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What are **SURDs** in Math?

A surd is an expression or term containing an irrational root. That is, a number that can't be
written as a whole number.

As they contain the radical symbol **√**, surds can also be referred to
as "radicals".

So the above question could also be phrased as what is a radical?

A couple of examples.

But

So

A **SURD** can be classed as pure, or mixed.

A "pure surd" contains only irrational terms, such as **√7**,  or
^{6}**√15**.

A "mixed surd" also contains rational terms, such as **2√3**,  or
**4√7**.

On occasions, when trying to solve certain Math problems. It can help to simplify expressions
involving surds.

Some general rules for more basic surds are:

- √a × √a = a

- \color{green}{\sqrt{a \space \times \space b}} = √a × √

- \color{blue}{\sqrt{\frac{a}{b}}} = \color{blue}{\frac{\sqrt{a}}{\sqrt{b}}}

Simplify

=> \bf{\sqrt{6 \times 10}} =

Simplify \bf\frac{\sqrt{450}}{ \sqrt{10}},

=> \bf\sqrt{\frac{450}{10}} =

Sometimes you can be required to rationalize a fraction/quotient involving a SURD.

Usually, this involves removing a SURD from the denominator on the bottom, and replacing it with a
whole number instead.

The

\bf{\frac{2}{\sqrt{7}}}.

Rationalizing the denominator was done by multiplying the top and bottom by **√7**.

Which eventually led to giving a fraction with a rational denominator, \bf{\frac{2\sqrt{7}}{7}}.

This multiplication didn't change the overall value of the fraction, as **√7** over itself is
equal to **1**.

So what was done, is the same as just multiplying by **1**. \bf{\frac{\sqrt{7}}{\sqrt{7}}} = **1**

The approach with denominators involving two terms where a surd is present, is very similar.

The section below explains how to deal with rationalizing such a denominator.

Conjugate Surd

As well as learning what are surds?

There is also something else to learn about which is known as the conjugate surd.

Say you wanted to rationalize the denominator in a fraction such as:

\bf{\frac{1}{2 \space + \space \sqrt{3}}}

Looking at the bottom line, something that can help here is knowing:

**(a + b)(a − b) = a ^{2} − b^{2}**.

To rationalize in situations like this, we can multiply the fraction we have by another fraction.

Which will be the denominator over itself, with a changed sign to minus instead of plus.

This original denominator with a changed sign is known as the "

\bf{\frac{1}{2 \space + \space \sqrt{3}}}

= \bf{\frac{2 \space - \space \sqrt{3}}{4 \space\space - \space\space 2\sqrt{3} \space\space + \space\space 2\sqrt{3} \space\space - \space\space \sqrt{3}^2}}

= \bf{\frac{2 \space - \space \sqrt{3}}{4 \space - \space 3}} = \bf{\frac{2 \space - \space \sqrt{3}}{1}} =

Rationalize \bf{\frac{2}{\sqrt{5} \space + \space \sqrt{3}}}

=> \bf{\frac{2}{\sqrt{5} \space + \space \sqrt{3}}}

= \bf{\frac{2\sqrt{5} \space\space - \space\space 2\sqrt{3}}{2}} = \bf{\frac{2\sqrt{5}}{2}} − \bf{\frac{2\sqrt{3}}{2}} =

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