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165 pagesrevision 46935
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24
The property being used in these steps is as follows:
√ a = √ x ∗ y = √ x ∗ √ y = x √ y (25)
Here a is basically the initial number, x and y are your factors and they can be
split into separate square roots. Assuming that x appeared in a pair, its
squared root can be cut and will be be only left as a single number outside the
square root. The remaining number y will remain inside in its initial form.
3.2 Rationalisation of Surds
Rationalisation refers to the process of changing a fraction that has surd in its
denominators into a denominator with a whole number. The procedure for it, in
general form, is as follows:
4 1. Look at the denominator and find its real conjugate:
conjugate refers to the alternate form of the denominator so for single surds
like √ 5 its the same √ 5 but for denominators like 1 + √ 5, always change the
sign in between to find the conjugate as 1 − √ 5 so as a rule of thumb:
for compound conjugates, if there’s a plus sign, change it to a minus and if
there’s a minus, change it to a plus sign.
2. Multiply the surd fraction’s numerator and denominator given in the
question with the conjugate your just found.
3. Use basic algebraic simplifications or sometimes surd simplification to give
the final answer in the simplest terms possible. Let’s consider an example as
follows:
Simplify 2 1+√ 5
Now we can simplify it as:
2 1 + √ 5 ∗ (1 − √ 5) 1 − √ 5 (26) 2(1 − √ 5) (1 + √ 5)(1 − √ 5) (27)
2(1 − √ 5) (1 + √ 5)(1 − √ 5) (28) 2(1 − √ 5) (12 − ( √ 5)2 (29) 2(1 − √ 5) (1 − 5
(30) 2(1 − √ 5) −4 (31) 1 − √ 5 −2 (32)
Multiplying out by -1 gives us the final answer as: √ 5 − 1 2 (33)