# Definition of Surds Rationalization of the denominator.

**WEEK THREE**

**SS1 FURTHER MATHS FIRST TERM**

**SURDSÂ **

**CONTENTS**

- Definition of Surds
- Rules for manipulating surds
- Rationalization of the denominator.
- Equality of surds.
- Equations in irrational forms

**SUB TOPIC: DEFINITION OF SURDS**

Certain numbers can be expressed as ratios of two integers, i. e . Where p and q belong to the set of integers and q â‰ 0, such numbers are called rational numbers.

Examples of rational numbers are; 3, 1^{1}/_{2}, 3.5, -7.1 etc. Each of them can be expressed in the form , where p and q are integers such that q â‰ 0 as follows:

- 3 = (b) 1
^{1}/_{2 }= (c) 3.5 = 3^{1}/_{2 }= (d) -7.1 =

Some numbers however, cannot be expressed as ratios of two integers, i.e p/q such that q â‰ 0, p and q belonging to the set of integers. Examples of such numbers are , , etc such numbers are said to be irrational. Other examples of irrational numbers are pi()and the number exponential (). Their exact values cannot be determined. Their approximate values can only be determined.

and **2.7183**.

**SURDS** are irrational numbers which are roots of rational numbers. Examples of surds are , etc.

We shall consider only expressions which contains one or more square roots of prime numbers of their multiples. Such expressions are called quadratic surds.

**CLASS ACTIVITIES:**

- What is a surd?
- Write out the examples of surd
- Differentiate between rational and irrational numbers.

**SUB TOPIC: RULES OF MANIPULATING SURDS**

- = x

**Example:**

- = x
- = 12 or

= = x

= 3 x 4 = 12

- =

**Example **1: = =

2: = = = 4

- = x

=

2: = x

- =

**Example** 1: = =

2: = = = 1^{1}/_{2}

**CLASS ACTIVITIES:**

- State and give examples of the rules of surds
- Express the following in basis form:
- (b) (c) (d)

**SUB TOPIC: RATIONALIZATION OF THE DENOMINATOR**

To rationalize the denominator of the fraction is to remove the radial (the square root sign) from the denominator. This is accomplished by multiplying the fraction by = 1

Thus =

Therefore

**Example **1: =

= (3 x /5

=

To rationalize the denominator of the fraction of the form , we multiplied by the numerator and denominator by the **conjugate** of the denominator.

Two surds are said to be **conjugate** of each other if their product gives rise to a rational number.

e.g, to simplify we rationalize by the conjugate of the denominator i.e multiplied the fraction by thus

= x

**= **. Thus we observed from the solution above that the conjugate of the denominator is just the changing of the basic arithmetic sign in between (i.e + is changed to â€“ and viz versa).

**Examples**

- Rationalize the denominator of

**Solution:**

- Express in the form p + , where p and q are integers

**Solution:**

= x

=

= =

**CLASS ACTIVITIES:**

- If = p + q, where p and q are constants, find the value of p and q (WAEC)
- Given that = p + q where p and q are constants, find the value of p â€“q
- Simplify

**SUB TOPIC: EQUALITY OF SURDS**

Given two surds

Suppose

The LHS of the equation is a rational number while the RHS is not and can only be equal if they are both equal to zero.

Thus p â€“ q = 0

Â

And

The concept of equality of two surds enables us to determine the square roots of a surdic number.

**EXAMPLE: **Find the square roots of

**SOLUTION**

Let the square root of be

Solving simultaneously, m= 5, n = 7

Hence the square roots of .

**CLASS ACTIVITIES:**

- Find the square roots of each of the following

**SUB TOPIC: EQUATIONS IN IRRATIONAL FORM**

**EXAMPLE: **solve the equation

**SOLUTION:**

Rearrange the equation

Square both sides

Â

Â

Â

Â

Squaring both sides again, we obtain

Â

Â

Â

Let

Â

let

Â

Â

NOTE: squaring an equation alters the equation and since we squared the equation twice above, it should not come as a surprise that one of the possible values of x does not satisfy the ORIGINAL equation.

**CLASS ACTIVITIES:**

- Solve the equation

Â

**PRACTICE EXERCISE:**

**Objective Test:**

- â€¦â€¦â€¦â€¦â€¦â€¦â€¦ are irrational numbers which are roots of rational numbers.

(a) Conjugate (b) Rationalization (c) Surds (d) Quadratics

- â€¦â€¦â€¦â€¦â€¦â€¦â€¦surds contain one or more square roots of prime numbers of their multiples.
- Rational (b) Quadratic (c) Rational (d) Simultaneous

**Essay Test:**

- Rational numbers are â€”â€”â€”-, â€¦â€¦â€¦â€¦, â€¦â€¦â€¦â€¦, â€¦â€¦â€¦â€¦.. etc
- Irrational numbers are â€¦â€¦â€¦â€¦.., â€¦â€¦â€¦â€¦â€¦, â€¦â€¦â€¦â€¦â€¦., â€¦â€¦â€¦â€¦. etc
- Other examples of irrational numbers that their exact values cannot be determined are â€¦., â€¦â€¦.
- Simplify (WAEC)
- Express in the form a + where a and b are integers (WAEC)
- Simplify â€“
- Find the square root of
- Solve

**ASSIGNMENT:**

- Find
- Rationalize
- Rationalize
- Given that
- Express in its basic form.

**KEY WORDS:**

- SURDS
- RATIONAL NUMBERS
- IRRATIONAL FORMS
- INTEGERS
- SIMILAR SURDS
- CONJUGATE SURDS
- RATIONALIZE
- EQUIVALENT
- EQUALITY
- RADICAL EQUATION
- RADICAND
- EXTRANEOUS ROOT