CHAPTER 4

SURDS

Surds are roots that are not possible to find exactly. They are irrational numbers eg √5, √7 etc. however the approximate values can be found from square root tables. We also have surds such as ³√5, ⁵√27 etc.

RULES OF SURDS

1. √mn = √m x √n

Example: √(4 x 9) = √4 x √9 = 2 x 3 = 6

2. √m/n = √m

                √n

Example: √(9/4) = √9 = 3

                             √4    2

Note: √(m + n) ≠ √m + √n 

And √(m - n) ≠ √m - √n 

REDUCTION OF SURDS TO BASIC FORM

√n is in basic form if n does not contain a factor which is a perfect square. Thus √10, √11, √13 etc are in basic form and cannot be simplified further. However √32 is not in basic form and can be simplified as      √(2 x 16) = √16 x √2 = 4√2

ADDITION AND SUBTRACTION OF SIMILAR SURDS

Simplify √50 - √18 + √32

Solution:

Reduce all the surds to their basic forms

√(2 x 25)  - √(2 x 9)  + √(2 x 16) 

√2 x √25 - √2 x √9 + √2 x √16

5√2 - 3√2 + 4√2 = 6√2

RATIONALIZING THE DENOMINATOR

A surd such as √3 , cannot be simplified, but  2 can be written in a more convenient form by

                         2                                           √3

 multiplying the numerator and denominator of  2 by √3

                                                                     √3

Eg  2  =  2  x √3 =      2√3    = 2√3 = 2√3. This process removes the irrational number (√3)

     √3    √3    √3     √(3 x 3)     √9        3

from the denominator and this is called rationalizing the denominator.

CONJUGATE SURDS

Given the surd expression a + √b, then a - √b is called its conjugate. The product of a surd and its conjugate is not a surd but a real number eg (a + √b) (a - √b ) = a² – b. We use the conjugate of a surd to rationalize the denominator. E.g.

Simplify     1     

            3 - 2√2

Solution:

     1      x  3 + 2√2

3 - 2√2     3 + 2√2

= 3 + 2√2 = 3 + 2√2

   9 – 4(2)

FINDING THE SQUARE ROOT OF A SURD EXPRESSION

Note the following rules in surd:

(√a + √b)² = (a + b) + 2√ab

(√a - √b)² = (a + b) – 2√ab

Example: Find the positive square root of 49 – 12√5

Solution:

49 – 12√5 = (√a - √b)² = (a + b) – 2√ab

Hence 49 = a + b

Or a + b = 49 ………………………. (1)

12√5 = 2√ab

Dividing both sides by 2

6√5 = √ab, hence

ab = 180 ……………………..…….. (2)

From (1), a = 49 - b

Substitute for a in (2)

(49 - b)b = 180

49b – b² = 180, hence

b² – 49b + 180 = 0

(b – 4) (b – 45) = 0

Hence b = 4 or 45

From (2), ab = 180

But a > b, since the root must be positive, hence

a = 45, b = 4

Thus, the square root of 49 – 12√5 = √a - √b

= √45 - √4 = [√(5 x 9) - √4]

= 3√5 – 2

QUESTIONS

1. Simplify the following:

(a) √72  (b) √20  (c) √1000  (d) √(1/8)  (e) √864  (f) √50 + √32 - √162  (g) (2√3 – 1)(2√3 + 1)

(h) √72 - √12 + √75  (i) √2 x √8  (j) (2√2 – 1)²  (k) (2 + √3)²   (l)  (2√2 – 1)² 

(m) √162 + √32 - √98   (n)    6  .   (o)  12   (p)     1       (q)      2     (r) √3 – 1 

                  √8                      √2           √80       √3 – 1         √5 – 2       √3 + 1

(s) 2√2 + 3  (t) 3√5 - √2   (u) a – b   (v)        1        -       1         (w) 3√2 + √3

     2√2 – 1      2√5+3√2       √a–√b          2√2 – 1      2√2 + 1           √2 + √3