c.
\(1+\dfrac{1}{n\left(n+2\right)}=\dfrac{n\left(n+2\right)+1}{n\left(n+2\right)}=\dfrac{n^2+2n+1}{n\left(n+2\right)}=\dfrac{\left(n+1\right)^2}{n\left(n+2\right)}\)
\(\Rightarrow C=\dfrac{2^2}{1.3}\times\dfrac{3^2}{2.4}\times\dfrac{4^2}{3.5}\times...\times\dfrac{100^2}{99.101}\)
\(=\dfrac{2.3.4...100}{1.2.3...99}\times\dfrac{2.3.4...100}{3.4.5...101}=\dfrac{100.2}{101}=\dfrac{200}{101}\)
d.
\(D=\dfrac{\left(2^2\right)^7.2^8.\left(\dfrac{1}{\left(-2\right)^5}\right)}{3.2^{15}.\left(2^4\right)^2-5.2^2.\left(2^{10}\right)^2}=\dfrac{2^{14}.2^8}{-2^5\left(3.2^{15}.2^8-5.2^2.2^{20}\right)}\)
\(=\dfrac{2^{22}}{-2^5.\left(3.2^{23}-5.2^{22}\right)}=\dfrac{2^{22}}{-2^5.2^{22}\left(3.2-5\right)}=\dfrac{2^{22}}{-2^5.2^{22}}=-\dfrac{1}{2^5}\)
\(=-\dfrac{1}{32}\)