# Surds

## What are surds?

Before we can try to solve surds, we first need to know what they are. Surds are square roots that cannot be reduced to give us a whole number. They
are irrational numbers. However, we can work with them by simplifying them or rationalising the denominator of the surd.
Simplifying Surds
First, we will try to simplify some surds. The concept of simplifying a surd is quite easy. However,
you will have to think out of the box a bit. On the site, there are three examples of simplification. In
the first one, we have converted the sqrt of 20 into a multiplication sum: sqrt of 5
x sqrt of 4. Since we know that the square root of 4 is 2, we can simplify this
expression by writing it as: 2 x sqrt 5. In the second expression, we have to
multiply two surds together. To gives us the square root of 81. Since the square
root of 81 is equal to 9, the answer to this sum is 9. In the last example, we have divided one square root
by another. This is possible because the sqrt of 16 over the sqrt of 8 is generally seen as the sqrt of (16/8).
Solving the fraction then allows us to give our answer as sqrt of 2. These example should have given you
a general idea of how simplifying surds works. You basically have to divide and multiply to see if you can
simplify the surd or even solve part of it as in example 1. Examples 2 and 3 were just there to show you how to multiply and divide surds to simplify
them. Underneath you will find some more examples.
Rationalising Denominators of Surds
In mathematics, we do not like to have surds as denominators of fractions.
Consequently, we rationalise them to make the fraction more sophisticated. Look at the
example on the right. We start off with the fraction sqrt 5 / sqrt 8. To rationalise the
denominator in this example, we just need to multiply it by sqrt 8 / sqrt 8 which is
equivalent to 1. Sqrt 5 x sqrt 8 gives us sqrt 40 and on the bottom, when we times sqrt 8 by sqrt 8 we get 8 because any square root times by itself is
equal to the number inside the square root. The rationalised fraction is sqrt 40 / 8. There are a few other situations in which the method is the same.
Look at the examples underneath.
Presentation
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Example 1
In this example, we split the square root
into two other square roots (9 and 5). We
can reduce sqrt 9 to 3 and are so able to
give a simplified answer.

Example 2
In this example, we multiplied two sqrts
together which gave us the sqrt of 144
that we can reduce to 12.

Example 3
In this example, we divided the two sqrts
by each other to give us the sqrt of 4.

Example 1
In this example, we have a similar
situation to the one in the example above.
We apply the same method to rationalise
the denominator.

Example 2
In this example, we have a number as the numerator and a number
and a surd as a denominator. To rationalise the denominator, we just
times the fraction by sqrt 2 / sqrt 2 (=1) which gives us 2 times 2
which is equal to 4 as the new denominator and then we just need to
solve the top by multiplying 5 and sqrt 2 which is 5 sqrt 2.

Example 3
In this last example, we have a number multiplied by a surd on the top and bottom of the fraction. The procedure is still,
the same, we times the fraction by sqrt 6 / sqrt 6 which gives us 7 sqrt 48 as the numerator and 30 as the denominator
of the rationalised fraction.

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Chapter 7.3: Learning Outcomes
Students will know what surds are!
Students will be able to simplify surds!
Students will be able to rationalise denominators in surd form!

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