The subtriplicate ratio of $(x^{4} - y^{4})^{3} : (x^{2} + y^{2})^{6}$ is ____

(x^{2} - y^{2}) : (x^{2} + y^{2})

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(x^{2} - y^{2}) : (x^{2} + y^{2})^{2}

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(x - y) : (x + y)

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(x^{2} + y^{2}) : x y

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Solution

The correct option is A $(x^{2} - y^{2}) : (x^{2} + y^{2})$
The subtriplicate ratio of $a : b$ is $\sqrt[3]{a} : \sqrt[3]{b}$
$∴$ The subtriplicate ratio of $(x^{4} - y^{4})^{3} : (x^{2} + y^{2})^{6}$ is $\sqrt[3]{(x^{4} - y^{4})^{3}} : \sqrt[3]{(x^{2} + y^{2})^{6}} = (x^{4} - y^{4}) : (x^{2} + y^{2})^{2}$
$= \frac{x^{4} - y^{4}}{(x^{2} + y^{2})^{2}} = \frac{(x^{2} - y^{2}) (x^{2} + y^{2})}{(x^{2} + y^{2})^{2}}$
$= \frac{x^{2} - y^{2}}{x^{2} + y^{2}} = x^{2} - y^{2} : x^{2} + y^{2}$

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