a) \( \dfrac{10^{2+ \sqrt{7}}}{2^{2 + \sqrt{7}}. 5^{1+\sqrt{7}}}\)
\( = \dfrac{ (2.5)^{2+ \sqrt{7}}}{2^{2 + \sqrt{7}}. 5^{1+\sqrt{7}}}\)
\(= \dfrac{2^{2+ \sqrt{7}}. 5^{2+ \sqrt{7}}}{2^{2 + \sqrt{7}}. 5^{1+\sqrt{7}}}\)
\( = \dfrac{5^{2+ \sqrt{7}}}{5^{1+\sqrt{7}}} \)
\( = 5^{(2+ \sqrt{7}) - ( 1+ \sqrt{7})} \)
\( = 5^1 =5 \).
b) \( ( 4^{2\sqrt{3}} - 4^{\sqrt{3} - 1}). 2^{-2\sqrt{3}}\)
\( = 4^{2\sqrt{3}}.2^{-2\sqrt{3}} - 4^{\sqrt{3} - 1}.2^{-2\sqrt{3}}\)
\(= \Big(2^{2}\Big)^{2\sqrt{3}}.2^{-2\sqrt{3}} - \Big(2^{2}\Big)^{\sqrt{3} - 1}. 2^{-2\sqrt{3}}\)
\(= 2^{4\sqrt{3}}.2^{-2\sqrt{3}} - 2^{2\sqrt{3} - 2}. 2^{-2\sqrt{3}}\)
\(= 2^{2\sqrt{3}} - 2^ {-2}\)
\(= 2^{2\sqrt{3}} - \dfrac{1}{4}\)
