Math Doubts

$a^3-b^3$ Identity

Formula

$a^3-b^3$ $\,=\,$ $(a-b)(a^2+b^2+ab)$

Introduction

Let $a$ and $b$ be two quantities in algebraic form.

  1. The subtraction of $b$ from the $a$ is written as $a-b$ in mathematical form.
  2. The subtraction of $b$ cube from $a$ cube is mathematically written as $a^3-b^3$.
  3. Add the sum of the squares of $a$ and $b$ to the product of $a$ and $b$, and their sum is written as $a^2+ab+b^2$.

As per the difference of cubes arithmetic property, the subtraction of $b$ cubed from $a$ cubed is equal to the product of the subtraction of $b$ from $a$ and the addition of sum of squares of $a$ and $b$ and the product of $a$ and $b$.

$\therefore\,\,\,$ $a^3-b^3$ $\,=\,$ $(a-b)$ $\times$ $(a^2+b^2+ab)$

Now, it is time to learn more about the $a$ cube minus $b$ cube algebraic identity.

Other form

The $a$ cube minus $b$ cube algebraic identity is alternatively written in mathematics as follows.

$x^3-y^3$ $\,=\,$ $(x-y)(x^2+y^2+xy)$

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Uses

The $a$ cube minus $b$ cube algebraic identity is used in two different cases mainly.

  1. In basic mathematics, it is used as a formula to evaluate the difference of cubes of two quantities.
  2. It is also used as a formula to write the difference of two quantities in cube form as a product of two factors.
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Proofs

The $a$ cube minus $b$ cube algebraic identity can be derived in two different methods.

Algebraic Method

Learn how to derive the $a$ cube minus $b$ cube formula by the algebraic identities.

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Geometric Method

Learn how to prove the $a$ cubed minus $b$ cubed identity geometrically by the volume of a cube.

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Verification

Assume that $a = 5$ and $b = 2$

$(1).\,\,$ $a-b$ $\,=\,$ $5-2$ $\,=\,$ $3$

$(2).\,\,$ $a^2+b^2+ab$ $\,=\,$ $5^2+2^2+5 \times 2$ $\,=\,$ $25+4+10$ $\,=\,$ $39$

$(3).\,\,$ $a^3-b^3$ $\,=\,$ $5^3-2^3$ $\,=\,$ $125-8$ $\,=\,$ $117$

Now, calculate the product of the $a-b$ and $a^2+b^2+ab$.

$\implies$ $(a-b)$ $\times$ $(a^2+b^2+ab)$ $\,=\,$ $3 \times 39$ $\,=\,$ $117$

It is calculated that the product of them is equal to $117$, and the difference of cubes of $a$ and $b$ is also equal to $117$.

$\,\,\,\therefore\,\,\,\,\,\,$ $a^3-b^3$ $\,=\,$ $117$ $\,=\,$ $(a-b)(a^2+b^2+ab)$

You too can verify the $a$ cube minus $b$ cube algebraic identity by taking any two numbers as explained above.