$B={27}^3{.9}^4.243=3^{22}$

$64:2^3=2^6:2^3=2^3$

$243:3^4=3^5:3^4=3^1$

$625:5^3=5^4:5^3=5^1$

$7^5:343=7^5:7^3=7^2$

$100000:{10}^3={10}^5:{10}^3={10}^2$

${11}^5:121={11}^5:{11}^2={11}^3$

${5.2}^2-18:3$

$=5.4-18:3$$=20-6$$=14$

$17.85+15.17-2^3\mathrm{.3.5}$

$=17.85+15.17-120$

$=17.\left(85+15\right)-120$

$=17.100-120$$=1700-120$$=1580$

$2^3.17-2^3.14$

$=2^3\left(17-14\right)$

$=2^3.3$

$\begin{array}{l}=8.3\\ =24\end{array}$

$20-\left[30-{\left(5-1\right)}^2\right]$

$=20-\left[30-4^2\right]$

$=20-\left(30-16\right)$

$=20-14=6$

$75-\left({3.5}^2-{4.2}^3\right)$

$=75-\left(3.25-4.8\right)$

$=75-\left(75-32\right)$

$\begin{array}{l}=75-75+32\\ =32\end{array}$

$150+50:5-{2.3}^2$ $=150+10-2.9$

$\begin{array}{l}=150+10-18\\ =142\end{array}$

${5.3}^2-32:4^2$$=5.9-32:16$

$\begin{array}{l}=45-2\\ =43\end{array}$

$27.75+25.27-{\mathrm{2.3.5}}^2$

$=27.\left(75+25\right)-150$

$=27.100-150$

$=2700$

$12:\left\{400:\left[500-\left(125+25.7\right)\right]\right\}$

$=12:\left\{400:\left[500-\left(125+175\right)\right]\right\}$

$=12:\left\{400:\left[500-300\right]\right\}$

$\begin{array}{l}=12:\left\{400:200\right\}\\ =12:2=6\end{array}$

$13.17-256:16+14:7-{2021}^0$

$=221-16+2-1$$=206$

${2.3}^2:3+182+3.\left(51:17\right)$

$=6+182+3.3$

$=6+182+9$

$=197$

$15-5^2{.2}^3:\left(100.2\right)$

$=15-25.8:200$

$=15-200:200$

$\begin{array}{l}=15-1\\ =14\end{array}$

$5^2{.2}^3-12.5+170:17-8$

$=1000-60+10-8$

$=942$

$2^3-5^3:5^2+{12.2}^2$

$=8-5+12.4$

$\begin{array}{l}=8-5+48\\ =51\end{array}$

$5.\left[\left(85-35:7\right):8+90\right]-5^2.2$

$=5\left[\left(85-5\right):8+90\right]-50$

$=5\left[80:8+90\right]-50$

$\begin{array}{l}=5.100-50\\ =450\end{array}$

$2.\left[\left(7-3^3:3^2\right):2^2+99\right]-100$

$=2.\left[\left(7-3\right):4+99\right]-100$

$=2.\left(4:4+99\right)-100$

$\begin{array}{l}=2.100-100\\ =100\end{array}$

$2^7:2^2+5^4:5^3{.2}^4-{3.2}^5$

$=2^5+{5.2}^4-{3.2}^5$

$\begin{array}{l}=2^4.\left(2+5-6\right)\\ =2^4\end{array}$

$\left(3^5{.3}^7\right):3^{10}+{5.2}^4-7^3:7$

$=3^{12}:3^{10}+{5.2}^4-7^2$

$=3^2+{5.2}^4-7^2$

$=9+5.16-49$

$\begin{array}{l}=9+80-49\\ =40\end{array}$

$142-\left[50-\left(2^3.10-2^3.5\right)\right]$

$=142-\left[50-2^3.5\right]$

$=142-5.(10-8)$

$\begin{array}{l}=142-10\\ =132\end{array}$

$375:\left\{32-\left[4+\left({5.3}^2-42\right)\right]\right\}-14$

$=375:\left\{32-\left[4+\left(45-42\right)\right]\right\}-14$

$=375:\left\{32-\left(4+3\right)\right\}-14$

$=375:\left\{32-7\right\}-14$

$=375:25-14$$=15-14=1$

$\left\{210:\left[16+3.\left(6+{3.2}^2\right)\right]\right\}-3$

$=\left\{210:\left[16+3.\left(6+12\right)\right]\right\}-3$

$=\left\{210:\left[16+3.18\right]\right\}-3$

$=\left\{210:70\right\}-3$

$=3-3=0$

$500-\left\{5.\left[409-{\left(2^3.3-21\right)}^2\right]-1724\right\}$

$=500-\left\{5\left[409-{\left(8.3-21\right)}^2\right]-1724\right\}$

$=500-\left\{5.\left[409-{\left(24-21\right)}^2\right]-1724\right\}$

$=500-\left\{5.\left[409-9\right]-1724\right\}$

$=500-\left\{5.400-1724\right\}$

$=500-276=224$

$80-\left({4.5}^2-{3.2}^3\right)$

$=80-\left(4.25-3.8\right)$

$=80-\left(100-24\right)$

$=80-76=4$

$5^6:5^4+2^3{.2}^2-1^{2017}$

$=5^2+2^5-1$

$\begin{array}{l}=25+32-1\\ =56\end{array}$

$5^3-2.\left[56-48:\left(15-7\right)\right]$

$=125-2.\left[56-48:8\right]$

$=125-2.\left(56-6\right)$

$\begin{array}{l}=125-2.50\\ =25\end{array}$

$23.75+5^2.10+5^2.13+180$

$=23.75+25.(10+13)+180$

$=23.75+25.23+180$

$=23.100+180$

$\begin{array}{l}=2300+180\\ =2480\end{array}$

$A=2002.20012001-2001.20022002$

$A=2002.\left(20010000+2001\right)-2001.\left(20020000+2002\right)$

$A=2002.\left({2001.10}^4+2001\right)-2001.\left({2002.10}^4+2001\right)$

$A={\mathrm{2002.2001.10}}^4+2002.2001-{\mathrm{2001.2002.10}}^4-2001.2002$$A=0$

$A=2+2^2+2^3+2^4+\mathrm{...}+2^{100}$

$2A=2.2+2^2.2+2^3.2+2^4.2+\mathrm{...}+2^{100}.2$

$2A=2^2+2^3+2^4+2^5+\mathrm{...}+2^{101}$

$2A-A=\left(2^2+2^3+2^4+2^5+\mathrm{...}+2^{101}\right)-\left(2+2^2+2^3+2^4+\mathrm{...}+2^{100}\right)$

$A=2^2+2^3+2^4+2^5+\mathrm{...}+2^{101}-2-2^2-2^3-2^4-\mathrm{...}-2^{100}$

$A=2^{101}-2$

$B=1+5+5^2+5^3+\mathrm{...}+5^{150}$

$5B=1.5+5.5+5^2.5+5^3.5+\mathrm{...}+5^{150}.5$

$5B=5+5^2+5^3+5^4+\mathrm{...}+5^{151}$

$5B-B=\left(5+5^2+5^3+5^4+\mathrm{...}+5^{151}\right)-\left(1+5+5^2+5^3+\mathrm{...}+5^{150}\right)$

$4B=5+5^2+5^3+5^4+\mathrm{...}+5^{151}-1-5-5^2-5^3-\mathrm{...}-5^{150}$

$4B=5^{151}-1$

$B=\frac{5^{151}-1}{4}$

$C=3+3^2+3^3+\mathrm{...}+3^{1000}$

$3C=3.3+3^2.3+3^3.3+\mathrm{...}+3^{1000}.3$

$3C=3^2+3^3+3^4+\mathrm{...}+3^{1001}$

$3C-C=\left(3^2+3^3+3^4+\mathrm{...}+3^{1001}\right)-\left(3+3^2+3^3+\mathrm{...}+3^{1000}\right)$

$2C=3^2+3^3+3^4+\mathrm{...}+3^{1001}-3-3^2-3^3-\mathrm{...}-3^{1000}$

$2C=3^{1001}-3$

$C=\frac{3^{1001}-3}{2}$