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# a^3+b^3 Formula

The a^{3 }+ b^{3} formula is called the sum of cubes (of two numbers) formula. The a cube plus b cube formula is used to find the sum of the two cubes without actually computing the cubes. Also, it is used to factorize the binomials of cubes. Here, we will be discussing the different aspects of the a^{3 }+ b^{3} formula, along with solved examples, and understand where this identity can be applied.

## What is a^3+b^3 Formula?

The **a ^{3} + b^{3} formula** or the difference of cubes formula is mentioned below:

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

Are the signs confusing? The following trick helps!

**☛Also Check:**

Let us learn the a^{3} + b^{3} formula with a few solved examples. The a^{3} + b^{3} formula can be verified, by multiplying (a + b) (a^{2} - ab + b^{2}) and see whether you get a^{3} + b^{3}.

### Proof of a Cube plus b Cube Formula

Let us verify the "a cube plus b cube" formula here. To prove or verify that a^{3 }+ b^{3 }= (a + b) (a^{2} - ab + b^{2}) we need to prove here LHS = RHS.

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

On simplifying the RHS side 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, a^{3} + b^{3} = (a + b) (a^{2} - ab + b^{2}).

## Examples on a^3 + b^3 Formula

**Example 1:** Find the value of 102^{3} + 8^{3 }by using the a^3+b^3 formula.

**Solution:** To find: 102^{3} + 8^{3}.

Let us assume that a = 102 and b = 8.

We will substitute these in a cube plus b cube formula i.e.,

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

102^{3} + 8^{3} = (102 + 8) (102^{2} - (102)(8) + 8^{2})

= (110) (10404 - 816 + 64)

= (110) (9652)

= 1061720

**Answer:** 102^{3} + 8^{3} = 1,061,720.

**Example 2:** Factorize the expression 8x^{3} + 27 by using the a^3+b^3 formula.

**Solution:** To factorize: 8x^{3} + 27.

We will use the a^{3} + b^{3} formula to factorize this.

We can write the given expression as

8x^{3} + 27 = (2x)^{3} + 3^{3}

We will substitute a = 2x and b = 3 in the formula of a^{3} + b^{3}.

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

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

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

**Answer:** 8x^{3} + 27 = (2x + 3) (4x^{2} - 6x + 9).

**Example 3:** Simplify 19^{3} + 20^{3} using a cube plus b cube formula

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

Let us assume a = 19 and b = 20

Using 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.

## FAQs on a^3^{ }+ b^3 Formula

### What is the Expansion of a^{3 }+ b^{3} Formula in Algebra?

a^{3} + b^{3} formula is read as a cube plus b cube. It is called the sum of cubes formula. The formula says a^{3 }+ b^{3 }= (a + b) (a^{2} - ab + b^{2}).

### How to Apply the a^{3 }+ b^{3} Formula?

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

**Example:** Evaluate the value of 18^{3} + 2^{3} using the a^{3 }+ b^{3} formula.

To find: 18^{3} + 2^{3}, let us assume that a = 18 and b = 2. We will substitute these in the formula of a^{3} + b^{3}.

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

18^{3 }+ 2^{3} = (18 + 2)(18^{2} - (18)(2) + 2^{2})

= (20) (324 - 36 + 4)

= 5840

**Answer:** 18^{3} + 2^{3} = 5840.

### What is a Cube Plus b Cube Plus c Cube Formula?

In algebra, we have a formula a^{3} + b^{3} + c^{3} - 3abc = (a + b + c)(a^{2} + b^{2} + c^{2} - ab - bc - ca). By adding 3abc on both sides, we get a^{3} + b^{3} + c^{3} = (a + b + c)(a^{2} + b^{2} + c^{2} - ab - bc - ca) + 3abc.

**☛Note:** When a + b + c = 0, a^{3} + b^{3} + c^{3} = 3abc from the above formula.

### What is a Cube Plus b Cube Formula in Algebra?

The a^{3} + b^{3} formula is one of the algebraic identities. It is pronounced as "a cube plus b cube". The a^{3} + b^{3} formula is: a^{3} + b^{3} = (a + b) (a^{2} - ab + b^{2}).

### How to Use the a^3 + b^3 Formula?

The following steps are followed while using a^3 + b^3 formula.

**Step 1:**Express the given expression as sum of cubes.**Step 2:**Recall the formula a^{3}+ b^{3 }= (a + b) (a^{2}- ab + b^{2})**Step 3:**Substitute the values of a and b in the a^{3 }+ b^{3}formula and simplify it.

### What are the Formulas of a Cube Plus b Cube and a Cube Minus b Cube?

Here are the formulas:

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

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