Surds

A surd is an irrational root that cannot be expressed exactly as a rational number.
Examples include:
However:
is not a surd, because it gives an exact rational number.
Similarly:
is not a surd.
Important point
A surd is an exact value, not a decimal approximation.
For example:
but � is the exact form.
8. Simplifying Surds
Look for a perfect-square factor.
Example 1
Simplify:
Break 12 into:
Therefore:
So:
Example 2
Simplify:
Therefore:
9. Rules for Manipulating Surds
Rule 1: Multiplication
Example
Rule 2: Division
where appropriate.
Example
10. Adding and Subtracting Surds
Only like surds can be added or subtracted directly.
Example 1
Add the coefficients:
Example 2
Important misconception
Students must not do:
This is incorrect.
In general:
11. Multiplication of Binomial Surds
Surds can be expanded using ordinary algebra.
Example
Simplify:
Using expansion:
The middle terms cancel:
This introduces the idea of conjugates, which are important when rationalizing denominators.
12. Rationalization of the Denominator
A fraction is said to have a rationalized denominator when there is no surd in the denominator.
Example 1
Rationalize:
Multiply the numerator and denominator by �:
Therefore:
Example 2
Rationalize:
Multiply by:
Therefore:
So:
13. Rationalizing a Binomial Denominator
When the denominator contains two terms, use the conjugate.
For:
the conjugate is:
Example
Rationalize:
Multiply by the conjugate:
Therefore:
Using difference of two squares:
Therefore:
Key rule
This makes the surd disappear from the denominator.