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The Laws Of Surds.
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What Is A Surd ? Calculate the following roots: = 3 = 5 = 2 = 6 = 2
All of the above roots have exact values and are called rational . Now use a calculator to estimate the following roots: All these roots do not have exact values and are called irrational . They are called surds.
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Adding & Subtracting Surds.
Because a surd such as 2 cannot be calculated exactly it can be treated in the same way as an “x” variable in algebra. The following examples will illustrate this point. Simplify the following: Treat this expression the same as : 4 x + 6x = 10x Treat this expression the same as : 16 x - 7x = 9x
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Simplifying Square Roots.
Some square roots can be broken down into a mixture of integer values and surds. The following examples will illustrate this idea: Simplify: To simplify 12 we must split 12 into factors but at least one of the factors used must have an exact square root. (1) 12 = 4 x 3 Now simplify the square root. From this example it can be appreciated that you must use the square numbers as factors in order to simplify the square root. = 2 3 The square numbers are: 4,9,16,25,36,49,64,81,100,121,144,169,196,225…
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(3) 72 (2) 45 (3) 32 = 4 x 18 = 9 x 5 = 16 x 2 = 2 x 9 x 2 = 42 = 35 = 2 x 3 x 2 (4) 2700 = 62 = 100 x 27 This example demonstrates the need to keep looking for further simplification. Using 36 would have saved time = 10 x 9 x 3 = 10 x 3 x 3 = 303
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What Goes In The Box ? 1 Simplify the following square roots: (2) 27
(3) 48 (1) 20 = 43 = 25 = 33 (6) 3200 (4) 75 (5) 4500 = 53 = 402 = 305
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Rationalising Surds. You may recall from your fraction work that the top line of a fraction is the numerator and the bottom line the denominator. Fractions can contain surds: If by using certain maths techniques we remove the surd from either the top or bottom of the fraction then we say we are “rationalising the numerator” or “rationalising the denominator”.
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To rationalise the denominator in the following multiply the top and bottom of the fraction by the square root you are trying to remove: 5 x 5 = 25 = 5
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What Goes In The Box ? 2. Rationalise the denominator of the following expressions:
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Conjugate Pairs. Consider the expression below:
This is a conjugate pair. The brackets are identical apart from the sign in each bracket . Now observe what happens when the brackets are multiplied out: = 3 X 3 - 6 3 + 6 3 - 36 = = -33 When the brackets are multiplied out the surds cancel out and we end up seeing that the expression is rational . This result is used throughout the following slide.
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Rationalise the denominator in the expressions below by multiplying top and bottom by the appropriate conjugate: In both of the above examples the surds have been removed from the denominator as required.
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What Goes In The Box ? 3. Rationalise the denominator in the expressions below : Rationalise the numerator in the expressions below :
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