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} ) 

Answer:

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 )

Answer:

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} ) 

Answer:

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 ) 

Answer:

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 ) 

Answer:

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 )

Answer:

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}

Answer:

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