»Lecture: Surd (Mathematics Tutorial)

**RE:**

Rules of Surds

The following are the basic rules mostly used when performing operations on surds.

Rule 1:√ab = √a × √b

Rule2 : √a/b = √a/√b

Rule3 : √a + b not equal to √a + √b

Rule4 : √a - b not equal to √a - √b

provided a and b are positive.

e.g √12 = √4 × √3 or √6 × √2 etc.

also √7/2 = √7/√2

Student should avoid the error of writing

√a+b as √a + √b or √a-b as √a - √b

as these are wrong methods of evaluations.

**RE:**

Basic Forms of Surds

√p is said to be in its basic form if p does not have a factor that is a perfect square.Thus,√p will only be basic if √p cannot be broken down further into two factors where one of them has an exact square root.e.g √6,√5,√7,√2,√3 are in basic forms.Here √18 is not in basic form because it can be broken into √9 × 2

√18 = √9 ×2 = 3√2

But here 3√2 is now in a basic form.

**RE:**

**RE:**

Conjugate Surds

conjugate surds are two surds whose product results in a rational number.in an expression containing difference of two squares,it is known thatbr /> (x +y)(x +y) = ×^2 - y^,

in a similiar manner

(√x + √y)(√x - √y) = {√x)^2 - (√y)^2 = x -y

e.g

i.the conjugate of √3 -√5 = √3 + √5

ii.the conjugate of -2√7 + √3 = -2√7 - √3

in general: The conjugate of √x + √y is √x -√y

The conjugate of √x - √y is √x + √y.

**RE:**

Simplification of Surds

Surds can be simplified either in their simplest form or as a single surd.

Examples

1. Simplify (a) √45 (b) √98

Solution

a. √45 = √9 × 5 = √9 × √5 = 3√5

b.√98 = √49 × 2 = √49 × √2 = 7√2

2.Express the following as single surds

a.2√5

b.17√2

Solution

(a). 2√5

=√4×5 = √20

(b). 17√2

=√17^2 ×2 = √578

**RE:**

Addition and subtraction of Surds

Note: only surds in the same basic form can either be added or subtracted.sometimes,there may be need to simplify such surds before doing either the addition or the subtraction.

Examples

1. evaluate √32 + 3√8

solution

firstly u simplify the surd,which is

√16 ×2 + 3√4×2

=4√2 +(3×2)√2

=4√2 + 6√2

since they are sharing similiar surds,you can add them which gives

=10√2

2.Evaluate 7√3 - √75

Solution:firstly simplify the surd,which gives

=7√3 - √25 × 3

=7√3 - 5√3

since they are sharing similiar surds,you subtract,which gives

=2√3

3.Evaluate 3√48 - √75 + 2√12

solution:3√16 × 3 - √25×3 + 2√4 ×3

= {3×4)√3 } - 5√3 + (2×2)√3

= 12√3 - 5√3 + 4√3 since they have similiar surds,you can add nd subtract.

7√3 + 4√3

=11√3

**RE:**

Multiplication and Division of surds

Examples

1.Simplify √45 × √28

solution : √9×5 × √4×7

=3√5 × 2√7

n/b:when dealing with multiplication nd division,when the surds are not similiar you can multiply or divide it irrespective of the difference in the surds.this is what am saying

3×2√5×7

=6√35

2.Evaluate √24 / √50

solution: √24 / √50

= √24/50

=bring the value to the lowest,which we will have

=√12 / 25

=√12 / √25

= √4×3 / 5

= 2√3 / 5

3. Simplify √60 × √180 / √75

solutions

√60 × √180 /√ 75

= √60× 180 / √75

=√10800/ √75

=√144 = 12

4.find the square of √3 + √2

Solution

square of (√3 + √2) = (√3 +√2)^2

simplify which gives

√3(√3 + √2) + √2 (√3 + √2)

3 + √6 + √6 + 2

= 3+ 2√6 +2

collect like terms

3 +2 +2√6

=5 + 2√6.

**RE:**

Surds Rationalisation

Rationaliasation of surds is concerned with the process of removing irrational number from the denominator.Consider the surd 2/√3 is an irrational number dividing a rational number.Here,both the numerator and the denominator must be multiplied by √3,the denominator,for it to be removed.

Examples

1.simplify 2/ 5√3

solution

2/5√3

use the denominator √3 to multiply both the numerator and the denominator,which will give

2/5√3 ×√3 / √3

we will have

2√3 / 5√9

= 2√3 / 5×3

=2/3 / 15

2. Evaluate 3/ √3 + √2

solution

using the conjugate of √3 +√2 to rationalise,but this surd √3 + √ 2 will be changed to √3 - √2 when we begin to rationalise.we re going to have

3/√3 + √2 × √3 - √2 / √3 - √2

we have

=3√3 - 3√2 divided by √ 9 - √6 + √6 - √4

=3√3 - 3√2 divided by 3 - 2

we have

3√3 - 3√2 divided by 1

so we have

3√3 - 3√2

both are sharing dsame factor so bring one out nd put them in a fractional way.what i mean is this

3√3 - 3√2

3 (√3 - √2)

**RE:**

more on examples

3. Evaluate 2 + √3 divided by √2 - √3

solution

rationalize.we have

2 + √3 / √2 - √3 × √2 + √3 / √2 + √3

multiply,we have

2√2 + 2√3 + √6 + √9 divided by √4 + √6 - √6 - √9

= 2√2 + 2√3 + √6 + 3 divided by 2 - 3

= 2√2 + 2√3 + √6 + 3 / (-1)

to eliminate the division sign.this is what we are going to do.

-1(2√2 + 2√3 + √6 + 3)

so therefore,the answer is

-1(2√2 + 2√3 + √6 + 3).

**RE:**

1.Simplify (√3 - √2) (2√3 + √2)

Solution

firstly factorize,we have

√3(2√3 + √2) -√2(2√3 + √2)

open bracket,we have

2√9 + √6 - 2√6 - √4

= 2×3 + √6 - 2√6 - 2

= 6 + √6 - 2√6 - 2

collect like terms,we have

6-2 +√6 - 2√6

4 + √6 - 2√6

applying the rules of similar surd,we have

4 - √6.so therefore the answer is

= 4 - √6

**RE:**

2. Simplify 2√2 - √3 divided by √2 + √3

Solution

Rationalize,we have

(2√2 - √3 / √2 + √3 ) divided by √2 - √3 /√2 - √3

= 2√4 - 2√6 - √6 + √9 divided by √4 - √6 + √6 - √9

= 2×2 - 2√6 - √6 + 3 / 2 - 3

= 4 - 2√6 - √6 + 3 / 2 - 3

Collect like terms,we have

4 + 3 - 2√6 - √6 / (-1)

= 7 - 2√6 - √6 / (-1)

apply the rules of similar surd,we have

7 - 3√6 /(-1)

eliminate the division by multiplying with (-1),we have

-1(7 - 3√6)

open bracket,we have

-7+ 3√6 or 3√6 -7.