解決{r}{(1/3)^{8}+8*(2/3)(1/3)^{7}+8!/6!2!(frac{2}{3})^{2}(frac{1}{3})^{6}+frac{8!}{5!3!}(frac{2}{3})^{3}(frac{1}{3})^{5}+frac{8!}{4!4!}(frac{2}{3})^4(frac{1}{3})^4}{quad+frac{8!}{3!5!}(frac{2}{3})^5(frac{1}{3})^3} |Microsoft數學求解器

$$\left. \begin{array} { r } { ( \frac { 1 } { 3 } ) ^ { 8 } + 8 \cdot ( \frac { 2 } { 3 } ) ( \frac { 1 } { 3 } ) ^ { 7 } + \frac { 8 ! } { 6 ! 2 ! } ( \frac { 2 } { 3 } ) ^ { 2 } ( \frac { 1 } { 3 } ) ^ { 6 } + \frac { 8 ! } { 5 ! 3 ! } ( \frac { 2 } { 3 } ) ^ { 3 } ( \frac { 1 } { 3 } ) ^ { 5 } + \frac { 8 ! } { 4 ! 4 ! } ( \frac { 2 } { 3 } ) ^ { 4 } ( \frac { 1 } { 3 } ) ^ { 4 } } \\ { \quad + \frac { 8 ! } { 3 ! 5 ! } ( \frac { 2 } { 3 } ) ^ { 5 } ( \frac { 1 } { 3 } ) ^ { 3 } } \end{array} \right.$$

評估

$$\frac{1163}{2187}\approx 0.531778692$$

解題步驟

將 4! 乘上 4! 得到 \left(4!\right)^{2}。

$$\left(\frac{1}{3}\right)^{8}+8\times \frac{2}{3}\times \left(\frac{1}{3}\right)^{7}+\frac{8!}{6!\times 2!}\times \left(\frac{2}{3}\right)^{2}\times \left(\frac{1}{3}\right)^{6}+\frac{8!}{5!\times 3!}\times \left(\frac{2}{3}\right)^{3}\times \left(\frac{1}{3}\right)^{5}+\frac{8!}{\left(4!\right)^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

計算 \frac{1}{3} 的 8 乘冪，然後得到 \frac{1}{6561}。

$$\frac{1}{6561}+8\times \frac{2}{3}\times \left(\frac{1}{3}\right)^{7}+\frac{8!}{6!\times 2!}\times \left(\frac{2}{3}\right)^{2}\times \left(\frac{1}{3}\right)^{6}+\frac{8!}{5!\times 3!}\times \left(\frac{2}{3}\right)^{3}\times \left(\frac{1}{3}\right)^{5}+\frac{8!}{\left(4!\right)^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

運算式 8\times \frac{2}{3} 為最簡分數。

$$\frac{1}{6561}+\frac{8\times 2}{3}\times \left(\frac{1}{3}\right)^{7}+\frac{8!}{6!\times 2!}\times \left(\frac{2}{3}\right)^{2}\times \left(\frac{1}{3}\right)^{6}+\frac{8!}{5!\times 3!}\times \left(\frac{2}{3}\right)^{3}\times \left(\frac{1}{3}\right)^{5}+\frac{8!}{\left(4!\right)^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

將 8 乘上 2 得到 16。

$$\frac{1}{6561}+\frac{16}{3}\times \left(\frac{1}{3}\right)^{7}+\frac{8!}{6!\times 2!}\times \left(\frac{2}{3}\right)^{2}\times \left(\frac{1}{3}\right)^{6}+\frac{8!}{5!\times 3!}\times \left(\frac{2}{3}\right)^{3}\times \left(\frac{1}{3}\right)^{5}+\frac{8!}{\left(4!\right)^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

計算 \frac{1}{3} 的 7 乘冪，然後得到 \frac{1}{2187}。

$$\frac{1}{6561}+\frac{16}{3}\times \frac{1}{2187}+\frac{8!}{6!\times 2!}\times \left(\frac{2}{3}\right)^{2}\times \left(\frac{1}{3}\right)^{6}+\frac{8!}{5!\times 3!}\times \left(\frac{2}{3}\right)^{3}\times \left(\frac{1}{3}\right)^{5}+\frac{8!}{\left(4!\right)^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

\frac{16}{3} 乘上 \frac{1}{2187} 的算法: 將分子和分子相乘以及將分母和分母相乘。

$$\frac{1}{6561}+\frac{16\times 1}{3\times 2187}+\frac{8!}{6!\times 2!}\times \left(\frac{2}{3}\right)^{2}\times \left(\frac{1}{3}\right)^{6}+\frac{8!}{5!\times 3!}\times \left(\frac{2}{3}\right)^{3}\times \left(\frac{1}{3}\right)^{5}+\frac{8!}{\left(4!\right)^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

在分數 \frac{16\times 1}{3\times 2187} 上完成乘法。

$$\frac{1}{6561}+\frac{16}{6561}+\frac{8!}{6!\times 2!}\times \left(\frac{2}{3}\right)^{2}\times \left(\frac{1}{3}\right)^{6}+\frac{8!}{5!\times 3!}\times \left(\frac{2}{3}\right)^{3}\times \left(\frac{1}{3}\right)^{5}+\frac{8!}{\left(4!\right)^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

因為 \frac{1}{6561} 和 \frac{16}{6561} 的分母相同，所以將分子相加即可相加這兩個值。

$$\frac{1+16}{6561}+\frac{8!}{6!\times 2!}\times \left(\frac{2}{3}\right)^{2}\times \left(\frac{1}{3}\right)^{6}+\frac{8!}{5!\times 3!}\times \left(\frac{2}{3}\right)^{3}\times \left(\frac{1}{3}\right)^{5}+\frac{8!}{\left(4!\right)^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

將 1 與 16 相加可以得到 17。

$$\frac{17}{6561}+\frac{8!}{6!\times 2!}\times \left(\frac{2}{3}\right)^{2}\times \left(\frac{1}{3}\right)^{6}+\frac{8!}{5!\times 3!}\times \left(\frac{2}{3}\right)^{3}\times \left(\frac{1}{3}\right)^{5}+\frac{8!}{\left(4!\right)^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

8 的階乘是 40320。

$$\frac{17}{6561}+\frac{40320}{6!\times 2!}\times \left(\frac{2}{3}\right)^{2}\times \left(\frac{1}{3}\right)^{6}+\frac{8!}{5!\times 3!}\times \left(\frac{2}{3}\right)^{3}\times \left(\frac{1}{3}\right)^{5}+\frac{8!}{\left(4!\right)^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

6 的階乘是 720。

$$\frac{17}{6561}+\frac{40320}{720\times 2!}\times \left(\frac{2}{3}\right)^{2}\times \left(\frac{1}{3}\right)^{6}+\frac{8!}{5!\times 3!}\times \left(\frac{2}{3}\right)^{3}\times \left(\frac{1}{3}\right)^{5}+\frac{8!}{\left(4!\right)^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

2 的階乘是 2。

$$\frac{17}{6561}+\frac{40320}{720\times 2}\times \left(\frac{2}{3}\right)^{2}\times \left(\frac{1}{3}\right)^{6}+\frac{8!}{5!\times 3!}\times \left(\frac{2}{3}\right)^{3}\times \left(\frac{1}{3}\right)^{5}+\frac{8!}{\left(4!\right)^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

將 720 乘上 2 得到 1440。

$$\frac{17}{6561}+\frac{40320}{1440}\times \left(\frac{2}{3}\right)^{2}\times \left(\frac{1}{3}\right)^{6}+\frac{8!}{5!\times 3!}\times \left(\frac{2}{3}\right)^{3}\times \left(\frac{1}{3}\right)^{5}+\frac{8!}{\left(4!\right)^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

將 40320 除以 1440 以得到 28。

$$\frac{17}{6561}+28\times \left(\frac{2}{3}\right)^{2}\times \left(\frac{1}{3}\right)^{6}+\frac{8!}{5!\times 3!}\times \left(\frac{2}{3}\right)^{3}\times \left(\frac{1}{3}\right)^{5}+\frac{8!}{\left(4!\right)^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

計算 \frac{2}{3} 的 2 乘冪，然後得到 \frac{4}{9}。

$$\frac{17}{6561}+28\times \frac{4}{9}\times \left(\frac{1}{3}\right)^{6}+\frac{8!}{5!\times 3!}\times \left(\frac{2}{3}\right)^{3}\times \left(\frac{1}{3}\right)^{5}+\frac{8!}{\left(4!\right)^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

運算式 28\times \frac{4}{9} 為最簡分數。

$$\frac{17}{6561}+\frac{28\times 4}{9}\times \left(\frac{1}{3}\right)^{6}+\frac{8!}{5!\times 3!}\times \left(\frac{2}{3}\right)^{3}\times \left(\frac{1}{3}\right)^{5}+\frac{8!}{\left(4!\right)^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

將 28 乘上 4 得到 112。

$$\frac{17}{6561}+\frac{112}{9}\times \left(\frac{1}{3}\right)^{6}+\frac{8!}{5!\times 3!}\times \left(\frac{2}{3}\right)^{3}\times \left(\frac{1}{3}\right)^{5}+\frac{8!}{\left(4!\right)^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

計算 \frac{1}{3} 的 6 乘冪，然後得到 \frac{1}{729}。

$$\frac{17}{6561}+\frac{112}{9}\times \frac{1}{729}+\frac{8!}{5!\times 3!}\times \left(\frac{2}{3}\right)^{3}\times \left(\frac{1}{3}\right)^{5}+\frac{8!}{\left(4!\right)^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

\frac{112}{9} 乘上 \frac{1}{729} 的算法: 將分子和分子相乘以及將分母和分母相乘。

$$\frac{17}{6561}+\frac{112\times 1}{9\times 729}+\frac{8!}{5!\times 3!}\times \left(\frac{2}{3}\right)^{3}\times \left(\frac{1}{3}\right)^{5}+\frac{8!}{\left(4!\right)^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

在分數 \frac{112\times 1}{9\times 729} 上完成乘法。

$$\frac{17}{6561}+\frac{112}{6561}+\frac{8!}{5!\times 3!}\times \left(\frac{2}{3}\right)^{3}\times \left(\frac{1}{3}\right)^{5}+\frac{8!}{\left(4!\right)^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

因為 \frac{17}{6561} 和 \frac{112}{6561} 的分母相同，所以將分子相加即可相加這兩個值。

$$\frac{17+112}{6561}+\frac{8!}{5!\times 3!}\times \left(\frac{2}{3}\right)^{3}\times \left(\frac{1}{3}\right)^{5}+\frac{8!}{\left(4!\right)^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

將 17 與 112 相加可以得到 129。

$$\frac{129}{6561}+\frac{8!}{5!\times 3!}\times \left(\frac{2}{3}\right)^{3}\times \left(\frac{1}{3}\right)^{5}+\frac{8!}{\left(4!\right)^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

透過找出與消去 3，對分式 \frac{129}{6561} 約分至最低項。

$$\frac{43}{2187}+\frac{8!}{5!\times 3!}\times \left(\frac{2}{3}\right)^{3}\times \left(\frac{1}{3}\right)^{5}+\frac{8!}{\left(4!\right)^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

8 的階乘是 40320。

$$\frac{43}{2187}+\frac{40320}{5!\times 3!}\times \left(\frac{2}{3}\right)^{3}\times \left(\frac{1}{3}\right)^{5}+\frac{8!}{\left(4!\right)^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

5 的階乘是 120。

$$\frac{43}{2187}+\frac{40320}{120\times 3!}\times \left(\frac{2}{3}\right)^{3}\times \left(\frac{1}{3}\right)^{5}+\frac{8!}{\left(4!\right)^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

3 的階乘是 6。

$$\frac{43}{2187}+\frac{40320}{120\times 6}\times \left(\frac{2}{3}\right)^{3}\times \left(\frac{1}{3}\right)^{5}+\frac{8!}{\left(4!\right)^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

將 120 乘上 6 得到 720。

$$\frac{43}{2187}+\frac{40320}{720}\times \left(\frac{2}{3}\right)^{3}\times \left(\frac{1}{3}\right)^{5}+\frac{8!}{\left(4!\right)^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

將 40320 除以 720 以得到 56。

$$\frac{43}{2187}+56\times \left(\frac{2}{3}\right)^{3}\times \left(\frac{1}{3}\right)^{5}+\frac{8!}{\left(4!\right)^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

計算 \frac{2}{3} 的 3 乘冪，然後得到 \frac{8}{27}。

$$\frac{43}{2187}+56\times \frac{8}{27}\times \left(\frac{1}{3}\right)^{5}+\frac{8!}{\left(4!\right)^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

運算式 56\times \frac{8}{27} 為最簡分數。

$$\frac{43}{2187}+\frac{56\times 8}{27}\times \left(\frac{1}{3}\right)^{5}+\frac{8!}{\left(4!\right)^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

將 56 乘上 8 得到 448。

$$\frac{43}{2187}+\frac{448}{27}\times \left(\frac{1}{3}\right)^{5}+\frac{8!}{\left(4!\right)^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

計算 \frac{1}{3} 的 5 乘冪，然後得到 \frac{1}{243}。

$$\frac{43}{2187}+\frac{448}{27}\times \frac{1}{243}+\frac{8!}{\left(4!\right)^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

\frac{448}{27} 乘上 \frac{1}{243} 的算法: 將分子和分子相乘以及將分母和分母相乘。

$$\frac{43}{2187}+\frac{448\times 1}{27\times 243}+\frac{8!}{\left(4!\right)^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

在分數 \frac{448\times 1}{27\times 243} 上完成乘法。

$$\frac{43}{2187}+\frac{448}{6561}+\frac{8!}{\left(4!\right)^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

2187 和 6561 的最小公倍數為 6561。將 \frac{43}{2187} 和 \frac{448}{6561} 轉換為分母是 6561 的分數。

$$\frac{129}{6561}+\frac{448}{6561}+\frac{8!}{\left(4!\right)^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

因為 \frac{129}{6561} 和 \frac{448}{6561} 的分母相同，所以將分子相加即可相加這兩個值。

$$\frac{129+448}{6561}+\frac{8!}{\left(4!\right)^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

將 129 與 448 相加可以得到 577。

$$\frac{577}{6561}+\frac{8!}{\left(4!\right)^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

8 的階乘是 40320。

$$\frac{577}{6561}+\frac{40320}{\left(4!\right)^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

4 的階乘是 24。

$$\frac{577}{6561}+\frac{40320}{24^{2}}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

計算 24 的 2 乘冪，然後得到 576。

$$\frac{577}{6561}+\frac{40320}{576}\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

將 40320 除以 576 以得到 70。

$$\frac{577}{6561}+70\times \left(\frac{2}{3}\right)^{4}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

計算 \frac{2}{3} 的 4 乘冪，然後得到 \frac{16}{81}。

$$\frac{577}{6561}+70\times \frac{16}{81}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

運算式 70\times \frac{16}{81} 為最簡分數。

$$\frac{577}{6561}+\frac{70\times 16}{81}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

將 70 乘上 16 得到 1120。

$$\frac{577}{6561}+\frac{1120}{81}\times \left(\frac{1}{3}\right)^{4}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

計算 \frac{1}{3} 的 4 乘冪，然後得到 \frac{1}{81}。

$$\frac{577}{6561}+\frac{1120}{81}\times \frac{1}{81}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

\frac{1120}{81} 乘上 \frac{1}{81} 的算法: 將分子和分子相乘以及將分母和分母相乘。

$$\frac{577}{6561}+\frac{1120\times 1}{81\times 81}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

在分數 \frac{1120\times 1}{81\times 81} 上完成乘法。

$$\frac{577}{6561}+\frac{1120}{6561}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

因為 \frac{577}{6561} 和 \frac{1120}{6561} 的分母相同，所以將分子相加即可相加這兩個值。

$$\frac{577+1120}{6561}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

將 577 與 1120 相加可以得到 1697。

$$\frac{1697}{6561}+\frac{8!}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

8 的階乘是 40320。

$$\frac{1697}{6561}+\frac{40320}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

3 的階乘是 6。

$$\frac{1697}{6561}+\frac{40320}{6\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

5 的階乘是 120。

$$\frac{1697}{6561}+\frac{40320}{6\times 120}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

將 6 乘上 120 得到 720。

$$\frac{1697}{6561}+\frac{40320}{720}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

將 40320 除以 720 以得到 56。

$$\frac{1697}{6561}+56\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}$$

計算 \frac{2}{3} 的 5 乘冪，然後得到 \frac{32}{243}。

$$\frac{1697}{6561}+56\times \frac{32}{243}\times \left(\frac{1}{3}\right)^{3}$$

運算式 56\times \frac{32}{243} 為最簡分數。

$$\frac{1697}{6561}+\frac{56\times 32}{243}\times \left(\frac{1}{3}\right)^{3}$$

將 56 乘上 32 得到 1792。

$$\frac{1697}{6561}+\frac{1792}{243}\times \left(\frac{1}{3}\right)^{3}$$

計算 \frac{1}{3} 的 3 乘冪，然後得到 \frac{1}{27}。

$$\frac{1697}{6561}+\frac{1792}{243}\times \frac{1}{27}$$

\frac{1792}{243} 乘上 \frac{1}{27} 的算法: 將分子和分子相乘以及將分母和分母相乘。

$$\frac{1697}{6561}+\frac{1792\times 1}{243\times 27}$$

在分數 \frac{1792\times 1}{243\times 27} 上完成乘法。

$$\frac{1697}{6561}+\frac{1792}{6561}$$

因為 \frac{1697}{6561} 和 \frac{1792}{6561} 的分母相同，所以將分子相加即可相加這兩個值。

$$\frac{1697+1792}{6561}$$

將 1697 與 1792 相加可以得到 3489。

$$\frac{3489}{6561}$$

透過找出與消去 3，對分式 \frac{3489}{6561} 約分至最低項。

$$\frac{1163}{2187}$$

因式分解

$$\frac{1163}{3 ^ {7}} = 0.5317786922725194$$