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Jan 27, 2018

$$49x^8-16y^{14}$$ can be written as a difference of squares.
 $$49x^8-16y^{14}\Rightarrow\left(\textcolor{red}{7x^4}\right)^2-\left(\textcolor{blue}{4y^7}\right)^2$$ This proves that this expression can be written as a difference of squares. Remember that a difference of squares can be factored using the following rule: $$\textcolor{red}{a}^2-\textcolor{blue}{b}^2=(\textcolor{red}{a}+\textcolor{blue}{b})(\textcolor{red}{a}-\textcolor{blue}{b})$$ $$\hspace{5mm}\textcolor{red}{a}^2\hspace{3mm}-\hspace{5mm}\textcolor{blue}{b}^2\hspace{4mm}=(\hspace{2mm}\textcolor{red}{a}\hspace{2mm}+\hspace{3mm}\textcolor{blue}{b})(\hspace{2mm}\textcolor{red}{a}\hspace{4mm}-\hspace{2mm}\textcolor{blue}{b})\\ \left(\textcolor{red}{7x^4}\right)^2-\left(\textcolor{blue}{4y^7}\right)^2=(\textcolor{red}{7x^4}+\textcolor{blue}{4y^7})(\textcolor{red}{7x^4}-\textcolor{blue}{4y^7})$$ Notice how the rule and this particular expression go hand in hand. I introduced colors to ease understanding. At this point, no more can be done.