Surds

A surd is a value in a root that cannot be determined unless you use a calculator.
Example: √2 (square root of 2) can't be simplified further so it is a surd

How to multiply surds

If you have two surds that are multiplied and their roots are the same, you put them under one root and multiply the values.

$\sqrt{3a}\times <!--\; \times \; -->\sqrt{3a}=\sqrt{3a\times <!--\; \times \; -->3a}=\sqrt{9a^2}=3a$


How to add and subtract surds

If you add and subtract surds, you always have to find a common factor and then simplify the expression.

$2\sqrt{2}+6\sqrt{8}=2\sqrt{2}+6\sqrt{2\times <!--\; \times \; -->4}\text{(write 8 as 2 times 4)}=2\sqrt{2}+6\times <!--\; \times \; -->2\sqrt{2}\text{(the root of 4 is 2)}=2\sqrt{2}+12\sqrt{2}=\sqrt{2}(2+12)\text{(take }\sqrt{2}\text{out as common factor)}=14\sqrt{2}$

Rationalizing denominators 

When fractions have a surd in the denominator we can rationalize the denominator so that it is not in surd form. We do this by multiplication.

Factors 

To simplify an expression you must always try to take out common factors.

$\sqrt{4-<!--\; -\; -->4x}=\sqrt{4(1-<!--\; -\; -->x)}=\sqrt{4}\times <!--\; \times \; -->\sqrt{1-<!--\; -\; -->x}\text{(}\sqrt{4}=2)=2\sqrt{1-<!--\; -\; -->x}$


Equations with surds 

Given an equation that contains a surd you must always first change the surd into a form that has no root sign and then solve the unknown values.

How to Simplify Radicals/Surds Summery