ICSE-VIII-Mathematics
12: Algebraic Identities Class 8 Maths
- #5Evaluate: (i) (6 - xy) (6 + xy) (ii) (7x + 2/3y)(7x - 2/3y) (iii) (a/2b + 2b/a) (a/2b - 2b/a) (iv) (3x - 1/2y)(3x + 1/2y) (ix) (2a - 3b)(3a + 4b) (v) (2a + 3)(2a - 3)(4a2 + 9) (vi) (a + bc)(a - bc)(a2 + b2c2) (vii) (5x + 8y)(3x + 5y) (viii) (7x + 15y)(5x - 4y) (x) (9a - 7b)(3a - b)Ans : (i) (6 - xy)(6 + xy) = 6(6 + xy) - xy(6 + xy)
= 36 + 6xy - 6xy + (xy)2
= 36 - x2y2 (ii) (7x + 2/3y)(7x - 2/3y)
= 7x(7x - 2/3y) + 2/3y(7x - 2/3y)
= 49x2 - 14/3xy + 14/3xy - 4/9y2
= 49x2- 4/9y2 (iii) (a/2b + 2b/a)(a/2b - 2b/a)
= a/2b(a/2b - 2b/a) + 2b/a(a/2b - 2b/a)= (a/2b)2 - (2b/a)2
= a2/4b2 - 4b2/a2 (iv) (3x - 1/2y)(3x + 1/2y)
= 3x(3x + 1/2y) - 1/2y(3x + 1/2y)
= 9x2 + 3x/2y - 3x/2y - 1/4y2
= 9x2 - 1/4y2 (ix) (2a - 3b)(3a + 4b)
= 2a(3a + 4b) - 3b(3a + 4b)
= 6a2 + 8ab - 9ab - 12b2
= 6a2 - ab - 12b2 (v) (2a + 3)(2a - 3)(4a2 + 9)
= [(2a)2 - (3)2] (4a2 + 9) [∵ (a + b)(a - b) = a2 - b2]
= (4a2 - 9)(4a2 + 9)
= (4a2)2 - (9)2
= 16a4 - 81 (vi) (a + bc)(a - bc)(a2 + b2c2)
= [
(a)2 - (bc)2](a2 + b2c2) [∵ (a + b)(a - b) = a2 - b2]
= (a2 - b2c2)(a2 + b2c2)
= (a2)2 - (b2c2)2 [∵ (a + b)(c - b) = a2 - b2]
= a4 - b4c4 (vii) (5x + 8y)(3x + 5y)
= 5x(3x + 5y) + 8y(3x + 5y)
= 15x2 + 25xy + 24xy + 40y2
= 15x2 + 49xy + 40y2 (viii) (7x + 15y)(5x - 4y)
= 7x(5x - 4y) + 15y (5x - 4y)
= 35x2 - 28xy + 75xy - 60y2
= 35x2 + 47xy - 60y2 (x) (9a - 7b)(3a - b)
= 9a(3a - b) - 7b(3a - b)
= 27a2 - 9ab - 21ab + 7b2
= 27a2 - 30ab + 7b2
- #5-i(6 - xy) (6 + xy)Ans : (6 - xy)(6 + xy) = 6(6 + xy) - xy(6 + xy)
= 36 + 6xy - 6xy + (xy)2
= 36 - x2y2
- #5-ii(7x + 2/3y)(7x - 2/3y)Ans : (7x + 2/3y)(7x - 2/3y)
= 7x(7x - 2/3y) + 2/3y(7x - 2/3y)
= 49x2 - 14/3xy + 14/3xy - 4/9y2
= 49x2- 4/9y2
- #5-iii(a/2b + 2b/a) (a/2b - 2b/a)Ans : (a/2b + 2b/a)(a/2b - 2b/a)
= a/2b(a/2b - 2b/a) + 2b/a(a/2b - 2b/a)= (a/2b)2 - (2b/a)2
= a2/4b2 - 4b2/a2
- #5-iv(3x - 1/2y)(3x + 1/2y)Ans : (3x - 1/2y)(3x + 1/2y)
= 3x(3x + 1/2y) - 1/2y(3x + 1/2y)
= 9x2 + 3x/2y - 3x/2y - 1/4y2
= 9x2 - 1/4y2
- #5-ix(2a - 3b)(3a + 4b)Ans : (2a - 3b)(3a + 4b)
= 2a(3a + 4b) - 3b(3a + 4b)
= 6a2 + 8ab - 9ab - 12b2
= 6a2 - ab - 12b2
- #5-v(2a + 3)(2a - 3)(4a2 + 9)Ans : (2a + 3)(2a - 3)(4a2 + 9)
= [(2a)2 - (3)2] (4a2 + 9) [∵ (a + b)(a - b) = a2 - b2]
= (4a2 - 9)(4a2 + 9)
= (4a2)2 - (9)2
= 16a4 - 81
- #5-vi(a + bc)(a - bc)(a2 + b2c2)Ans : (a + bc)(a - bc)(a2 + b2c2)
= [
(a)2 - (bc)2](a2 + b2c2) [∵ (a + b)(a - b) = a2 - b2]
= (a2 - b2c2)(a2 + b2c2)
= (a2)2 - (b2c2)2 [∵ (a + b)(c - b) = a2 - b2]
= a4 - b4c4
- #5-vii(5x + 8y)(3x + 5y)Ans : (5x + 8y)(3x + 5y)
= 5x(3x + 5y) + 8y(3x + 5y)
= 15x2 + 25xy + 24xy + 40y2
= 15x2 + 49xy + 40y2
- #5-viii(7x + 15y)(5x - 4y)Ans : (7x + 15y)(5x - 4y)
= 7x(5x - 4y) + 15y (5x - 4y)
= 35x2 - 28xy + 75xy - 60y2
= 35x2 + 47xy - 60y2
- #5-x(9a - 7b)(3a - b)Ans : (9a - 7b)(3a - b)
= 9a(3a - b) - 7b(3a - b)
= 27a2 - 9ab - 21ab + 7b2
= 27a2 - 30ab + 7b2