\( \scriptsize \sqrt{a}\) 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 – y2
In a similar manner
\( \left ( \scriptsize \sqrt{x} + \sqrt{y}\right) \left (\scriptsize \sqrt{x} \: – \: \sqrt{y} \right) \\ \scriptsize = \left (\scriptsize \sqrt{x^2} \right) \scriptsize \: – \: \left (\scriptsize \sqrt{y^2} \right) \\ \scriptsize = x \: – \: y \)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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