# CLASS-9PROBLEM & SOLUTION OF SURDS

There are some example of problem and solution of Surds are given below for better understanding.

(√5 + √3)

Example.1)  Simplify ------------- by rationalizing the denominator.

(√5 - √3)

Ans.) On multiplying the numerator and denominator of the given number by (√5 + √3), we get as follows –

(√5 + √3)         (√5 + √3) X (√5 + √3)           (√5 + √3)²

----------- = ----------------------- = ---------------

(√5 - √3)         (√5 - √3) X (√5 + √3)           √5² - √3²

5 + 2√15 + 3          8 + 2√15

= ---------------- = -------------  =  ( 4 + √15)         (Ans.)

5 – 3                    2

7

Example.2)  Simplify ----------- by rationalizing the denominator.

(2√5 + √3)

Ans.) On multiplying the numerator and denominator of the given number by (2√5 - √3), we get as follows –

7                 7 X (2√5 - √3)              14√5 - 7√3

------------ = ---------------------- = ----------------

(2√5 + √3)       (2√5 + √3) X (2√5 - √3)        (2√5)² - (√3)²

14√5 - 7√3          (14√5 - 7√3)

= --------------- = ----------------      (Ans.)

20 – 3                   17

(3 - √2)

Example.3) If, x & y are rational numbers and --------- = x + y√2,

(3 + √2)

find the value of x & y

(3 - √2)

Ans.) we have = -----------, On multiplying the numerator and

(3 + √2)

denominator of the given number

by (3 - √2), we get as follows –

(3 - √2)          (3 - √2) X (3 - √2)          9 - 6√2 + 2

------------ = --------------------- = ---------------

(3 + √2)          (3 + √2) X (3 - √2)             9 – 2

=  (11 - 6√2)/7

(3 - √2)      (11 - 6√2)

As per the given condition, x + y√2 = ---------- = -----------………(1)

(3 + √2)           7

7x + 7y√2  =  11 - 6√2

(11 - 6√2 – 7y√2)

x  =  --------------------- ……………………….(2)

7

Now put the value of x in equation number 1, then we find that –

(3 - √2)           (11 - 6√2)

x + y√2 = ------------ = ---------------

(3 + √2)               7

(11 - 6√2 – 7y√2)                 (11 - 6√2)

=> -----------------  +  y√2 = -------------

7                                7

=>    11 - 6√2 – 7y√2 + 7 y√2  =  (11 - 6√2)        (Ans.)

3√2            √7             4√3

Example.4) Simplify { --------- + --------- - --------- }

√6 + √3       √3 + √2       √6 + √2

3√2              √7                 4√3

Ans.)  { ----------- + -----------  - ----------- }

√6 + √3         √3 + √2           √6 + √2

3√2 X (√6 - √3)          √7 X (√3 - √2)            4√3 X (√6 - √2)

= {------------------- + ------------------- - -------------------}

(√6 + √3) X (√6 - √3)     (√3 + √2) X (√3 - √2)     (√6 + √2) X (√6 - √2)

(3√2 X √3 X √2) – (3√3 X √2)        (√7 X √3) – (√7 X √2)       (4√3 X √3 X √2) – (4√3 X √2)

= {-------------------------- + ---------------------- - --------------------------- }

(√6 X √6) – (√3 X √3)             (√3 X √3) – (√2 X √2)           (√6 X √6) – (√2 X √2)

(6√3 - 3√6)         (√21 - √14)         (12√2 - 4√6)

= { -------------- + -------------- - --------------- }

(√6)² - (√3)²        (√3)² - (√2)²        (√6)² - (√2)²

(6√3 - 3√6)          (√21 - √14)          (12√2 - 4√6)

= { --------------- + -------------- - ---------------- }

3                     1                       4

=   { 4 X (6√3 - 3√6) + 12 X (√21 - √14) – 3 X (12√2 - 4√6) }

=     24√3 - 12√6 + 12√21 - 12√14 - 36√2 + 12√6

=      24√3 + 12√21 - 12√14 - 36√2               (Ans.)