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.

\(\sqrt{12} \cdot \sqrt{ 8}\)

\(\sqrt{4 \cdot 3} \cdot \sqrt{4 \cdot 2}\)

\(2\sqrt{3} \cdot 2\sqrt{2}\)

\(2 \cdot 2 \cdot \sqrt{3 \cdot 2}\)

\(4 \sqrt{6}\)

**Adding and Subtracting Surds
**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.

2. Simplify the radicals.

**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.