a³ − b³ and a³ + b³ Formula: Proof, Examples & Common Mistakes

The a^3 – b^3 formula is (a – b) (a^2 + ab + b^2). This formula is used to calculate the difference of two cubes.

Cube Formulas at a Glance

Expression	Formula	Name
a3 − b3	(a − b)(a2 + ab + b2)	Difference of two cubes
a3 + b3	(a + b)(a2 − ab + b2)	Sum of two cubes
(a + b)3	a3 + 3a2b + 3ab2 + b3	Cube of a sum
(a − b)3	a3 − 3a2b + 3ab2 − b3	Cube of a difference

Tip to remember the signs: in both a³ − b³ and a³ + b³, the first bracket has the same sign as the question, and the middle term (ab) of the second bracket has the opposite sign. The last term (b²) is always positive.

What is a^3-b^3 Formula?

The a³-b³ formula is (a – b) (a² + ab + b²)

Prove a Cube Minus b Cube Formula (a^3-b^3)

Let’s prove a cube minus b cube formula. To prove that a³ – b³ = (a – b) (a² + ab + b²).

LHS = RHS

LHS = a³ – b³

On Solving RHS side we get,

= (a – b) (a² + ab + b²)

On multiplying a and b separately with (a² + ab + b²)

= a (a² + ab + b²) – b(a² + ab + b²)
= a³ + a²b + ab² – a²b – ab² – b³
= a³ + a²b – a²b + ab²– ab² – b³
= a³ – 0 – 0 – b³
= a³ – b³

Hence proved, LHS = RHS

a³ + b³ Formula (A Cube Plus B Cube)

The a³ + b³ formula is a³ + b³ = (a + b)(a² − ab + b²). It is used to factorise the sum of two cubes.

Proof of a³ + b³ Formula

Start with the right-hand side and multiply each term of (a + b) with (a² − ab + b²):

(a + b)(a² − ab + b²)
= a(a² − ab + b²) + b(a² − ab + b²)
= a³ − a²b + ab² + a²b − ab² + b³
= a³ + b³

The middle terms cancel out, so LHS = RHS. Hence proved.

Example: Find 5³ + 3³ Using the a³ + b³ Formula

Let a = 5 and b = 3.
a³ + b³ = (a + b)(a² − ab + b²)
= (5 + 3)(25 − 15 + 9)
= 8 × 19
= 152
Check: 5³ + 3³ = 125 + 27 = 152 ✔

Example: Find 11³ + 9³

Let a = 11 and b = 9.
(a + b)(a² − ab + b²) = (11 + 9)(121 − 99 + 81) = 20 × 103 = 2060
Check: 1331 + 729 = 2060 ✔

(a + b)³ and (a − b)³ Formulas

Students often confuse a³ + b³ with (a + b)³. They are different:

• (a + b)³ = a³ + b³ + 3ab(a + b) = a³ + 3a²b + 3ab² + b³
• (a − b)³ = a³ − b³ − 3ab(a − b) = a³ − 3a²b + 3ab² − b³
• So a³ + b³ = (a + b)³ − 3ab(a + b) and a³ − b³ = (a − b)³ + 3ab(a − b)

Example: if a = 2 and b = 3, then (a + b)³ = 5³ = 125, while a³ + b³ = 8 + 27 = 35. The difference is 3ab(a + b) = 3 × 2 × 3 × 5 = 90 ✔

Prove that (a + b + c)³ − a³ − b³ − c³ = 3(a + b)(b + c)(c + a)

Group the terms as [(a + b + c)³ − a³] − [b³ + c³].

Step 1: Use the a³ − b³ formula with x = (a + b + c) and y = a:
(a + b + c)³ − a³ = (b + c)[(a + b + c)² + a(a + b + c) + a²]

Step 2: Use the a³ + b³ formula:
b³ + c³ = (b + c)(b² − bc + c²)

Step 3: Take (b + c) common:
= (b + c)[(a + b + c)² + a(a + b + c) + a² − b² + bc − c²]

Step 4: Expand the bracket:
(a² + b² + c² + 2ab + 2bc + 2ca) + (a² + ab + ac) + a² − b² + bc − c²
= 3a² + 3ab + 3ac + 3bc
= 3(a² + ab + ac + bc)
= 3(a + b)(a + c)

Step 5: So (a + b + c)³ − a³ − b³ − c³ = (b + c) × 3(a + b)(a + c) = 3(a + b)(b + c)(c + a). Hence proved.

Examples on a^3-b^3 Formula

Here are a some examples on a³-b³.

Example 1: Find the Value of 83³-53³ Using the a^3-b^3 Formula.

Solution:

To find 83³− 53³ using the a3−b³ formula:

Let us assume that a = 83 and b = 53

Apply a³−b³ formula: a³ – b³ = (a – b) (a² + ab + b²)

(83−53)(83²+(83×53)+53²)

=(83-53)(83²+4399+53²)

=(83-53)(6889+4399+2809)

=(30) (14097)

Now, multiply (a – b) by (a² + ab + b²)

= 422910

Therefore, 83³−53³=422910

Example 2: Simplify 19^3-20^3 Using a Cube Minus b Cube Formula

Solution

To find 19^3 – 20^3

Let us assume a = 19 and b = 20

Using formula a³ – b³ = (a – b) (a² + ab + b²)

a³ – b³ = (a – b) (a² + ab + b²)

19³ – 20³ = (19 – 20) (19² + (19)(20) + 20²)

= (-1) (361 + 380 + 400)

= (-1) (1141)

= -1141

Therefore, 19^3-20^3 = -1141

Common Mistakes to Avoid

• Writing a³ − b³ = (a − b)³. This is wrong: for a = 3, b = 1, a³ − b³ = 26 but (a − b)³ = 8.
• Writing the second bracket as (a² − ab + b²) for a³ − b³. The middle sign must be +ab for a difference of cubes.
• Writing (a² + b²) and forgetting the ab term completely.
• Mixing up a³ + b³ (sum of cubes) with (a + b)³ (cube of a sum).

FAQ’s Related a^3-b^3 Formula

What is the Expansion of a³ – b³ Formula?

a³– b³ formula is read as a cube minus b cube. The a³– b³ expansion is expressed as (a – b) (a2 + ab + b2).

What is the Formula for a^3-b^3?

The formula for a^3 – b^3 is (a – b) (a^2 + ab + b^2)

What is a3 – b3 Formula in Algebra?

The formula for a3-b3 in algebra is a^3- b^3= (a – b) (a^2 + ab + b^2). This formula is used for factoring and simplifying the expressions involving the difference of two cubes.

What are a^3 – b^3 and a^3 + b^3 Formulas?

The formula of a³-b³ = (a – b) (a² + ab + b²)
The formula for a³+b³ = (a + b)(a² – ab + b²)

What is a Cube Minus b Cube Formula?

The formula of a Cube Minus b Cube is a3-b3 = (a – b) (a2 + ab + b2)

Also Read:

• What is the Formula of X^3+y^3+z^3-3xyz?
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