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# (a+b)^3 Formula

The (a + b)^3 formula is used to find the cube of a binomial. This formula is also used to factorize some special types of polynomials. This formula is:

- one of the algebraic identities.
- the formula for the cube of the sum of two terms.

Let us understand (a + b)^{3 }formula in detail in the following section.

## What is the (a + b)^3 Formula?

**(a + b) whole cube formula** says: (a + b)^{3} = a^{3} + 3a^{2}b + 3ab^{2} + 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.

### Derivation of a plus b whole cube Formula

The (a + b)^{3} formula can be derived as follows:

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

= (a^{2} + 2ab + b^{2})(a + b) [∵ (a + b)^{2} = a^{2} + 2ab + b^{2}]

= a^{3} + a^{2}b + 2a^{2}b + 2ab^{2} + ab^{2} + b^{3} [∵ By distribution property]

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

= 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}

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## Examples on (a + b)^{3} Formula

**Example 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: **(2x + 3y)^{3} = 8x^{3} + 36x^{2}y + 54xy^{2} + 27y^{3}

**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

xy = 2

Using a plus b whole cube formula,

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

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}

216 = x

^{3}+ 6 × 2 × 6 + 8y

^{3}

x

^{3}+ 8y

^{3}= 144

**Answer: **x^{3} + 8y^{3} = 144

**Example 3:** Solve the following expression using (a + b)^{3} formula: (5x + 2y)^{3}.

**Solution:**

To find: (5x + 2y)^{3}

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

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

= (5x)^{3} + 3 × (5x)^{2} × 2y + 3 × (5x) × (2y)^{2} + (2y)^{3}

= 125x^{3} + 150x^{2}y + 60xy^{2} + 8y^{3}

**Answer: **(5x + 2y)^{3} = 125x^{3} + 150x^{2}y + 60xy^{2} + 8y^{3}

## FAQ's on (a + b)^3 Formula

### What is the Expansion of (a + b)^{3} Formula?

**(a + b) ^{3} formula** is read as a plus b whole cube. Its expansion is expressed as (a + b)

^{3}= a

^{3}+ 3a

^{2}b + 3ab

^{2}+ b

^{3}

### What is the Formula of a Plus b Plus c Whole Cube?

The formula of a plus b plus c whole cube is: (a + b + c)^{3} = a^{3} + b^{3} + c^{3} + 3 (a + b) (b + c) (c + a).

**☛Note: **If a + b + c = 0, then we can write a + b = -c, b + c = -a, and c + a = -b. Then the above formula becomes: 0^{3} = a^{3} + b^{3} + c^{3} + 3 (-c) (-a) (-b) ⇒ a^{3} + b^{3} + c^{3} = 3abc. This is also a widely used formula in algebra.

### What is the (a + b)^{3} Formula in Algebra?

The (a + b)^{3} formula is also known as one of the important algebraic identities. It is read as a plus b whole cube. The (a + b)^{3} formula is says: (a + b)^{3} = a^{3} + 3a^{2}b + 3ab^{2} + b^{3}

### How To Simplify Numbers Using the (a + b)^{3} Formula?

Let us understand the use of the (a + b)^{3} formula with the help of the following example.

**Example:** Find the value of (20 + 5)^{3} using the (a + b)^{3} formula.

To find: (20 + 5)^{3}

Let us assume that a = 20 and b = 5.

We will substitute these in the formula of (a + b)^{3}.

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

(20+5)^{3} = 20^{3} + 3(20)^{2}(5) + 3(20)(5)^{2} + 5^{3}

=8000 + 6000 + 1500 + 125

= 15625

**Answer:** (20 + 5)^{3} = 15625.

### How To Use the (a + b)^{3} Formula Give Steps?

The following steps are followed while using (a + b)^{3} formula.

- Firstly observe the pattern of the numbers whether the numbers have whole ^3 as power or not.
- Write down the formula of (a + b)
^{3}: (a + b)^{3}= a^{3}+ 3a^{2}b + 3ab^{2}+ b^{3}. - Substitute the values of a and b in the (a + b)
^{3}formula and simplify.

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