Đáp án:
$\begin{array}{l}
5)a)\left( {{2^{17}} + {{17}^2}} \right).\left( {{9^{15}} - {3^{15}}} \right).\left( {{2^4} - {4^2}} \right)\\
= \left( {{2^{17}} + {{17}^2}} \right).\left( {{9^{15}} - {3^{15}}} \right).\left( {{2^4} - {2^{2.2}}} \right)\\
= \left( {{2^{17}} + {{17}^2}} \right).\left( {{9^{15}} - {3^{15}}} \right).\left( {{2^4} - {2^4}} \right)\\
= \left( {{2^{17}} + {{17}^2}} \right).\left( {{9^{15}} - {3^{15}}} \right).0\\
= 0\\
b)\left( {{7^{1997}} - {7^{1995}}} \right):\left( {{7^{1994}}.7} \right)\\
= \left( {{7^2} - 1} \right){.7^{1995}}:{7^{1995}}\\
= {7^2} - 1\\
= 48\\
c)\left( {{1^2} + {2^3} + {3^4} + {4^5}} \right).\left( {{1^3} + {2^3} + {3^3} + {4^3}} \right).\left( {{3^8} - {{81}^2}} \right)\\
= \left( {{1^2} + {2^3} + {3^4} + {4^5}} \right).\left( {{1^3} + {2^3} + {3^3} + {4^3}} \right).\left( {{3^8} - {3^{4.2}}} \right)\\
= \left( {{1^2} + {2^3} + {3^4} + {4^5}} \right).\left( {{1^3} + {2^3} + {3^3} + {4^3}} \right).\left( {{3^8} - {3^8}} \right)\\
= \left( {{1^2} + {2^3} + {3^4} + {4^5}} \right).\left( {{1^3} + {2^3} + {3^3} + {4^3}} \right).0\\
= 0\\
d)\left( {{2^8} + {8^3}} \right):\left( {{2^5}{{.2}^3}} \right)\\
= \left( {{2^8} + {2^9}} \right):{2^8}\\
= \left( {1 + 2} \right){.2^8}:{2^8}\\
= 3\\
6)a)A = {2^0} + {2^1} + ... + {2^{2020}}\\
\Rightarrow 2A = {2^1} + {2^2} + ... + {2^{2021}}\\
\Rightarrow 2A - A = A = {2^{2021}} - {2^0}\\
\Rightarrow A = {2^{2021}} - 1\\
b)B = 1 + 3 + {3^2} + ... + {3^{100}}\\
\Rightarrow 3B = 3 + {3^2} + {3^3} + ... + {3^{101}}\\
\Rightarrow 3B - B = 2B = {3^{101}} - 1\\
\Rightarrow B = \dfrac{{{3^{101}} - 1}}{2}
\end{array}$
