Page 22 - Mathematics Class - IX

3. Make a cuboid of dimensions ab , i.e. 3 × 1 × 1 (Fig. (d)). Add this cuboid three times in Fig. (c) as shown
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in Fig. (e).
4. Make a cube of dimensions b , i.e. 1 × 1 × 1 (Fig. (f)). Add this cube in Fig. (e) as shown in
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Fig. (g).

Fig. (e) Fig. (f) Fig. (g)

5. The total number of cubes will be 64 = 4 , i.e. (a + b) as shown in Fig. (g).
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OBSERVATIONS
1. Number of unit cubes in a = 3 = 27
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2. Number of unit cubes in a b = 9
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3. Number of unit cubes in a b = 9
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4. Number of unit cubes in a b = 9
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5. Number of unit cubes in ab = 3
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6. Number of unit cubes in ab = 3
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7. Number of unit cubes in ab = 3
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8. Number of unit cubes in b = 1
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9. Total cubes = 64

10. 64 = 4 3

INFERENCE

It is verified that, (a + b) = a + a b + a b + a b + ab + ab + ab + b = a + 3a b + 3ab + b 3
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EXTENDED TASK
1. Verify the identity (a + b) = a + 3a b + 3ab + b for a cube of 10 units. [Hint: use (7 + 3) or (6 + 4) ]
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2. Using the identity (x + y) = x + 3x y + 3xy + y represent it geometrically and algebraically.
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APPLICATION
The result can be used in
1. Calculating cube of a number expressed as the sum of two convenient numbers.
2. To simplify and factorise the algebraic expressions.
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