# Surds

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## What are Surds?

Surds are an expression in root form such as square root, cube root and other in a root symbol.  A surd cannot be simplified to remove the root symbol. It does not have an exact decimal value and cannot be represented by a fraction. The decimal value just continues on and on to infinity, neither a terminating nor recurring decimal. It is also called an irrational number. 4 is not a surd as it can be simplified $$\sqrt{4}$$ = 2. $$\sqrt{2}$$ is an irrational numbers. $$\sqrt{2}$$ =1.41421356… and does not have an exact decimal form.

Manipulating surds using operations, there are rules to follow in order to perform a mathematical operation. Let’s first learn a way to simplify a surd.

Example:
Simplify $$\sqrt{200}$$                      Think of a factor of 200, it can be 100 multiplied by 2.
$$\sqrt{100 \cdot 2}$$                                    The square root of one hundred is 10
$$10\sqrt{2}$$                                         is the simplified form of $$\sqrt{200}$$.

Multiplying Surds
Multiplying surds is simply combining surds. However, if like surds are multiplied, an answer is a rational number.
Example 1: $$\sqrt{2} \sqrt{ (2)}$$
$$\sqrt{2} \sqrt{ (2)}$$  They are like surds since their radicand is both $$\sqrt{2}$$.
$$\sqrt{2 \cdot 2}$$
$$\sqrt{4}$$The answer is always in simplest form:

Example 2: $$\sqrt{12} \cdot \sqrt{ 8}$$
$$\sqrt{12} \cdot \sqrt{ 8}$$
$$\sqrt{12 \cdot 8}$$
$$\sqrt{96}$$
$$\sqrt{12} \cdot \sqrt{ 8}$$
$$4 \sqrt{6}$$

Another way to answer this is to simplify both equations first.
$$\sqrt{12} \cdot \sqrt{ 8}$$
$$\sqrt{4 \cdot 3} \cdot \sqrt{4 \cdot 2}$$
$$2\sqrt{3} \cdot 2\sqrt{2}$$
$$2 \cdot 2 \cdot \sqrt{3 \cdot 2}$$
$$4 \sqrt{6}$$

A surd that has the same radicand can be added or subtracted. Before adding or subtracting surds, simplify the expressions.
Example: $$\sqrt{5}+ 3 \sqrt{5}$$
$$\sqrt{5}+ 3 \sqrt{5}$$  Notice that they have same radicand which is $$\sqrt{5}$$.
$$\sqrt{5}+ 3 \sqrt{5}$$   Add like surds.
$$4 \sqrt{5}$$

Example 2: $$\sqrt{8} – \sqrt{12}$$
$$\sqrt{8} – \sqrt{12}$$  Simplify the surds
$$2\sqrt{2} – 2 \sqrt{3}$$ Unlike radicands, which is $$\sqrt{2}$$ and $$\sqrt{3}$$, cannot be subtracted.

Example 3: $$\sqrt{8} + \sqrt{18}$$
$$\sqrt{8} + \sqrt{18}$$     Simplify the given surds.
$$\sqrt{4 \cdot 2} + \sqrt{9 \cdot 2}$$
$$2\sqrt{2} + 3\sqrt{2}$$     There radicand is same so it can be added.
$$5\sqrt{2}$$

Rationalising the Denominator
A surd is cannot be a denominator. If the denominator is a surd, it is not the simplest form.
To rationalise a denominator, we need to clear the radicals.
1. Multiply the numerator and the denominator that make the denominator a perfect root.

Example:
$$\frac{1}{\sqrt{3}}$$
$$\frac{1}{\sqrt{3}}$$    To be a perfect root, multiply the numerator and denominator of $$\sqrt{3}$$.
$$\frac{1}{\sqrt{3}}\cdot \frac{\sqrt{3}}{\sqrt{3}}$$
$$\frac{1 \cdot\sqrt{3}}{\sqrt{3\cdot 3}}$$
$$\frac{\sqrt{3}}{\sqrt{9}}$$     Simplify the surds.
$$\frac{\sqrt{3}}{3}$$

Example 2:
$$\frac{2}{1 +\sqrt{2}}$$
$$\frac{2}{1 +\sqrt{2}}$$  To make the denominator a perfect root conjugate the denominator, which is 1-2.
$$\frac{2}{1 +\sqrt{2}}$$ $$\cdot$$ $$\frac{1- \sqrt{2}}{1 -\sqrt{2}}$$
$$\frac{2(1-\sqrt{2})}{(1+\sqrt{2})(1-\sqrt{2})}$$    Multiply using distributive law.
$$\frac{2 – 2 \sqrt{2}}{1 + \sqrt{2} -\sqrt{ 2} – \sqrt{2 \cdot 2}}$$   Simplify the surds.
$$\frac{2-2\sqrt{2}}{1-2}$$
$$\frac{2(1 -\sqrt{ 2})}{-1}$$
$$-2 (1-\sqrt{2})$$

Note: The conjugate of a + b is a – b and vice versa. The product of the sum and difference of the two terms.