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Composite Function

Simplify i ...

Question

Simplify (i) $(3 x + 4)^{3} - (3 x - 4)^{3}$ (ii) ${(x + \frac{2}{x})}^{3} + {(x - \frac{2}{x})}^{3}$

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Solution

(i) $(3 x + 4)^{3} - (3 x - 4)^{3}$
$= [(3 x)^{3} + (4)^{3} + 3 (3 x) (4) (3 x + 4)] - [(3 x)^{2} - (4)^{3} - 3 (3 x) (4) (3 x - 4)]$
$= [27 x^{3} + 64 + 36 x (3 x + 4)] - [27 x^{3} - 64 - 36 x (3 x - 4)]$
$= [27 x^{3} + 64 + 108 x^{2} + 144 x] - [27 x^{3} - 64 - 108 x^{2} + 144 x]$
$= 27 x^{3} + 64 + 108 x^{2} + 144 x - 27 x^{3} + 64 + 108 x^{2} - 144 x$
$= 128 + 216^{2}$
(ii) ${(x + \frac{2}{x})}^{3} + {(x - \frac{2}{x})}^{3}$
$= x^{3} + {(\frac{2}{x})}^{3} + 3 (x) (\frac{2}{x}) (x + \frac{2}{x}) + x^{3} - {(\frac{2}{x})}^{3} - 3 (x) (\frac{2}{x}) (x - \frac{2}{x})$
$= x^{3} + \frac{8}{x^{3}} + 6 x + \frac{12}{x} + x^{3} - \frac{8}{x^{3}} - 6 x + \frac{12}{x} = 2 x^{3} + \frac{24}{x}$

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