Exponentiation problems: Very Difficult Problems with Solutions

Solution:
[tex]5^{2n+3}=5^3\times (5^2)^n=125\times 25^n[/tex]
[tex]5\times 25^{n+1}=5\times 25\times 25^n=125\times 25^n[/tex]
[tex]3^{n+3}\times 2^n=27\times 3^n\times 2^n=27\times 6^n[/tex]
[tex]B=125\times 25^n+27\times 6^n-125\times 25^n-27\times 6^n=0[/tex]

Solution:
[tex]5^{n+1}+3\times 5^n=5^n(5+3)=8\times 5^n[/tex]
[tex](5^2)^{n+1}-3^2\times (5^2)^n=25^n(25-9)=16\times 25^n=16\times 5^{2n}[/tex]
[tex]A=\frac{8^4\times 5^{4n}}{16^2\times 5^{4n}}=\frac{2^{12}}{2^8}=2^4=16[/tex]

Solution:
Write each [tex]6[/tex] as [tex]5+1[/tex]:
[tex]\frac{6}{5^k}=\frac{5+1}{5^k}=\frac{1}{5^{k-1}}+\frac{1}{5^k}[/tex]
[tex]\left(\frac{1}{5}+\frac{1}{5^2}\right)-\left(\frac{1}{5^2}+\frac{1}{5^3}\right)+\left(\frac{1}{5^3}+\frac{1}{5^4}\right)-\ldots+\left(\frac{1}{5^{99}}+\frac{1}{5^{100}}\right)-\frac{1}{5^{100}}[/tex]
All terms except the first cancel in pairs:
[tex]=\frac{1}{5}[/tex]

Solution:
[tex]A=2^{2013}(2^2-2-1)=2^{2013}[/tex]
Note that [tex]2013=3\times 671[/tex] and [tex]1342=2\times 671[/tex]:
[tex]A=(2^3)^{671}=8^{671}[/tex], [tex]B=(3^2)^{671}=9^{671}[/tex]
[tex]8^{671}\lt 9^{671} \Rightarrow A\lt B[/tex]