Basic Forms of Surds

\( \scriptsize \sqrt{a}\) is said to be in its basic form if a does not have a factor that is a perfect square.

Thus \( \scriptsize \sqrt{a}\) will only be basic if it cannot be broken down further into two factors where one of them is an exact square root.

E.g. √6, √5, √17, √2, √3, are in basic form.

 \( \scriptsize \sqrt{18}\) is not in its basic form because

⇒ \( \scriptsize \sqrt{18} \\ \scriptsize = \sqrt {9 \: \times \: 2} \\ \scriptsize = \sqrt{9}\: \times \:\sqrt {2} \)

= \( \scriptsize 3 \sqrt{2} \) which is now in its basic form.

Similar Surds:

Surds are similar if their irrational parts contain the same number e.g.

i.  \(\scriptsize 3 \sqrt{a} \: and \: 3 \sqrt{a}\) are similar
ii. \(\scriptsize 6 \sqrt{x} \: and \: 4 \sqrt{x}\) are similar

Conjugate Surds:

Conjugate surds are two surds whose product results in a rational number.

In an expression containing a difference of two squares, it is known that:

(x + y)(x – y)
= x² – y² 

In a similar manner

\( \left ( \scriptsize \sqrt{x} + \sqrt{y}\right) \left (\scriptsize \sqrt{x} \: – \: \sqrt{y} \right) \\ \scriptsize = \left (\scriptsize \sqrt{x^2} \right) \scriptsize \: – \: \left (\scriptsize \sqrt{y^2} \right) \\ \scriptsize = x \: – \: y \)

Simplification of Surds:

Surds can be simplified either in their simplest forms or as a single surd.

Example 1.2.1:

i. \( \scriptsize \sqrt{50} \)
ii. \( \scriptsize \sqrt{200} \)

Solution

i. \( \scriptsize \sqrt{50} \\ \scriptsize = \sqrt{2 \: \times \: 25} \\ \scriptsize = \sqrt{2} \: \times \: \sqrt{25} \\ \scriptsize = \sqrt{2} \: \times \: 5\)

= \( \scriptsize 5 \sqrt{2} \)

ii. \( \scriptsize \sqrt{200} \\ \scriptsize = \sqrt{2 \: \times \: 100} \\ \scriptsize = \sqrt{100} \: \times \: \sqrt{2} \)

= \( \scriptsize 10 \sqrt{2} \)

Example 1.2.2:

Express the following as surds;

i. \( \scriptsize3 \sqrt{7} \)
ii. \( \scriptsize 7 \sqrt{6} \)

Solution

i. \(\scriptsize 3 \sqrt{7} \\ \scriptsize = \sqrt{9} \: \times \: \sqrt{7} \\ \scriptsize = \sqrt{3^2} \: \times \: \sqrt{7}\\ \scriptsize = \sqrt{3^2 \: \times \: 7}\\ \scriptsize = \sqrt{9 \: \times \: 7} \\ \scriptsize =\sqrt{63} \)

ii. \( \scriptsize 7 \sqrt{6}\\ \scriptsize = \sqrt{7^2 \: \times \: 6}\\ \scriptsize = \sqrt{49 \: \times \: 6} \\ \scriptsize = \sqrt{294} \)