Math Formula (a+b)3

Understanding the Mathematical Formula $(a + b)^3$

The expression $(a + b)^3$ is a common algebraic formula that represents the cube of a binomial. This formula is frequently encountered in algebra and is useful in various mathematical problems and real-life applications.

Expansion of $(a + b)^3$

To expand $(a + b)^3$, we use the binomial theorem, which provides a systematic way of expanding powers of binomials. The binomial theorem states that for any positive integer $n$:

\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]

For $n=3$, the binomial expansion of $(a + b)^3$ is:

\[ (a + b)^3 = \binom{3}{0} a^3 b^0 + \binom{3}{1} a^2 b^1 + \binom{3}{2} a^1 b^2 + \binom{3}{3} a^0 b^3 \]

Simplifying this, we get:

\[ (a + b)^3 = 1 \cdot a^3 + 3 \cdot a^2 b + 3 \cdot a b^2 + 1 \cdot b^3 \]

Therefore, the expanded form of $(a + b)^3$ is:

\[ (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \]

Derivation Using the Distributive Property

Another way to understand this expansion is by using the distributive property step-by-step:

\[ (a + b)^3 = (a + b)(a + b)(a + b) \]

First, we expand $(a + b)(a + b)$:

\[ (a + b)(a + b) = a^2 + ab + ab + b^2 = a^2 + 2ab + b^2 \]

Now, we multiply this result by $(a + b)$:

\[ (a^2 + 2ab + b^2)(a + b) \]

Distribute each term in $(a^2 + 2ab + b^2)$ to $(a + b)$:

\[ = a^2(a + b) + 2ab(a + b) + b^2(a + b) \] \[ = a^3 + a^2b + 2a^2b + 2ab^2 + ab^2 + b^3 \] \[ = a^3 + 3a^2b + 3ab^2 + b^3 \]

Thus, we reach the same expanded form:

\[ (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \]

Example

Let's apply this formula to a specific example. Suppose we want to expand $(2 + 3)^3$.

First, identify $a = 2$ and $b = 3$.

Using the formula:

\[ (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \]

Substitute $a$ and $b$:

\[ (2 + 3)^3 = 2^3 + 3(2^2)(3) + 3(2)(3^2) + 3^3 \]

Calculate each term:

\[ 2^3 = 8 \] \[ 3(2^2)(3) = 3 \cdot 4 \cdot 3 = 36 \] \[ 3(2)(3^2) = 3 \cdot 2 \cdot 9 = 54 \] \[ 3^3 = 27 \]

Combine these results:

\[ (2 + 3)^3 = 8 + 36 + 54 + 27 \] \[ (2 + 3)^3 = 125 \]

Therefore, $(2 + 3)^3 = 125$.

Conclusion

The formula $(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$ provides a straightforward way to expand the cube of a binomial. This expansion is useful in algebraic manipulations and problem-solving. Understanding and applying this formula allows for efficient simplification of complex expressions and enhances mathematical fluency.

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