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Factorize the following expressions:

(a - 2b)^{3} - 512b^{3}

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#### Solution

(a - 2b)^{3} - 512b^{3}

= (a - 2b)^{3} - (8b)^{3}

= (a - 2b - 8b)((a - 2b)^{2} + (a - 2b)8b + (8b)^{2}) [ ∵ a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2})]

= (a -10b)(a^{2} + 4b^{2} - 4ab + 8b (a - 2b) + (8b)^{2} ) [ ∵ (a - b)^{2} = a^{2} + b^{2} - 2ab]

= (a -10b)(a^{2} + 4b^{2} - 4ab + 8ab -16b^{2} + 64b^{2})

= (a = 10b)(a^{2} + 68b^{2} -16b^{2} - 4ab + 8ab)

= (a -10b)(a^{2} + 52b^{2} + 4ab)

∴ (a - 2b)^{3} - 512b^{3} = (a -10b)(a^{2} + 4ab + 52b^{2})

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