## a plus b cube formula

One of the Algebraic Identities, a plus b cube formula is used to find the cube of a binomial. This algebraic expression is also used to factorize some special types of trinomials.

In this article, the learners will get to know the formula (a+b)^{3} and comprehend it with supportive examples at the end.

**A plus B cube formula**

**(a+b) ^{3} = a^{3 }+ 3a^{2}b + 3ab^{2} + b^{3}**

Derivation of the (a+b)^{3}

To find the cube of a binomial,

we will just multiply (a + b)(a + b)(a + b). (a + b)^{3} formula is also an identity.

It holds true for every value of a and b.

The (a + b)^{3} is given as,

(a + b)^{3 }= (a + b)(a + b)(a + b)

= (a^{2} + 2ab + b^{2})(a + b)

= a^{3} + a^{2}b + 2a^{2}b + 2ab^{2} + ab^{2} + b³

= a^{3} + 3a^{2}b + 3ab^{2} + b³

= a^{3} + 3ab(a+b) + b^{3}

Therefore, (a + b)^{3} formula is:

(a + b)^{3} = a^{3} + 3a^{2}b + 3ab^{2} + b^{3}.

## Proof of formula in Algebraic Method

The a plus b whole cube formula can be derived in algebraic approach by multiplying three same sum basis binomials. .

### Product form of Binomials

Multiplying the binomial by itself three times is the mathematical meaning of the cube of the binomial . So, the A plus B whole cube can be expressed in product form by multiplying three same binomials.

Multiplying three same binomials is a special case in mathematics. Hence, the product of them is often called the special product of binomials.

### Multiplying the Algebraic expressions

If It is not possible to multiply all three same binomials at a time, then another method is to

multiply any two binomials first and then multiply the remaining two factors for getting expansion of identity in algebraic approach.

### Simplified form of the Expansion

The expansion of the cube of the sum of terms can also be written as the following simplified form.

(a + b)^{3} = a^{3} + 3a^{2}b + 3ab^{2} + b^{3}.

(a + b)^{3} – Solved Examples using the given formula :

**Ques 1: Solve the following expression using suitable algebraic identity:**

(2x + 3y)^{3}

**Solution: **

To find: (2x + 3y)^{3}

Using (a + b)^{3} Formula,

(a + b)^{3} = a^{3} + 3a^{2}b + 3ab^{2} + b^{3}

= (2x)^{3} + 3 × (2x)^{2} × 3y + 3 × (2x) × (3y)^{2} + (3y)^{3}

= 8x^{3} + 36x^{2}y + 54xy^{2} + 27y^{3} (answer).

**Example 2: Find the value of x ^{3}+ 8y^{3}**

if x + 2y = 6 and xy = 2.

**Solution:**

To find: x^{3} + 8y^{3}

Given: x + 2y = 6 and xy = 2

Using (a + b)^{3} formula,

Here, a = x; b = 2y

Therefore,

(x + 2y)^{3 }= x^{3}+ 3 × x^{2} × (2y) + 3 × x × (2y)^{2} + (2y)^{3}

(x + 2y)^{3} = x^{3} + 6x^{2}y + 12xy^{2} + 8y^{3}

6^{3} = x^{3}+ 6xy(x + 2y) + 8y^{3} ( putting the value of (x+2y)

216 = x^{3} + 6 × 2 × 6 + 8y^{3}

x^{3} + 8y^{3} = 144