# Sum of Cubes Formula

The formula to find the addition of two polynomials, a^{3 }+ b^{3} is known as the sum of cubes formula. Let's learn more about the sum of cubes formula with a few solved examples. This factoring formula comes in very handy when solving algebraic expressions of various types. Memorizing this formula is also easy and can be done within a matter of minutes. It is very similar to the difference in cubes formula as well.

## What Is the Sum of Cubes Formula?

In this section, let us go further and understand what exactly does it mean when some is referring to the sum of cubes. The formula to the sum of cubes formula is given as:

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

where,

- a is the first variable
- b is the second variable

## Proof of Sum of Cubes Formula

To prove or verify that sum of cubes formula that is, a^{3 }+ b^{3 }= (a + b) (a^{2} - ab + b^{2}) we need to prove here LHS = RHS.

LHS term = a^{3 }+ b^{3}

On Solving RHS term we get,

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

On multiplying the a and b separately with (a^{2} + ab + b^{2}) we get

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

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

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

= a^{3} - 0 + 0 + b^{3}

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

Hence proved, LHS = RHS

## Examples on Sum of Cubes Formula

**Example1:** Use the sum of cubes formula to find the factor of 216x^{3 }+ 64.

To find: Factor of 216x^{3 }+ 64, using the sum of cubes formula.

216x^{3}+ 64 = (6x)^{3} + 4^{3}

Using the sum of cubes formula,

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

Put the values,

(6x)^{3} + 4^{3} = (6x + 4)((6x)^{2 }- 6x × 4 + 4^{2})

(6x)^{3} + 4^{3} = (6x + 4)(36x^{2 }- 24x +16)

(6x)^{3} + 4^{3} = 8(3x + 2)(9x^{2 }- 6x + 4)

**Answer: The factor of 216x ^{3} + 64 is 2(3x + 2)(9x^{2 }- 6x + 4).**

**Example 2: **Find the factor of 8x^{3} + 125y^{3}.

To find: Factor of 8x^{3} + 125y^{3}, using the sum of cubes formula.

8x^{3} + 125y^{3} = (2x)^{3} + (5y)^{3}

Using the sum of cubes formula,

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

Put the values,

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

(2x)^{3} + (5y)^{3 }= (2x + 5y)(4x^{2} – 10xy + 25y^{2})

**Answer: The factor of 8x ^{3} + 125y^{3} is (2x + 5y)(4x^{2} – 10xy + 25y^{2}).**

**Example 3: **Simplify 19^{3} + 20^{3} using the sum of cubes formula.

**Solution: **To find 19^{3} + 20^{3}

Let us assume a = 19 and b = 20

Using sum of cubes formula a^{3 }+ b^{3 }= (a + b) (a^{2} - ab + b^{2})

We will substitute these in the a^{3} + b^{3} formula

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

19^{3}+20^{3} = (19+20)(19^{2} - (19)(20)+20^{2})

= (39)(361-380+400)

= (39)(381)

= 14,859

**Answer:** 19^{3} + 20^{3} = 14859.

## FAQ's on Sum of Cubes Formula

### What Is the Expansion of Sum of Cubes Formula?

a^{3 }+ b^{3} formula is known as the sum of cubes formula it is read as a cube plus b cube. Its expansion is expressed as a^{3 }+ b^{3 }= (a + b) (a^{2} - ab + b^{2}).

### What Is the Sum of Cubes Formula in Algebra?

The sum of cubes formula is one of the important algebraic identity. It is represented by a^{3} + b^{3} and is read as a cube plus b cube. The sum of cubes (a^{3 }+ b^{3}) formula is expressed as a^{3 }+ b^{3} = (a + b) (a^{2} - ab + b^{2}).

### How To Simplify Numbers Using the Sum of Cubes Formula?

Let us understand the use of the sum of cubes formula i.e., a^{3 }+ b^{3} formula with the help of the following example.

**Example:** Find the value of 100^{3} + 2^{3} using the sum of cubes formula.

To find: 100^{3} + 2^{3}

Let us assume that a = 100 and b = 2.

We will substitute these in the formula of the sum of cubes formula that is, a^{3} + b^{3}

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

100^{3}+2^{3} = (100+2)(100^{2} - (100)(2)+2^{2})

= (102) (10000-200+4)

= (102)(9804)

= 1000008

**Answer:** 100^{3} + 2^{3} = 1000008.

### How To Use the Sum of Cubes Formula Give Steps?

The following steps are followed while using the sum of cubes formula.

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

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