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# Binomial Products with Surds

Like the distributive law of numbers, binomial products with surds is given as:

(a + b)(c + d)  =  ac + ad + bc + bd

We use this procedure to look at a few examples of products of surds

Example 1:

=

=  2 – 25

=  -23

Example 2

=

=

=

=

Remember these rules/formulae from Algebra – they will be useful in expanding and simplifying surds

(a + b)2  =  a2 + 2ab + b2

(a – b)2  =  a2 – 2ab + b2

a2 – b2  =  (a – b)(a + b)

Example 3:

=

We can use the distributive law to solve the above equation, but instead we will use the formula (a + b)2  =  a2 + 2ab + b2 to solve it. So

=    +    +

=  2 +  + 3

=  5 +

Example 4

=    –    +

Notice we have used the formula (a – b)2  =  a2 – 2ab + b2 above

=  9×5  –    +  9×3

=  45 + 18 –

=  63 –

Example 5

=   –   (using the formula a2 – b2  =  (a – b)(a + b))

=  5 – 3

=  2

and  are called conjugate surds. Two surds are called conjugate surds when they multiply to give a rational number. This process of getting a rational number is called rationalisation of surds, which we will see in the next section.