### Video Transcript

In this video, weโll learn how to
solve problems involving operations and properties of operations on real
numbers. These will include calculations
involving surds or radicals, such as the square root of two or the square root of
five, and will involve us considering inverse operations. In other words, opposite
operations.

We begin by defining some
terms. When we talk about the inverse in
mathematics, weโre talking about something opposite in effect. An inverse operation is an
operation which undoes what was done by the previous operation. Weโre going to be looking at the
additive inverse and the multiplicative inverse. Now, the additive inverse of a
number ๐ is the number that when added to ๐ gives zero. And a multiplicative inverse of a
number ๐ is the number that when multiplied by ๐ gives one. Another way of considering this is
as the reciprocal, one over ๐. Letโs have a look at how we can
calculate these.

Find the additive inverse of eight
minus the square root of 135.

Remember, the additive inverse of a
number ๐ is the number that when added to ๐ gives zero. So, we need to find a number that
when we add it to eight minus the square root of 135, we get zero. And one way to answer this is to
use algebra. Letโs let ๐ฅ be the additive
inverse of eight minus the square root of 135. Then, we know that the sum of ๐ฅ
and eight minus the square root of 135 is zero. And since weโre simply adding here,
we donโt actually need these parentheses or brackets.

We want to find the value of
๐ฅ. Remember, weโre trying to find the
additive inverse of our number. And so, weโre going to solve this
equation. We have ๐ฅ plus eight minus the
square root of 135. So, we begin by subtracting eight
from both sides of our equation. On the left-hand side, that leaves
us with ๐ฅ minus the square root of 135. And on the right-hand side, we get
negative eight. So, ๐ฅ minus the square root of 135
is equal to negative eight. The opposite of subtracting is
adding. So, next, we add the square root of
135 to both sides. And so, we see that ๐ฅ is equal to
negative eight plus the square root of 135. And itโs quite usual to write the
positive number first.

And so, we can say that the
additive inverse of eight minus the square root of 135 is the square root of 135
minus eight. Now, it follows that since we know
that the additive inverse of a number and that number sum to zero. We can check our solution by adding
eight minus the square root of 135 and the square root of 135 minus eight. So, thatโs eight minus the square
root of 135 plus the square root of 135 minus eight. So, we see that eight minus eight
is zero and negative root 135 plus root 135 is zero. So, we get zero as required.

Next, weโll look at an example of
how to find the multiplicative inverse.

Find the multiplicative inverse of
the square root of six over 30.

Remember, the multiplicative
inverse of a number ๐ is the number that when we multiply it by ๐ gives us
one. Another way of considering this is
as the reciprocal of that number. So, if we have a number ๐, its
reciprocal is one over ๐. So, here, we need to find the
number that when we multiply it by the square root of six over 30, we get one. If we let ๐ฅ be the multiplicative
inverse of the square root of six over 30, then we could say that ๐ฅ times the
square root of six over 30 is equal to one.

Now, equivalently, we would achieve
this by solving this equation. We said that it is also the
reciprocal of the original number. Thatโs one over the number. So, itโs one over the square root
of six over 30. This doesnโt look very nice,
though. So, weโre going to recall how we
divide fractions. Really, weโre wanting to divide one
by the square root of six over 30. So, we write one as one over
one. And then recall that to divide by a
fraction, we multiply by the reciprocal of that fraction. This is sometimes called keep,
change, flip. So, ๐ฅ is equal to one over one
times 30 over root six.

And if we multiply the numerators
and then separately multiply the denominators of our fractions, we get ๐ฅ is equal
to 30 over root six. Now, in fact, we really didnโt need
to perform this step. Given a fraction in the form ๐
over ๐, its reciprocal is simply ๐ over ๐. But of course, itโs always good to
understand where these things come from. So, we found the multiplicative
inverse to be 30 over the square root of six.

But weโre really not finished. We need to rationalize the
denominator. In other words, we want the
denominator of our fraction to be rational. At the moment, itโs an irrational
number. The square root of six cannot be
written as a fraction where the numerator and denominator are integers. So, how do we achieve this? Well, we multiply the numerator and
denominator of our fraction by the square root of six. Thatโs the same as multiplying by
the square root of six over the square root of six, or just by multiplying by
one.

And in doing so, all weโre doing is
creating an equivalent fraction. 30 times the square root of six is
30 root six. Then, the square root of six times
itself is, of course, simply six. Multiplying a number by itself is
squaring it and squaring is the inverse to square rooting. So, we see that our multiplicative
inverse is 30 root six over six. Finally, we spot that both 30 and
six have a common factor of six. And so, dividing through by six, we
get five root six over one, which is simply five root six. The multiplicative inverse of the
square root of six over 30 is five root six.

In our next example, weโll look at
how to find the multiplicative inverse of the sum of two radicals.

Find the multiplicative inverse of
root six plus root seven, expressing your answer in simplest form.

Remember, the multiplicative
inverse of a number ๐ is the number that when multiplied by ๐ gives one. Itโs the reciprocal of that number,
one over ๐. Weโre looking to find the
multiplicative inverse of root six plus root seven. So, thatโs the reciprocal of root
six plus root seven. Itโs one over that expression. The problem is, weโre not quite
finished. We need to give our answer in
simplest form. In other words, we need to
rationalize the denominator.

Currently, our denominator is an
irrational expression. It cannot be written as a fraction
where both the numerator and denominator of that fraction are integers; theyโre
whole numbers. So, how do we rationalize the
denominator of our fraction? We might recall that multiplying a
radical expression by its conjugate, that is, multiplying it by an expression where
we change the sign between the two terms, gives us a rational result. So, if we change the sign here, we
see that the conjugate of root six plus root seven is root six minus root seven.

We canโt just multiply the
denominator of our fraction though. We have to do the same to the
numerator. This is essentially like
multiplying by one. So, we create an equivalent
fraction. Letโs begin by evaluating the
denominator of our fraction. Weโre going to multiply root six
plus root seven by root six minus root seven. There are a number of techniques we
can use. Letโs use the FOIL method for
distributing parentheses or expanding brackets.

We multiply the first term in each
expression. The square root of six times the
square root of six is simply six. We then multiply the outer
terms. Thatโs the square root of six times
negative the square root of seven. By recalling that for real numbers
๐ and ๐, the square root of ๐ times the square root of ๐ is the square root of
๐๐. We see that the square root of six
times the square root of seven is root 42. So, our second term is negative
root 42. We then multiply the inner terms,
and we get positive root 42. Finally, we multiply the last terms
in each expression. And since root seven times itself
is simply seven, we get negative seven.

We notice now that negative root 42
plus root 42 is zero. And so, weโre left with six minus
seven, which is simply negative one. Now, distributing the parentheses
on our numerator is a little bit easier. One times root six and one times
negative root seven gives us root six minus root seven. So, we have root six minus root
seven over negative one.

Thereโs just one more step. Weโre going to divide each term on
our numerator by negative one. Root six divided by negative one is
negative root six and negative root seven divided by negative one is positive root
seven. So, we get negative root six plus
root seven, which we can write as root seven minus root six. So, we see that the multiplicative
inverse of root six plus root seven is root seven minus root six.

Now, of course, we can always check
our result by finding the product by multiplying these two values together. We use the FOIL method again. Root six times root seven is root
42. We then multiply root six by
negative root six to get negative six. Root seven times root seven is
seven. Then, root seven times negative
root six is negative root 42. The root 42s cancel and weโre left
with one as required.

Weโll now look at how to apply
these processes to a geometric problem.

Given that the dimensions of a
rectangle are 57 plus seven root two centimeters and 57 minus seven root two
centimeters, find the length of its perimeter.

Remember, the perimeter of a
rectangle is the total distance around the outside of the shape. Letโs sketch the rectangle out and
label its dimensions. We know that opposite sides in a
rectangle are equal in length. This means we can label our
opposite sides as shown. The perimeter is then the sum of
all of these dimensions. Itโs 57 minus seven root two plus
57 plus seven root two plus 57 minus seven root two plus 57 plus seven root two. And in fact, since weโre simply
finding the sum, we donโt actually need these brackets or parentheses.

Next, we notice that negative seven
root two and seven root two are additive inverses of one another. An additive inverse of a number ๐
is the number that when we add to ๐ gives zero. This means the sum of negative
seven root two and seven root two is zero. So, we have zero here and zero
here. The perimeter is, therefore, simply
57 plus 57 plus 57 plus 57 or 57 times four. Thatโs 228 or 228 centimeters. And so, the length of the perimeter
of our shape is 228 centimeters.

Weโll now consider the effect of
squaring on algebraic expressions involving square roots.

Given that ๐ is equal to the
square root of two and ๐ is equal to the square root of six, find the value of ๐
squared over ๐ squared.

Recall the order of operations. This is sometimes called BIDMAS or
PEMDAS. These tell us the order in which we
perform a series of operations. And so, looking at our expression
๐ squared over ๐ squared, we see we have exponents or indices. And this line here means
division. In each expression, BIDMAS and
PEMDAS, the exponents or indices are calculated before any divisions. So, weโre simply going to begin by
calculating the value of ๐ squared and ๐ squared.

๐ is the square root of two and ๐
is the square root of six, which means that ๐ squared must be root two squared and
๐ squared must be root six squared. We could rewrite each of these as
the square root of two times the square root of two and root six times root six,
respectively. Alternatively, we recall that
square rooting and squaring are inverse operations. They are the opposite of one
another, and they undo what the other one does.

This means the square root of two
squared is simply two, whilst the square root of six squared is six. So, weโve calculated ๐ squared and
๐ squared. We replace ๐ squared with two and
๐ squared with six in our original expression. And we see that ๐ squared over ๐
squared is two over six. And since both the numerator and
denominator of our fraction share a common factor of two, we divide them both by
two. Two divided by two is one and six
divided by two is three. So, the value of ๐ squared over ๐
squared is one-third.

In our very final example, weโll
consider one more real-life application of these processes.

A square has a side length of ๐
centimeters and an area of 63 square centimeters. Find the area of a square whose
side length is six ๐ centimeters.

Remember, the area of a rectangle
is calculated by multiplying its width by its height. A square is simply a rectangle
whose sides are the same length. And so, we can say that the area of
a square is its side length multiplied by itself or its side length squared. Now, weโre told that our square has
an area of 63 square centimeters. Weโre also told that its side
length is ๐. So, replacing area with 63 and side
length with ๐, we form an equation. We get 63 is equal to ๐
squared.

Now, the question wants us to find
the area of a square whose side length is six ๐ centimeters. So, what weโre going to do is begin
by calculating the value of ๐. In other words, weโre going to
solve this equation for ๐. To do so, we perform an inverse
operation. Currently, ๐ is being squared. The opposite of squaring is square
rooting a number. This essentially undoes the
previous operation.

And so, if we square both sides of
our equation, we get simply ๐ on the right-hand side. Then, the left-hand side is equal
to the square root of 63. Now, itโs worth recalling that when
we find the square root in an equation, we look to find the positive and negative
square roots of the number. But this is a side length, so we
canโt have a negative value. And ๐ is equal to the square root
of 63. We might be interested in
simplifying this radical or surd. But in fact, weโre not quite done
with it. So, weโll leave it as it is for
now.

Our new square has a side length of
six ๐ centimeters. We calculated ๐ to be equal to the
square root of 63. So, six ๐ must be six times
this. Itโs six root 63. Since this is the new side length
of our square, the area of this square is this value squared. Itโs six root 63 times six root
63. We distribute the two over both
parts of this value. So, we get six squared times the
square root of 63 squared. Six squared is 36.

And of course, squaring and finding
the square root are inverse operations of one another. They undo the other operation. So, the square root of 63 squared
is just 63. This means the area of our square
is 36 times 63, which is 2,268. And since weโre working in
centimeters, the units here are square centimeters.

In this video, weโve learned that
when we talk about the inverse in mathematics, weโre talking about something
opposite in effect. An inverse operation is one which
undoes what was done by the previous operation. We also learned about additive
inverses. We said the additive inverse of a
number ๐ is the number that when we add it to the original number ๐ gives
zero. We also saw that multiplicative
inverses of a number ๐ are the numbers that when multiplied by ๐ give one. And another way of considering this
is as the reciprocal, one over ๐.