Find the product. $$\left( a ^ { 4 n } + t ^ { 4 n } \right) \left( a ^ { 4 n } - t ^ { 4 n } \right)$$
A) $a ^ { 8 n } - t ^ { 4 n }$
B) $a ^ { 8 n } + t ^ { 8 n }$
C) $a ^ { 4 n } - t ^ { 8 n }$
D) $t ^ { 8 n } - a ^ { 8 n }$
E) $a ^ { 8 n } - t ^ { 8 n }$

Question

Quizplus · Accepted Answer

This is a difference of squares, so we can use the formula $(a^2 - b^2) = (a+b)(a-b)$. Letting $a = a^{2n}$ and $b = t^{2n}$, we get:
\begin{align*}
(a^{4n} + t^{4n})(a^{4n} - t^{4n}) &= (a^{4n})^2 - (t^{4n})^2 \
&= (a^{2n})^4 - (t^{2n})^4 \
&= (a^{2n} + t^{2n})(a^{2n} - t^{2n})(a^{2n} + t^{2n})(a^{2n} - t^{2n}) \
&= (a^{2n} + t^{2n})^2 (a^{2n} - t^{2n})^2 \
&= (a^{4n} + 2a^{2n}t^{2n} + t^{4n})(a^{4n} - 2a^{2n}t^{2n} + t^{4n}) \
&= \boxed{a^{8n} - t^{8n}}.
\end{align*}

Find the Product (a4n+t4n)(a4n−t4n)\left( a ^ { 4 n } + t ^ { 4 n } \right) \left( a ^ { 4 n } - t ^ { 4 n } \right)(a4n+t4n)(a4n−t4n)

Find the Product $\left( a ^ { 4 n } + t ^ { 4 n } \right) \left( a ^ { 4 n } - t ^ { 4 n } \right)$