SURDS (RADICALS)

SURDS (RADICALS)

Surds are roots of arithmetic numbers that cannot be determined exactly e.g are

surds. While are not surds since

RULES OF SURDS

If a, b, c, d are integers then

Example

Simplify the following, leaving your answers in surd form

Solution

BASIC FORM OF SURD

When a rational number under square root contain a factor which is a square of a number, the surd can be reduced to a simple form or its basic form.

Example

Reduce the following to its basic form

Solution

The transformation which reduced a surd to its form. can also be reversed to form a single surd

Example

Write the following basic form in their single surd form

Solution

So then we can say that s are similar, so also

Example

Simplify the following

Solution

CONJUGATE SURDS

Two surds are said to be conjugate of each other, if their product gives a difference of their square. That is

Example

Solution

Example

Simplify each of the following products

Solution

RATIONALISING DENOMINATORS

(i) To rationalize the denominator of the fraction

ii. To rationalize the denominator of the fraction

(iii) To rationalize the denominator of the

fraction

Example

By rationalizing the denominator, simplify each of the following

Example

Express as equivalent fraction with rational denominator

Solution

Example

If x = 5 − , find the value of x² +

Solution

Example

If x =

Find the value of 3x² – 5xy + 3y²

Solution

Example

Simplify

Solution

Example

Simplify

Solution

= [Difference of two squares]

= = =

Example

Simplify

= = −1

SQUARE ROOT OF SURD

Given a surd x + then the square root of the given surd is

Let

Comparing the equation

x = a + b …………{i}

y = 4ab ………….{ii}

Solving equation {i}and{ii}simultaneously will yield the value of a and b

Example

Find the square root of the following

1. 12+ 2

2. 31−4

Solution

1. 12 + 2

Let the square root of 12+2 be

12 +2

12 +

From equality of surds

a + b = 12………{i}

ab = 35……..{ii}

From{i} a = 12 − b

Substitute a = 12 −b into {11}

b = 35

12b – b² = 35

b²−12b −35 =0

b =7 or b = 5

Substituting the values of b

a = 12−7 =or a =12−5 =7

Hence the Square root is twice

(ii) 31 −4

Let the square root 31−4

Square both sides

31− 4

31 − 4

From the equality of surd

a + b = 31………..(i)

−2

ab = 84………(ii)

From (i) a = 31−b

Substituting a = 31– b into (11)

b = 84

b² – 31b + 84 =0

= 0

b= 28 or b = 3

Substituting the values of b

a = 31−28 =3 or a = 31 -3 = 28

Hence the square root of

EQUATION INVOLVING SURDS

Example

Solve for x in

Solution

Squaring both sides

6x – 3 = 49

6x = 52

x =

EXAMPLE

Solve for x in

With these types of surds, it is easier to transpose one to the opposite side

Squaring both sides

64(3x-5) = 1+10x + 25x²

192x – 320 = 1+10x+25x²

25x² – 182x + 321 = 0

(25x – 107) (x−3) = 0

x = 3 or x =

By substituting these two x-values in the original equation we find that x= is not a solution. Thus, the given equation has 3 as its solution

Example

Solve for x in the equation

Solution

Squaring both sides

9x² + 54x + 81 = 12x² + 40x + 32

3x² – 14x – 49 = 0

(3x+7) (x-7) = 0

x = 7 or x =

By substituting these two x-values in the original equation, we see that x = does not satisfy the given equation. Thus, the equation has one solution x = 7