factor. So, 80 divided by two is 40, and
that’s not a square number. 80 doesn’t divide exactly by
three. 80 divided by four is 20, and
that’s not a square number. And 80 divided by five is 16. And that is a square number. So, 16 is gonna be our larger
square factor.
So, root 80 is root 16 times
five. And then, still, we don’t have to
write out all the that equals root 16 times root five, and that equals four times
root five. We can jump straight to the answer
cause we can do that bit in our head. So, all of this stuff here would
probably have been done in our head. And this is the only working out
we’d have had to write down.
