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,
(a + b)3 = a3 + b3 + 3ab(a + b), Now, taking a3 + b3 to the right hand side, we get
a3 + b3 = (a + b)3 – 3ab(a + b) = (a + b) {(a + b)2 – 3ab}
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)