Prime Powers

$\begin{aligned} 2^2 & = 4 \\ 3^3 & = 27 \\ 5^5 & = 3125 \\ 7^7 & = 343\cdot 343\cdot7 = 823,543 \\ \text{Sum } & = 826,699 \end{aligned}$

$\text{ Calculating last five digits of }$ $11^{11}$

$\begin{aligned} 11^{11} & = \dbinom{11}{7}+\dbinom{11}{8}+\dbinom{11}{9}+\dbinom{11}{10}+\dbinom{11}{11} \quad \quad \small\color{#3D99F6}{\text{Adding priority from right}} \\ & = 330|\underbrace{16}_{\text{16 carry over to next column}}5|\underbrace{55}_{\text{5 carry over to next column}}|\underbrace{11}_{\text{1 carry over to next column}}|\underbrace{1}_{\text{last digit}} \\ & = 6|\underbrace{5+5}_{\text{1 carry over to next column}}|5+1|1|1 \\ & = 7|0|6|1|1\end{aligned}$

$\text{Last 5 digits: } 26699+70611 = 97310$

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See Pascal's triangle for better understanding .

$\begin{aligned} 11^{0} & = \text{1st row } \\ 11^{1} & = \text{2nd row } \\ 11^{2} & = \text{3rd row } \\ & \text{...And so on }\end{aligned}$