Core 1: Surds
This section summarises all the
key ideas required for the surd section at AS-level.
Contents
1)
Adding, subtracting and
multiplying simple surds
2)
Simplifying surds
3)
Rationalising surds
4)
Surd problems
Adding, subtracting and
multiplying surds
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Treat like algebra: Consider: |
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Consider: |
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Consider: Treat each surd like a different algebraic term |
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When multiplying or dividing surds, the numbers can be
placed under one square root sign |
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Sometimes the result can be easily worked out |
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Multiply the coefficients and surds separately. |
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Consider Treat surds in the same way |
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Generally, write the coefficient before the surd sign |
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Consider |
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Notice how the surds vanish. This is an important result Generally: |
Simplifying surds
This is a very important section.
Take a while to read and understand the examples
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Method 1: |
Look for the largest square factor of the number under the
square root. Then break it up into two surds multiplied together. |
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Method 2: |
Write the number as the product of its prime factors. Every number has a unique set of prime factors: |
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Method 1: |
36 is the highest square factor of 72 |
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Method 2: To speed up the next part, you can use the fact that: So, |
Write 72 as the unique product of its prime factors |
Rationalising surds:
Learn these two important facts.
Rationalising effectively means
‘move’ any surd from the denominator of a fraction to the numerator.
To do this we use ideas of
equivalent fractions
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We do not like the surd as a denominator. We therefore multiply top and bottom by Remember: |
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We do not like the surd as a denominator. We therefore multiply top and bottom by Remember: |
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When the denominator is of the form |
We will encounter more of these
in detail next...
Surd problems
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Simplify: |
Simplify the surds where possible: |
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So the question becomes: |
Rationalise each surd separately: Make the denominators the same. Write over the same denominator Simplify Write as two separate fractions. |