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
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