Difference of cubes Property

Property

The difference of any two quantities in cube form is equal to the product of their difference and the addition of sum of their squares and their product.

Introduction

The quantities are sometimes involved in cube form in subtraction and the difference of the quantities in cube form can be calculated after expanding the cubic quantities as per the exponentiation.

Evaluate $4^3-2^3$

$=\,\,$ $4 \times 4 \times 4$ $-$ $2 \times 2 \times 2$

$=\,\,$ $64-8$

$\therefore\,\,\,$ $4^3-2^3$ $\,=\,$ $56$

It is a very common method in basic mathematics, but it is not useful sometimes in advanced mathematics to find the difference of cubic quantities. Alternatively, the difference of quantities in cube form can be converted as a product of two factors by an arithmetic property.

Now, let us learn how to find the difference of cubic quantities on the basis of this property by the below three steps.

1. Calculate the difference of the quantities.
2. Add the product of quantities to the sum of their squares for finding the sum of them.
3. Evaluate the product of the difference and sum.

Now, let’s discuss the above three steps in detail by the same arithmetic example.

Find the Difference of cubic quantities

Firstly, let us calculate the difference of the numbers $4$ and $2$.

$\therefore\,\,\,$ $4-2$ $\,=\,$ $2$

Add the Product to the Sum of squares

Add the product of the numbers $4$ and $2$ to the sum of their squares, and then find the sum of them.

$4^2+2^2+4 \times 2$

$=\,\,$ $4 \times 4$ $+$ $2 \times 2$ $+$ $4 \times 2$

$=\,\,$ $16+4+8$

$\therefore\,\,\,$ $4^2+2^2+4 \times 2$ $\,=\,$ $28$

Calculate the Product of Difference and Sum

We have calculated the following two results in the above two steps.

$(1).\,\,\,$ $4-2$ $\,=\,$ $2$

$(2).\,\,\,$ $4^2+2^2+4 \times 2$ $\,=\,$ $28$

Now, find the product of the difference and the addition of the sum of the squares and product of the quantities.

$(4-2)$ $\times$ $(4^2+2^2+4 \times 2)$

$=\,\,\,$ $2 \times 28$

$=\,\,\,$ $56$

$\therefore\,\,\,$ $(4-2)$ $\times$ $(4^2+2^2+4 \times 2)$ $\,=\,$ $56$

Property

Observe the following two arithmetic equations.

$(1).\,\,\,$ $4^3-2^3$ $\,=\,$ $56$

$(2).\,\,\,$ $(4-2)$ $\times$ $(4^2+2^2+4 \times 2)$ $\,=\,$ $56$

It is clear that the difference of quantities in cube form is equal to the product of difference of quantities and sum of the squares of quantities and product of them.

$\therefore.\,\,\,$ $4^3-2^3$ $\,=\,$ $(4-2)$ $\times$ $(4^2+2^2+4 \times 2)$ $\,=\,$ $56$

It is an arithmetic property and is called the difference of cubes property.

You can test this property by taking any two real numbers and the following are some more examples to understand the difference of cubes property in mathematics.

$(1).\,\,$ $6^3-1^3$ $\,=\,$ $(6-1)(6^2+1^2+6 \times 1)$ $\,=\,$ $215$

$(2).\,\,$ $8^3-4^3$ $\,=\,$ $(8-4)(8^2+4^2+8 \times 4)$ $\,=\,$ $448$

$(3).\,\,$ $10^3-3^3$ $\,=\,$ $(10-3)(10^2+3^2+10 \times 3)$ $\,=\,$ $973$

Algebraic form

The difference of cubes property is generally written as a product of two factors in two popular algebraic forms. It is used as a formula in mathematics.

$(1).\,\,$ $a^3-b^3$ $\,=\,$ $(a-b)(a^2+b^2+ab)$

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

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