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 Treat like algebra: Consider:  Consider:   Consider: Treat each surd like a different algebraic term When multiplying or dividing surds, the numbers can be placed under one square root sign Sometimes the result can be easily worked out  Multiply the coefficients and surds separately. Consider Treat surds in the same way  Generally, write the coefficient before the surd sign   Consider     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

 Method 1:  Look for the largest square factor of the number under the square root. Then break it up into two surds multiplied together. Method 2:   Write the number as the product of its prime factors. Every number has a unique set of prime factors: Method 1:  36 is the highest square factor of 72 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  We do not like the surd as a denominator. We therefore multiply top and bottom by Remember:   We do not like the surd as a denominator. We therefore multiply top and bottom by Remember:    When the denominator is of the form multiply top and bottom by and vice versa. We will encounter more of these in detail next...

Surd problems

 Simplify:     Simplify the surds where possible:    So the question becomes:     Rationalise each surd separately: Make the denominators the same.     Write over the same denominator   Simplify     Write as two separate fractions.