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Lesson 1, Topic 2
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# Basic forms of Surds

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$$\scriptsize \sqrt{a}$$ 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.  3√a and 5√a are similar
ii. 6√x and 4√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 – y

In a similar manner

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

Examples:

i. conjugate of √3 – √5 = √3 + √5

ii. conjugate of -2√7 + √3 = -2√7 – √3

In general, conjugate of √x + √y = √x – √y

conjugate of √x – √y = √x + √y

### Simplification of Surds:

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

### Example 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 2:

Express the following as surds;

i. $$\scriptsize3 \sqrt{7}$$
ii. $$\scriptsize 7 \sqrt{6}$$

Solution

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

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

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