a/
\(\left(-\frac{1}{16}\right)^{1000}=\left(-\frac{1}{2^4}\right)^{1000}=\left(-\frac{1}{2}\right)^{4000}.\)
Do \(\left(\frac{1}{2}\right)^{4000}>\left(\frac{1}{2}\right)^{5000}\Rightarrow\left(-\frac{1}{2}\right)^{4000}< \left(-\frac{1}{2}\right)^{5000}\Rightarrow\left(-\frac{1}{16}\right)^{1000}< \left(-\frac{1}{2}\right)^{5000}\)
b/
\(3^{400}=\left(3^4\right)^{100}=81^{100}\)
\(4^{300}=\left(4^3\right)^{100}=64^{100}\)
\(\Rightarrow81^{100}>64^{100}\Rightarrow3^{400}>4^{300}\)