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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. âˆš6, âˆš5, âˆš17, âˆš2, âˆš3, 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.  3âˆša and 5âˆša are similar

ii. 6âˆšx and 4âˆšx 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 – y2Â

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 âˆš3 – âˆš5 = âˆš3 + âˆš5

ii. conjugate of -2âˆš7 + âˆš3 = -2âˆš7 – âˆš3

In general, conjugate of âˆšx + âˆšy = âˆšx – âˆšy

conjugate of âˆšx – âˆšy = âˆšx + âˆšy

### Simplification of surds

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

Example 1:

i. $$\scriptsize \sqrt{50}$$

ii. $$\scriptsize \sqrt{200}$$

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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