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.
\(\sqrt{12} \cdot \sqrt{ 8}\)
\(\sqrt{4 \cdot 3} \cdot \sqrt{4 \cdot 2}\)
\(2\sqrt{3} \cdot 2\sqrt{2}\)
\(2 \cdot 2 \cdot \sqrt{3 \cdot 2}\)
\(4 \sqrt{6}\)

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.