Question 1: Simplify each of the following:
Solution:
(i)
Using the formula:
Here,
(ii)
Using the formula:
Here,
Question 2: Simplify the following expressions:
(i) (4 + √7) (3 + √2)
(ii) (3 + √3)(5- √2 )
(iii) (√5 -2)( √3 – √5)
Solution:
(i) (4 + √7) (3 + √2)
= 12 + 4√2 + 3√7 + √14
(ii) (3 + √3)(5- √2)
= 15 – 3√2 + 5√3 – √6
(iii) (√5 – 2)(√3 – √5)
= √15 – √25 – 2√3 + 2√5
= √15 – 5 – 2√3 + 2√5
Question 3: Simplify the following expressions:
(i) (11 + √11) (11 – √11)
(ii) (5 + √7) (5 –√7)
(iii) (√8 – √2 ) (√8 + √2)
(iv) (3 + √3) (3 – √3)
(v) (√5 – √2) (√5 + √2)
Solution:
Using Identity: (a – b)(a + b) = a2 – b2
(i) (11 + √11) (11 – √11)
= 112 – (√11)2
= 121 – 11
= 110
(ii) (5 + √7) (5 –√7)
= (52 – (√7)2)
= 25 – 7 = 18
(iii) (√8 – √2) (√8 + √2)
= (√8)2 – (√2)2
= 8 – 2
= 6
(iv) (3 + √3) (3 – √3)
= (3)2 – (√3)2
= 9 – 3
= 6
(v) (√5 – √2) (√5 + √2)
= (√5)2 – (√2)2
= 5 – 2
= 3
Question 4: Simplify the following expressions:
(i) (√3 + √7)2
(ii) (√5 – √3)2
(iii) (2√5 + 3√2 )2
Solution:
Using identities: (a – b)2 = a2 + b2– 2ab and (a + b)2 = a2+ b2 + 2ab
(i) (√3 + √7)2
= (√3)2 + (√7)2 + 2(√3)(√7)
= 3 + 7 + 2√21
= 10 + 2√21
(ii) (√5 – √3)2
= (√5)2 + (√3)2 – 2(√5)(√3)
= 5 + 3 – 2√15
= 8 – 2√15
(iii) (2√5 + 3√2)2
= (2√5)2 + (3√2)2 + 2(2√5)( 3√2)
= 20 + 18 + 12√10
= 38 + 12√10