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Solved Example 1

Expand (x -2y)^3

Solution

Step 1: Compare the given expression with (a - b)^3

Comparing  (x - 2y)^3 with (a - b)^3, we get
a = x and b = 2y

Step 2: Apply the formula (a - b)^3 = a^3 - b^3 - 3ab(a - b)

Putting x in place of a and 2y in place of b in the above formula, we get,
(x - 2y)^3 = x^3 - (2y)^3 - 3(x)(2y)(x - 2y)
Simplifying the left hand expression,
(x - 2y)^3 = x^3 - 8y^3 - 6xy(x - 2y)
= x^3 - 8y^3 - 6x^2y + 12xy^2

Solved Example 2

Expand (3x -2y)^3

Solution

Step 1: Compare the given expression with (a - b)^3

Comparing  (3x - 2y)^3 with (a - b)^3, we get
a = 3x and b = 2y

Step 2: Apply the formula (a - b)^3 = a^3 - b^3 - 3ab(a - b)

Putting 3x in place of a and 2y in place of b in the above formula, we get,
(3x - 2y)^3 = (3x)^3 - (2y)^3 - 3(3x)(2y)(3x - 2y)
Simplifying the left hand expression,
(3x - 2y)^3 = 27x^3 - 8y^3 - 18xy(3x - 2y)
= 27x^3 - 8y^3 - 54x^2y + 36xy^2

Solved Example 3

Expand (x/3 - 2y)^3

Solution

Step 1: Compare the given expression with (a - b)^3

Comparing  (x/3 - 2y)^3 with (a - b)^3, we get
a = x/3 and b = 2y

Step 2: Apply the formula (a - b)^3 = a^3 - b^3 - 3ab(a - b)

Putting x/3 in place of a and 2y in place of b in the above formula, we get,
(x/3 - 2y)^3 = (x/3)^3 - (2y)^3 - 3(x/3)(2y)(x/3 - 2y)
Simplifying the left hand expression,
(x/3 - 2y)^3 = (x^3)/27 - 8y^3 - 2xy(x - 2y)
= (x^3)/27 - 8y^3 - 2x^2y + 4xy^2

Solved Example 4

Expand (a/2 - b/3)^3

Solution

Step 1: Compare the given expression with (a - b)^3

Comparing  (a/2 - b/3)^3 with (a - b)^3, we get
a = a/2 and b = b/3

Step 2: Apply the formula (a - b)^3 = a^3 - b^3 - 3ab(a - b)

Putting a/2 in place of a and b/3 in place of b in the above formula, we get,
(a/2 - b/3)^3 = (a/2)^3 - (b/3)^3 - 3(a/2)(b/3)(a/2 - b/3)
Simplifying the left hand expression,
(a/2 - b/3)^3 = (a^3)/8 - (b^3)/27 - (ab)/2(a/2 - b/3)
= (a^3)/8 - (b^3)/27 - (a^2b)/4 + (ab^2)/6

Solved Example 5

Expand (x - 1)^3

Solution

Step 1: Compare the given expression with (a - b)^3

Comparing  (x - 1)^3 with (a - b)^3, we get
a = x and b = 1

Step 2: Apply the formula (a - b)^3 = a^3 - b^3 - 3ab(a - b)

Putting x in place of a and 1 in place of b in the above formula, we get,
(x - 1)^3 = x^3 - 1^3 - 3(x)(1)(x - 1)
Simplifying the left hand expression,
(x - 1)^3 = x^3 - 1 - 3x(x - 1)
= x^3 - 1 - 3x^2 + 3x