Question

A expressão a^-4 -b^-4/a^-2 - b^-2 equivale a:

Answer 1

$a^{-4}-b^{-4}=(\frac{1}{a})^4-(\frac{1}{b})^4=\frac{1}{a^4}-\frac{1}{b^4}=(\frac{1}{a^2}+\frac{1}{b^2})\cdot(\frac{1}{a^2}-\frac{1}{b^2})$
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$a^{-2}-b^{-2}=(\frac{1}{a})^2-(\frac{1}{b})^2=\frac{1}{a^2}-\frac{1}{b^2}$
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$\\ \ \ \frac{a^{-4}-b^{-4}}{a^{-2}-b^{-2}}\\ \\= \frac{\frac{1}{a^4}- \frac{1}{b^4}}{ \frac{1}{a^2}- \frac{1}{b^2}} \\ \\ =\frac{(\frac{1}{a^2}+\frac{1}{b^2})\cdot(\frac{1}{a^2}-\frac{1}{b^2})}{\frac{1}{a^2}-\frac{1}{b^2}} \\ \\=\frac{(\frac{1}{a^2}+\frac{1}{b^2})}{1} \\ \\=\frac{1}{a^2}+\frac{1}{b^2}$
ou
$\frac{1}{a^2}+\frac{1}{b^2}=\frac{b^2}{a^2b^2}+\frac{a^2}{a^2b^2}=\frac{b^2+a^2}{a^2b^2}=\frac{a^2+b^2}{a^2b^2}$