a) \( (5x -7)^2 \)

\( = (5x)^2 -2.5x.7 + 7^2 = 25x^2 -70x + 49 \)

b) \( ( 3x +4y )^2 \)

\( = (3x)^2 + 2.3x.4y + ( 4y)^2 = 9x^2 + 24xy +16y^2 \)

c) \( \left (  xy + \cfrac{ 2}{3 } y \right )^2  \)

\( = (xy)^2 + 2. xy . \cfrac{2 }{ 3} y + \left (  \cfrac{2 }{ 3} y\right )^2  = x^2 y^2 + \cfrac{4 }{3 }xy^2 + \cfrac{4 }{ 9}y^2   \) 

d) \( (3x -8 ) (3x+8 )\)

\( = (3x)^2 - 8^2 = 9x^2 - 64 \)

e) \( ( xy + 1 )^3 \)

\( = (xy)^3 + 3.(xy)^2.1 + 3.xy.1^2 +1^3 = x^3 y^3 + 3x^2 y^2 + 3xy + 1 \)

a) \( x^2 -2x +1 \)

\( = ( x -1 )^2 \)

b) \( x^2 +12x + 36 \)

\( = x^2 +2.x.6 + 6^2 = ( x + 6)^2 \)

c) \( 4x^2 + \cfrac{1 }{ 4} -2x \)

\( = (2x)^2 - 2. 2x. \cfrac{1 }{2 } + \left ( \cfrac{1}{2 }  \right )^2 = \left ( 2x - \cfrac{1 }{ 2} \right )^2   \)

a) \( 38. 42 \) = \( ( 40-2 ).( 40+2 ) = 40^2 - 2^2 = 1600 -4 = 1596 \)

b) \( 102^2 \) \( = (100 + 2 )^2 = 100^2 +2.100.2 + 2^2 = 10000 +400 +4 = 10404 \)

c) \( 198^2 \) \( = ( 200 - 2 )^2 = 200^2 - 2.200.2 + 2^2 = 40000 - 800+4 = 39204 \)

d) \( 75^2 -25^2 \) \( = ( 75-25)(75 + 25 ) = 50 . 100 = 5000 \)