## What Is the (a + b)^{3} Formula?

**In algebra, the (a + b) ^{3} formula allows us to find the cube of a binomial and helps us to simplify **

**expressions**

**.**

The formula for the cube of the sum of two terms, **(a + b)^{3}**, can be expanded using the binomial theorem or by applying the distributive property multiple times. The result is:

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

**+ 3a**

^{3}^{2}b + 3 ab

**+ b**

^{2}

^{3}This formula shows how to find the cube of the sum of a and b by raising both terms to the power of 3, along with the combinations involving a and b raised to the power of 2 and 1.

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

^{3}

The (a + b) whole cube formula or (a + b)** ^{3}** formula represents the cube of a binomial. It represents the result of cubing the sum of ‘a’ and ‘b’.

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

**+ 3a**

^{3}**b + 3ab**

^{2}**+ b**

^{2}

^{3}It’s important to note that this formula holds true for any values assigned to both ‘a’ and ‘b.’

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## Derivation of (a+b)^{3} Formula

^{3}

To understand the derivation of the a + b whole cube formula, let’s start by expanding the expression (a + b)** ^{3}**.

We can achieve this by multiplying the binomial (a + b) by itself three times:

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

On expanding this expression, we get

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

**+ 2ab + b**

^{2}**)(a + b)**

^{2}Let’s use the distributive property to expand the above expression.

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

**+ a**

^{3}**b + 2a**

^{2}**b + 2ab**

^{2}**+ ab**

^{2 }**+ b**

^{2 }

^{3}On combining like terms, we arrive at the final form of the (a + b)** ^{3}** formula:

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

**+ 3a**

^{3}**b + 3ab**

^{2}**+ b**

^{2}

^{3}Alternatively, we can express the (a + b)** ^{3}** formula as:

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

**+ 3ab (a + b) + b**

^{3}

^{3}This derivation showcases the algebraic manipulation required to arrive at the (a + b)** ^{3}** formula.

## How to Use the (a+b)^{3} Formula

^{3}

Let’s use the formula (a + b)** ^{3}** = a

**+ 3a**

^{3}**b + 3ab**

^{2}**+ b**

^{2 }**to expand some expressions.**

^{3}**Example**: (x + 2y)^{3}

Here, a = x, b = 2x

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

**+ 3a**

^{3}**b + 3ab**

^{2}**+ b**

^{2}

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

**+ 3x**

^{3}**(2y) + 3x(2y)**

^{2 }**+ (2y)**

^{2}

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

**+ 6x**

^{3}**y + 3x(4y**

^{2}**) + 8y**

^{2}

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

**+ 6x**

^{3}**y +12xy**

^{2}**+ 8y**

^{2}

^{3}## Facts about the (a+b)3 Formula

- The formula (a + b)
represents the cube of the sum of two terms, “a” and “b.” (a + b)^{3}= a^{3}+ 3a^{3}b + 3ab^{2}+ b^{2}^{3}

- The a plus b whole cube formula represents the cube of the sum of “a” and “b.” It can also be expressed as (a + b)
= (a + b)$\times$(a + b)$\times$(a + b).^{3}

- By the commutative property of addition, we have (a + b)
= (b + a)^{3}.^{3}

- The (a + b)
formula is part of a group of related formulas known as the binomial coefficients, which are used to expand powers of binomials.^{3}

For example, (a + b)** ^{2 }**= a

**+ 2ab + b**

^{2}

^{2}- The coefficients of the terms in the expansion of (a + b)
correspond to the numbers in the fourth row of Pascal’s Triangle. Pascal’s Triangle is a triangular arrangement of numbers where each number is the sum of the two numbers directly above it.^{3}

In the expansion (a + b)** ^{3}** = a

**+ 3a**

^{3}**b + 3ab**

^{2}**+ b**

^{2}**, the coefficients (1, 3, 3, 1) are the same as the fourth row of Pascal’s Triangle. This relationship between binomial expansions and Pascal’s Triangle extends to higher powers as well.**

^{3}## Conclusion

The (a+b)3 formula is a powerful tool in algebra for expanding and simplifying cubic binomials. By understanding its derivation, properties, and applications, you can enhance your problem-solving skills and tackle a wide range of mathematical challenges. Let’s solve some (a + b)** ^{3}** formula examples and practice problems to understand the concept better.

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

^{3}

**Example 1. Expand (x + 2)^{3}.**

**Solution:**

Using the (a + b)** ^{3}** formula, we have:

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

**+ 3a**

^{3}**b + 3ab**

^{2}**+ b**

^{2 }

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

**+ 3x**

^{3}**(2) + 3x(2)**

^{2}**+ 2**

^{2}

^{3}Simplifying each term, we get

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

**+ 6x**

^{3}**+ 12x + 8**

^{2}**Example 2: Evaluate 143 ^{ }using the **(a + b)

^{3}**formula.**

**Solution:**

Hare, we can write** **

**14^{3} = (10 + 4)^{3}**

Here, a = 10, b = 4

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

**+ 3a**

^{3}**b + 3ab**

^{2}**+ b**

^{2}

^{3}**(10 + 4)^{3} **= 10

**+ 3(10)**

^{3}**(4) + 3(10)(4)**

^{2}**+ (4)**

^{2}

^{3}**(10 + 4)^{3} **= 1000 + 3(100)(4) + 3(10)(16) + 64

**(10 + 4)^{3} **= 1000 + 1200 + 480 + 64

**(10 + 4)^{3} **= 2744

**14^{3}**

**= 2744**

**Example 3: Find the coefficient of the term ****a²b**** in the expansion of ****(3a + 2b)³****.**

**Solution:**

Using the (a + b)** ^{3}** formula, we get

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

**+ 3(3a)**

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

^{2}**+ (2b)**

^{2}

^{3}(3a + 2b)** ^{3}** = 27a

**+ 3(9a**

^{3}**)2b + 3(3a)(4b**

^{2}**) + 8b**

^{2}

^{3}(3a + 2b)** ^{3}** = 27a

**+ 54a**

^{3}**b +36ab**

^{2}**+ 8b**

^{2 }

^{3}So, the coefficient of the term involving “**a²b**” is 54.

## Practice Problems on the (a + b)^{3} Formula

^{3}

## $(A+B)^{3}$ Formula - Solved Examples, FAQS, Practice Problems

### What is the expansion of $(a + b)^{3}$?

The correct expansion of $(a + b)^{3}$ is $a^{3} + 3a^{2}b + 3ab^{2} + b^{3}$.

### What does $(a + b)^{3}$ represent?

The $(a + b)^{3}$ formula represents the cube of the sum of a and b

### Which pattern do the coefficients in the expansion of $(a + b)^{3}$ follow?

The coefficients in the expansion of $(a + b)^{3}$ follow the arrangement of Pascal's triangle, where each coefficient is obtained by adding the two coefficients directly above it.

### What is the expanded form of $(x + 2y)^{3}$?

The expansion of $(x + 2y)^{3}$ is obtained by applying the $(a + b)^{3}$ formula, resulting in $x^{3} + 6x^{2}y + 12xy^{2} + 8y^{3}$

## Frequently Asked Questions about the (a + b)^{3} Formula

^{3}

**What is the purpose of the (a + b)^{3} formula?**

The formula is used to expand and simplify cubic binomials, making algebraic manipulations easier.

**Can the (a + b)^{3} formula be applied to other powers?**

Yes, the formula is a specific case of the binomial theorem and can be extended to any positive integer power.

**How can the (a + b)^{3} formula be proven geometrically?**

By visualizing a cube with side lengths (a+b) and observing how its faces, edges, and corners correspond to the expanded terms.

**What is the coefficient pattern in the expansion of (a + b)^{3}?**

The coefficients follow Pascal’s triangle: 1, 3, 3, 1.

**How is the (a + b)^{3} formula related to real-world problems?**

**Can the (a + b)^{3} formula be used with negative values of “a” and “b”?**

Yes, the **(a + b)^{3}** formula can be used with negative values of “a” and “b.” The formula works regardless of the signs of the variables.