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Solved Examples: Cube of a Sum `(a + b)^3`

Solved Example 1

Expand the expression `(x + 2y)^3`

Solution

Compare the given expression with `(a + b)^3`. Thus,
`a = x` and `b = 2y`
Write the formula,
`(a + b)^3 = a^3 + b^3 + 3ab(a + b)`
Put 'x' in place of 'a' and '2y' in place of 'b',
`(x + 2y)^3 = x^3 + (2y)^3 + 3(x)(2y)(x + 2y)`
Simplify the expression,
`(x + 2y)^3 = x^3 + 8y^3 + 6x^2y + 12xy^2`

Solved Example 2

Expand the expression `(3a + 4b)^3`

Solution

Compare the given expression with `(a + b)^3`. Thus,
`a = 3a` and `b = 4b`
Write the formula,
`(a + b)^3 = a^3 + b^3 + 3ab(a + b)`
Put '3a' in place of 'a' and '4b' in place of 'b',
`(3a + 4b)^3 = (3a)^3 + (4b)^3 + 3(3a)(4b)(3a + 4b)`
Simplify the expression,
`(3a + 4b)^3 = 27a^3 + 64b^3 + 36ab(3a + 4b)`
`= 27a^3 + 64b^3 + 108a^2b + 144ab^2`

Solved Example 3

Expand the expression `(x + y/2)^3`

Solution

Compare the given expression with `(a + b)^3`. Thus,
`a = x` and `b = y/2`
Write the formula,
`(a + b)^3 = a^3 + b^3 + 3ab(a + b)`
Put `x` in place of 'a' and `y/2` in place of 'b',
`(x + y/2)^3 = (x)^3 + (y/2)^3 + 3(x)(y/2)(x + y/2)`
Simplify the expression,
`(x + y/2)^3 = x^3 + (y^3)/8 + (3xy)/2(x + y/2)`
`= x^3 + (y^3)/8 + (3x^2y)/2 + (3xy^2)/4`

Solved Example 4

Expand the expression `(x/3 + y/4)^3`

Solution

Compare the given expression with `(a + b)^3`. Thus,
`a = x/3` and `b = y/4`
Write the formula,
`(a + b)^3 = a^3 + b^3 + 3ab(a + b)`
Put `x/3` in place of 'a' and `y/4` in place of 'b',
`(x + y/2)^3 = (x)^3 + (y/2)^3 + 3(x)(y/2)(x + y/2)`
Simplify the expression,
`(x + y/2)^3 = x^3 + (y^3)/8 + (3xy)/2(x + y/2)`
`= x^3 + (y^3)/8 + (3x^2y)/2 + (3xy^2)/4`

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