Exponents and Radicals: Very Difficult Problems with Solutions

Solution:
[tex]2^{100}=1024^{10}=10^{30}\times 1. 024^{10}[/tex]
[tex]1\lt 1. 024^{10}\lt 1. 25^{10}=\dfrac{5^{10}}{4^{10}}=\dfrac{9765625}{1048576}\lt 10[/tex]
[tex]10^{30}\lt 2^{100}\lt 10^{31}[/tex]
Answer: [tex]31[/tex]

Solution:
[tex]\left(3+2\sqrt{2}\right)\left(3-2\sqrt{2}\right)=9-8=1 \Rightarrow \dfrac{1}{3-2\sqrt{2}}=3+2\sqrt{2}, 6-4\sqrt{2}=2\left(3-2\sqrt{2}\right)[/tex]
[tex]2\left(3+2\sqrt{2}\right)^{2007}\times \dfrac{2^{2007}\left(3-2\sqrt{2}\right)^{2007}}{2^{2008}}=\dfrac{2^{2008}}{2^{2008}}=1[/tex]

Solution:
[tex]\left(\dfrac{a^3bd^2}{c}\right)^3\times \left(\dfrac{a^4b}{c^2d^2}\right)^{-2}=\dfrac{a^9b^3d^6}{c^3}\times \dfrac{c^4d^4}{a^8b^2}=abcd^{10}[/tex]
[tex]abcd^{10}=\dfrac{384^3}{3^2}=\dfrac{\left(3\times 2^7\right)^3}{9}=3\times 2^{21}=6291456[/tex]

Solution:
[tex]u=a^{-\frac{1}{3}}, v=b^{-\frac{1}{3}}, a^{-1}=u^3, b^{-1}=v^3[/tex]
[tex]\dfrac{uv}{u^3-v^3}-\dfrac{1}{u-v}=\dfrac{uv-\left(u^2+uv+v^2\right)}{u^3-v^3}=-\dfrac{u^2+v^2}{(u-v)\left(u^2+uv+v^2\right)}[/tex]
[tex]-\dfrac{u^2+v^2}{(u-v)\left(u^2+uv+v^2\right)}\times \dfrac{u^2+uv+v^2}{u^2+v^2}=-\dfrac{1}{u-v}=\dfrac{1}{b^{-\frac{1}{3}}-a^{-\frac{1}{3}}}=\dfrac{\sqrt[3]{ab}}{\sqrt[3]{a}-\sqrt[3]{b}}[/tex]

Solution:
[tex]5^{56}=25^{28}\lt 31^{28}[/tex]
[tex]31^{28}\lt 32^{28}=2^{140}=16^{35}\lt 17^{35}[/tex]
[tex]17^{35}\lt 20^{35}=2^{35}\times 10^{35}, 2^{35}\lt 2^{40}=1024^4\lt 10^{16} \Rightarrow 17^{35}\lt 10^{51}[/tex]
Answer: [tex]5^{56}\lt 31^{28}\lt 17^{35}\lt 10^{51}[/tex]