If \(3^{-22}+3^{-19}+3^{-17}+3^{-16}=k\cdot 3^{-24}\), then find the value of \(k\).
9000
Factor \(3^{-24}\) from every term on the left side.
\(3^{-22}=3^{-24}\cdot 3^{2},\ 3^{-19}=3^{-24}\cdot 3^{5},\ 3^{-17}=3^{-24}\cdot 3^{7},\ 3^{-16}=3^{-24}\cdot 3^{8}\).
So the sum equals \(3^{-24}(3^{2}+3^{5}+3^{7}+3^{8})=3^{-24}(9+243+2187+6561)\).
\(=3^{-24}\times 9000\), therefore \(k=9000\).
Hence, the required value is 9000.
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