Recap that a rational number is a number that can be expresses as a fraction , where a and b are integers, b. Examples of rational numbers are 1, 0.5 and .

When a number cannot be written as a rational number, it is a irrational number. Examples of irrational numbers are , and .

When an irrational number is expresses using the radical () or root symbol, it is called a surd.

Surds can be simplified using the following properties.

For a, b > 0,a) x = b) c) x =

Adding and subtracting like surds is similar to adding and subtracting like terms in algebraic expressions. For example,

+ = = and - = = .

Unlike surds can be combined only if they can be converted to like surds.

The product of two expression involving surds can be simplified using the following property.

, where b, d > 0

Example 1

Simplify each of the following :-

a)

Example 2

Simplify each of the following :-

a) b)

Example 3

Simplify each of the following :-

a)))

Rationalise fractions with surds in their denominators

A fraction with surds in its denominator is often simplified such that the resulting denominator will no longer contain the radical () symbol. For example, the fraction is usually simplified this way,

This process of simplifying such a fraction is called rationalising the denominator.

and are conjugate to each other, and

+ and - are conjugate to each other.

The product of conjugate surds is always a rational number. The concept of conjugate surds is used in rationalising denominators.

Eg ) = = , which is rational

Example 4

By rationalising the denominator, simplify each of the following:-

Example 5

A triangle has a base ) cm and an area ) cm2. Find the height in the form ) cm, where a and b are rational numbers.

Equations Involving surds The following properties are often used to solve simple surd equations for unknowns. a) (squaring both sides)b) = => (Equality Property)

where a, b, c and d are rational numbers, and is a irrational number.

Example 6

Solve each of the following equations :-

a) = b) - = 0

Example 7

Given that )( )= where a and b are integers, find the value of a and b

Homework Assignment

Textbook Page 41Q11 Textbook Page 42Q14, Q15 and Q17