What is the Formula of a³ + b³

What is the formula of a³ + b³?

In mathematics, there are many formulas and algebraic identities that play an important part in everyday calculations. Though they may seem to be negligible and tiny, the part they play in mathematics is no joke. One such identity is a³ + b³. Let us look at a few ways to express and write this identity. 

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

Let us look at the derivation of this identity. 

For that, we will find the cube of the expression (a + b)

So, 

1. (a + b)³ = a³ + b³ + 3ab(a + b), Now, taking a³ + b³ to the right hand side, we get

2. a³ + b³ = (a + b)³ – 3ab(a + b) = (a + b) {(a + b)² – 3ab}

3. a³ + b³ = (a + b) {a² + b² + 2ab – 3ab} = (a + b) (a² + b² – ab)

Hence, 

a³ + b³ = (a + b) (a² – ab + b²)

Let us understand this identity with the help of a few examples. 

• Example 1

If a = 7 and b = 2, then find the value of a³ + b³

We use the identity a³ + b³ = (a + b) (a² – ab + b²) here. 

So, a³ + b³ = (7+2)(7² – 2×7 + 2²) = 9 (49-14+4) = 9 × 39 = 351

Hence, the value of a³ + b³ is 351.

• Example 2

If a = 7 and b = -4, then find the value of a³ + b³

We use the identity a³ + b³ = (a + b) (a² – ab + b²) here. 

So, a³ + b³ = (7 + (-4))(7² – 7×(-4) + (-4)²) = 3 (49 + 28 + 16) = 3×93 = 279

Hence, the value of 7³+(-4)³ is 279.

• Example 3

Factorize: 125x³ + 8a³

To factorize the given expression, we will use the identity 

a³ + b³ = (a + b) (a² – ab + b²).

Here, 125x³ = (5x)³ and 8a³ = (2a)³

So, 125x³ + 8a³ = (5x)³ + (2a)³ = (5x + 2a)((5x)² – (5x)(2a) + (2a)²)

Hence, 125x³ + 8a³ = (5x + 2a)(25x² – 10ax + 4a²)

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