Simplify the Following Products
\[ \left(m + \frac{n}{7}\right)^3 \left(m – \frac{n}{7}\right) \]
Solution:
\[ \left(m + \frac{n}{7}\right)^3 \left(m – \frac{n}{7}\right) \]
\[ = \left(m + \frac{n}{7}\right)^2 \left(m + \frac{n}{7}\right) \left(m – \frac{n}{7}\right) \]
Using identity:
\[ (a+b)(a-b)=a^2-b^2 \]
\[ = \left(m + \frac{n}{7}\right)^2 \left(m^2 – \frac{n^2}{49}\right) \]
Again using identity:
\[ (a+b)^2=a^2+2ab+b^2 \]
\[ = \left(m^2 + \frac{2mn}{7} + \frac{n^2}{49}\right) \left(m^2 – \frac{n^2}{49}\right) \]
\[ = m^4 + \frac{2m^3n}{7} – \frac{2mn^3}{343} – \frac{n^4}{2401} \]