\(已知：ab=1,a+b=3,求解：a^4-b^4的值。\)
\(已知：ab=1,a+b=3,求解：a^4-b^4的值。\)
\[
\begin{align*}
& 解：a^4-b^4 \\
& = (a^2+b^2)(a^2-b^2) \\
& = (a^2+2ab+b^2-2ab)(a+b)(a-b) \\
& = [(a+b)^2-2ab](a+b)\sqrt{(a-b)^2} \\
& = [(a+b)^2-2ab](a+b)\sqrt{a^2-2ab+b^2} \\
& = [(a+b)^2-2ab](a+b)\sqrt{a^2+2ab+b^2-4ab} \\
& = [(a+b)^2-2ab](a+b)\sqrt{(a+b)^2-4ab} \\
& = [(3)^2-2\cdot1](3)\sqrt{(3)^2-4\cdot1} \\
& = [9-2](3)\sqrt{9-4} \\
& = 21\sqrt{5} \\
& 求得：a^4-b^4 = 21\sqrt{5}
\end{align*}
\]