Write the following in expanded form: 2 x + 3 3

Use identity ( a + b ) 3 = a 3 + b 3 + 3 a b ( a + b ) Consider, a = 2 x and b = 3 Hence, we have, 2 x + 3 3 = 2 x 3 + 3 3 + 3 2 x 3 2 x + 3 = 2 2 x 3 + 27 + 18 x 2 + 27 2 x = 2 2 x 3 + 18 x 2 + 27 2 x + 27 Therefore, the required answer is 2 2 x 3 + 18 x 2 + 27 2 x + 27 .

Write the following in expanded form: 3 + 4 y 3

Use identity ( a + b ) 3 = a 3 + b 3 + 3 a b ( a + b ) Consider, a = 3 and b = 4 y Hence, we have, 3 + 4 y 3 = 3 3 + 4 y 3 + 3 3 4 y 3 - 4 y = ( 3 ) 3 + ( 4 y ) 3 + 12 3 y ( 3 - 4 y ) = 3 3 + 64 y 3 + 36 y + 48 3 y 2 = 3 3 + 36 y + 48 3 y 2 + 64 y 3 Therefore, the required answer is 3 3 + 36 y + 48 3 y 2 + 64 y 3 .

Write the following in expanded form: 1 - 2 y 3

Use identity ( a - b ) 3 = a 3 - b 3 - 3 a b ( a - b ) Consider, a = 1 and b = 2 y Hence, we have, 1 - 2 y 3 = 1 3 - 2 y 3 - 3 1 2 y 1 - 2 y = 1 - 2 2 y 3 - 3 2 y + 6 y 2 = 1 - 3 2 y + 6 y 2 - 2 2 y 3 Therefore, the required answer is 1 - 3 2 y + 6 y 2 - 2 2 y 3 .

Write the following in expanded form: a 3 - b 2 3

Use identity ( x - y ) 3 = x 3 - y 3 - 3 x y ( x - y ) Consider, x = a 3 and y = b 2 Hence, we have, a 3 - b 2 3 = a 3 3 - b 2 3 - 3 a 3 b 2 a 3 - b 2 = a 3 3 3 - b 3 8 - 3 2 a b a 3 - b 2 = a 3 3 3 - b 3 8 - a 2 b 2 + 3 a b 2 4 = a 3 3 3 - a 2 b 2 + 3 a b 2 4 - b 2 8 Therefore, the required answer is a 3 3 3 - a 2 b 2 + 3 a b 2 4 - b 2 8 .

Write the following in expanded form: 3 - 1 y 3

Use identity ( a - b ) 3 = a 3 - b 3 - 3 a b ( a - b ) Consider, a = 3 and b = 1 y Hence, we have, 3 - 1 y 3 = 3 3 - 1 y 3 - 3 3 1 y 3 - 1 y = 27 - 1 y 3 - 27 y + 9 y 2 = 27 - 27 y + 9 y 2 - 1 y 3 Therefore, the required answer is 27 - 27 y + 9 y 2 - 1 y 3 .

Simplify: a 3 + b 2 3 - a 3 - b 2 3

Use identity ( x + y ) 3 = x 3 + y 3 + 3 x y ( x + y ) and ( x - y ) 3 = x 3 - y 3 - 3 x y ( x - y ) . Consider x = a 3 and y = b 2 . We have, a 3 + b 2 3 - a 3 - b 2 3 = a 3 27 + b 3 8 + 3 a 3 b 2 a 3 + b 2 - a 3 27 - b 2 8 - 3 a 3 b 2 a 3 - b 2 = a 3 27 + b 3 8 + a 2 b 6 + a b 2 4 - a 3 27 - b 2 8 - a 2 b 6 + a b 2 4 = a 3 27 + b 3 8 + a 2 b 6 + a b 2 4 - a 3 27 + b 2 8 + a 2 b 6 - a b 2 4 = b 3 4 + a 2 b 3 = b b 2 4 + a 2 3 = b a 2 3 + b 2 4 Therefore, the required answer is b a 2 3 + b 2 4 .