Math Labs with Activity – Verify the Identity a³ – b³ =(a-b)(a² + ab+b²)

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Math Labs with Activity – Verify the Identity a³ – b³ =(a-b)(a² + ab+b²)

OBJECTIVE

To verify the identify a³ – b³ =(a-b)(a² + ab+b²) using a set of unit cubes

Materials Required
A set of 53 plastic cubes where each cube has dimensions (1 unit x 1 unit x 1 unit).

Procedure
We shall verify the identity for a = 3 and b = 1.
Step 1: We shall make Arrangement 1 for 26 cubes.
Take 27 cubes and place them to form a stack consisting of 9 columns, each column having 3 cubes [see Figure 5.1(a)]. From this stack remove one cube [see Figure 5.1(b)] to form a stack consisting of 26 cubes as shown in Figure 5.1(c). The stack in Figure 5.1(c) forms Arrangement 1. The total volume of this arrangement of cubes is calculated (see the calculations).
Math Labs with Activity - Verify the Identity a³ - b³ =(a-b)(a² + ab+b²) 1
Step 2: We shall make Arrangement 2 for 26 cubes.
This arrangement consists of three stacks—the first stack [shown in Figure 5.2(a)] consists of 9 columns of two cubes each, the second stack [shown in Figure 5.2(b)] consists of two rows of three cubes each, and the third stack consists of 1 row of 2 cubes.
The total volume of this arrangement of cubes is calculated (see the calculations).
Math Labs with Activity - Verify the Identity a³ - b³ =(a-b)(a² + ab+b²) 2

Observations
Since the two arrangements have equal number of cubes (each arrangement has 26 cubes) and all the cubes have the same volume (1 cubic unit), the total volumes in both the arrangements must be equal.

Calculations

  1. Volume of Arrangement 1
    Volume of the stack in Figure 5.1(a) = a³.
    Volume of the stack in Figure 5.1(b) =b³.
    =volume of Arrangement 1 volume of the stack in Figure 5.1(c)
    = volume of the stack in Figure 5.1(a)
    =volume of the stack in Figure 5.1(b) = a³-b³.
  2. Volume of Arrangement 2
    Volume of the stack in Figure 5.2(a)=(a-b)a².
    Volume of the stack in Figure 5.2(b) =(a-b)ab.
    Volume of the stack in Figure 5.2(c) = (a-b)b².
    volume of Arrangement 2 = (a-b)a² + (a -b)ab + (a -b)b².
    =(a-b)(a² + ab+b²).
    Since the total volume in the two arrangements must be the same, therefore
    a³ – b³ =(a-b)(a² + ab+b²).

Result
It is verified that a³ – b³ =(a-b)(a² + ab+b²).

Remarks: The students must try to verify this identity for other values of a and b by taking required number of cubes and arranging them suitably.

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