Question 1: Find the following products:

i)  $( y+9 ) ( y-9 )$          ii)  $( 4+b ) ( 4-b )$

iii)  $( 3x-5 ) ( 3x+5 )$          iv)  $( a-$ $\frac{2}{3}$ $) ( a+$ $\frac{2}{3}$ $)$

i)  $( y+9 ) ( y-9 ) = y^2-81$

ii)  $( 4+b ) ( 4-b ) = 16-b^2$

iii)  $( 3x-5 ) ( 3x+5 ) = 9x^2-25$

iv)  $( a-$ $\frac{2}{3}$ $) ( a+$ $\frac{2}{3}$ $) = a^2-$ $\frac{4}{9}$

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Question 2: Find the following products:

i)  $( 3x-5 ) ( 3x+5 )$          ii)  $( 2+7x ) ( 2-7x )$

iii)  $($ $\frac{a}{2}$ $+3 ) ($ $\frac{a}{2}$ $-3 )$         iv)  $( 4x+3y ) ( 4x-3y )$

i)  $( 3x-5 ) ( 3x+5 ) = 9x^2-25$

ii)  $( 2+7x ) ( 2-7x ) = 4-49x^2$

iii)  $($ $\frac{a}{2}$ $+3 ) ($ $\frac{a}{2}$ $-3 ) =$ $\frac{a^2}{4}$ $-9$

iv)  $( 4x+3y ) ( 4x-3y ) = 16x^2-9y^2$

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Question 3: Find the following products:

i)  $($ $\frac{a}{3}$ $-$ $\frac{b}{4}$ $) ($ $\frac{a}{3}$ $+$ $\frac{b}{4}$ $)$           ii)  $($ $\frac{t}{2}$ $-$ $\frac{1}{3}$ $) ($ $\frac{t}{2}$ $+$ $\frac{1}{3}$ $)$

i)  $($ $\frac{a}{3}$ $-$ $\frac{b}{4}$ $) ($ $\frac{a}{3}$ $+$ $\frac{b}{4}$ $) =$ $\frac{a^2}{9}$ $-$ $\frac{b^2}{16}$

ii)  $($ $\frac{t}{2}$ $-$ $\frac{1}{3}$ $) ($ $\frac{t}{2}$ $+$ $\frac{1}{3}$ $) =$ $\frac{t^2}{4}$ $-$ $\frac{1}{9}$

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Question 4: Find the following products:

i)  $($ $\frac{2}{x}$ $+$ $\frac{3}{y}$ $) ($ $\frac{2}{x}$ $-$ $\frac{3}{y}$ $)$      ii)  $($ $\frac{1}{a}$ $-$ $\frac{1}{b}$ $) ($ $\frac{1}{a}$ $+$ $\frac{1}{b}$ $)$

iii)  $($ $\frac{1}{3x}$ $+$ $\frac{2}{5y}$ $) ($ $\frac{1}{3x}$ $-$ $\frac{2}{5y}$ $)$      iv)  $( 1.1x-0.3y ) ( 1.1x+0.3y )$

i)  $($ $\frac{2}{x}$ $+$ $\frac{3}{y}$ $) ($ $\frac{2}{x}$ $-$ $\frac{3}{y}$ $) =$ $\frac{4}{x^2}$ $-$ $\frac{9}{y^2}$

ii)  $($ $\frac{1}{a}$ $-$ $\frac{1}{b}$ $) ($ $\frac{1}{a}$ $+$ $\frac{1}{b}$ $) =$ $\frac{1}{a^2}$ $-$ $\frac{1}{b^2}$

iii)  $($ $\frac{1}{3x}$ $+$ $\frac{2}{5y}$ $) ($ $\frac{1}{3x}$ $-$ $\frac{2}{5y}$ $) =$ $\frac{1}{9x^2}$ $-$ $\frac{2}{25y^2}$

iv)  $( 1.1x-0.3y ) ( 1.1x+0.3y ) = 1.21x^2-0.09y^2$

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Question 5: Find the following products:

i)  $( a^2+2b^2 ) ( a^2-2b^2 )$          ii)  $( 6x^2-7y^2 ) ( 6x^2+7y^2 )$

iii)  $( 4x^2+2yz ) ( 2x^2-yz )$          iv)  $( ab-$ $\frac{3}{2}$ $cd ) ( 2ab+3cd )$

i)  $( a^2+2b^2 ) ( a^2-2b^2 ) = a^4-4b^4$

ii)  $( 6x^2-7y^2 ) ( 6x^2+7y^2 ) = 36x^4-49y^4$

iii)  $( 4x^2+2yz ) ( 2x^2-yz ) = 8x^4+4x^2yz-4x^2yz-2y^2z^2 = \ 8x^4-2y^2z^2$

iv)  $( ab-$ $\frac{3}{2}$ $cd ) ( 2ab+3cd ) = 2a^2b^2-3abcd+3abcd-$ $\frac{9}{2}$ $c^2d^2 = 2a^2b^2-$ $\frac{9}{2}$ $c^2d^2$

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Question 6: Find the following products:

i)  $( 2x+3 ) ( 2x-3 ) ( 4x^2+9 )$          ii)  $( x+2y ) ( x-2y ) ( x^2+4y^2 )$

iii)  $( a+bc ) ( a-bc ) ( a^2+b^2c^2 )$          iv)  $($ $\frac{2}{5}$ $+x ) ($ $\frac{2}{5}$ $-x ) ($ $\frac{4}{25}$ $+x^2 )$

i)  $( 2x+3 ) ( 2x-3 ) ( 4x^2+9 ) = \ ( 4x^2-9 ) ( 4x^2+9 ) = \ 16x^2-81$

ii)  $( x+2y ) ( x-2y ) ( x^2+4y^2 ) = \ ( x^2-4y^2 ) ( x^2+4y^2 ) = \ x^4-16y^4$

iii)  $( a+bc ) ( a-bc ) ( a^2+b^2c^2 ) = \ ( a^2-b^2c^2 ) ( a^2+b^2c^2 ) = {\ a}^4-b^4c^4$

iv)  $($ $\frac{2}{5}$ $+x ) ($ $\frac{2}{5}$ $-x ) ($ $\frac{4}{25}$ $+x^2 ) = \ ($ $\frac{4}{25}$ $-x^2 ) ($ $\frac{4}{25}$ $+x^2 ) =$ $\frac{16}{625}$ $-x^4$

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Question 7: Using the identity $( a+b ) ( a-b ) = ( a^2-b^2 )$ , evaluate the following:

i)  $88 \times 112$      ii)  $153 \times 167$       iii)  $10.8 \times 9.2$

iv)  $3$ $\frac{1}{3}$ $\times 4$ $\frac{2}{3}$      v)  $9$ $\frac{1}{4}$ $\times 15$ $\frac{3}{4}$

i)  $88 \times 112 = ( 100-12 ) ( 100+12 ) = 1000-144 = 9856$
ii)  $153 \times 167 = ( 160-7 ) ( 160+7 ) = 25600-49 = 25551$
iii)  $10.8 \times 9.2 = ( 10+0.8 ) ( 10-0.8 ) = {10}^2-{0.8}^2 = 100-0.64 = 99.36$
iv)  $3$ $\frac{1}{3}$ $\times 4$ $\frac{2}{3}$ $= ( 4-$ $\frac{2}{3}$ $) ( 4+$ $\frac{2}{3}$ $) = 16-$ $\frac{4}{9}$ $= 15$ $\frac{5}{9}$
v)  $9$ $\frac{1}{4}$ $\times 15$ $\frac{3}{4}$ $= ($ $\frac{25}{2}$ $-3$ $\frac{1}{4}$ $) ($ $\frac{25}{2}$ $+3$ $\frac{1}{4}$ $) = 145.6875$