Video Transcript
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 centimetres. 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.
