Manipulation of Surds

For any positive numbers a and b, can you observe the relationship between Can you evaluate the following expressions?  36 6 2 3 6 100 10 2 5 10 144 12 3 4 12 225 15 3 5 15

Properties of Surds (I) This property is useful in evaluating square roots. For example,

Follow-up question Evaluate the following expressions without using a calculator. Solution

For any positive numbers a and b, can you observe the relationship between Now, try to evaluate the following expressions. 4 2 2 16 4 4

Properties of Surds (II) We can also use this property to evaluate square roots. For example,

Follow-up question Evaluate the following expressions without using a calculator. Solution

Rationalization of Denominators For example, . the called is 2) (e.g. number rational a into ) 2 irrational an from denominator changing of process The r denominato ation rationaliz

Follow-up question Rationalize the denominators of the following expressions. Solution

Surd in its Simplest Form For a surd with an integer inside the radical sign, we can simplify the surd by using its properties. For example, The surd obtained is said to be in its simplest form. i.e. 25 is the simplest form of 20.

To simplify a surd with a fraction inside the radical sign, we have to rationalize denominator first. For example,

For surds in their simplest form, 15 For surds in their simplest form, the number inside each radical sign does not have a square number (other than 1) as its factor.

Follow-up question Determine whether each of the following surds is in its simplest form. If not, express it in its simplest form. Solution

Like surds and unlike surds Some surds contain the same number inside the radical signs when expressed in their simplest forms. They are called like surds. For example, 3 ,3 and 4 contain 3.  They are like surds.

Are they like surds? Let’s simplify them first.  They are like surds.

However, unlike surds cannot be combined. called are They signs. radical the inside number same contain not do 7 4 and 5 2 surds unlike However, unlike surds cannot be combined.

Addition, Subtraction, Multiplication and Division of Surds In adding or subtracting surds, we should express all the surds in their simplest forms and then combine the like surds. For example,

In multiplication and division of surds, we can applied the properties of surds to do the calculation. For example,

Follow-up question Simplify the following expressions. Solution