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.)