What is the formula for {a³} – {b³}

Answer:

The formula for {a³} – {b³} is {a – b}{a² + b² + ab}

The (a³ – b³) formula is referred to as the difference of cubes formula. This particular formula is used to compute the difference between the two cubes, cubes of a and cubes of b, without actually calculating the same cubes. Additionally, the formula is used to factorize the binomials of the cubes.

Proof of the formula

To prove that (a³ – b³) = (a – b)(a² + b² + ab), it is to be proven that LHS = RHS.

Here,

LHS = (a³ – b³)

RHS = (a – b)(a² + b² + ab)

On multiplying the a and b separately with (a² + ab + b²), the following is obtained

= 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, LHS = RHS

The same formula can also be proven by letting the values of a = 4 and b = 2.

(4³ – 2³) = (4 – 2)(4² + 2² + 4 x 2)

LHS = (a³ – b³)

LHS = (4³ – 2³)

LHS = (64 – 8)

LHS = 56

RHS (a – b)(a² + b² + ab)

RHS = (4 – 2)(4² + 2² + 4 x 2)

RHS = (2)(16 + 4 + 8)

RHS = (2)(28)

RHS = 56

Therefore, LHS = RHS

(a³ – b³) = (a – b)(a² + b² + ab)

Hence, it is proved that the formula for ${a^3} – {b^3}$ is {a – b}{a² + b² + ab}.