The given expression #160; 27a^3 #8722;(3a #8722;b)^3 #160;can be solved as shown below: 27a^ 3 #8722;(3a #8722;b)^ 3 =(3a)^ 3 #8 ...

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Algebraic Identities

27a3-3a-b3.

Question

$27 a^{3} - (3 a - b)^{3}$ .

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Solution

The given expression $27 a^{3} - (3 a - b)^{3}$ can be solved as shown below:

$27 a^{3} - (3 a - b)^{3} = (3 a)^{3} - (3 a - b)^{3} = [3 a - (3 a - b)] [(3 a)^{2} + (3 a - b)^{2} + 3 a (3 a - b)] (∵ x^{3} - y^{3} = (x - y) (x^{2} + y^{2} + x y) = (3 a - 3 a + b) [9 a^{2} + [(3 a)^{2} + b^{2} - (2 \times 3 a \times b)] + 9 a^{2} - 3 a b] (∵ (x - y)^{2} = x^{2} + y^{2} - 2 x y) = b (9 a^{2} + 9 a^{2} + b^{2} - 6 a b + 9 a^{2} - 3 a b) = b (27 a^{2} + b^{2} - 9 a b)$

Hence, $27 a^{3} - (3 a - b)^{3} = b (27 a^{2} + b^{2} - 9 a b)$

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