$\text{a)}$

$2^2\cdot {25}^2={\left(2\cdot 25\right)}^2={50}^2=2500$

${16}^2\cdot 5^2={\left(16\cdot 5\right)}^2={80}^2=6400$

$2^4\cdot {50}^4={\left(2\cdot 50\right)}^4={100}^4=100000000$

$4^3\cdot {25}^3={\left(4\cdot 25\right)}^3={100}^3=1000000$

${35}^2\cdot 2^2={\left(35\cdot 2\right)}^2={70}^2=4900$

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$\text{b)}$

${\left(-\frac{1}{4}\right)}^2\cdot {16}^2={\left(-\frac{1}{{\sout{4}}^1}\cdot {\sout{16}}^4\right)}^2={\left(-4\right)}^2=16$

${\left(-2\frac{1}{2}\right)}^3\cdot {\left(-4\right)}^3={\left(\left(-\frac{5}{2}\right)\cdot \left(-4\right)\right)}^3={\left(\left(-5\right)\cdot \left(-2\right)\right)}^3={10}^3=1000$

${\left(-0,1\right)}^5\cdot {10}^5={\left(-0,1\cdot 10\right)}^5={\left(-1\right)}^5=-1$

${\left(-1,5\right)}^2\cdot {\left(-8\right)}^2={\left(\left(-1,5\right)\cdot \left(-8\right)\right)}^2={12}^2=144$

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$\text{c)}$

${\left(\frac{1}{6}\right)}^5\cdot {12}^5={\left(\frac{1}{6}\cdot 12\right)}^5=2^5=32$

${7,5}^3\cdot {\left(\frac{2}{3}\right)}^3={\left(7\frac{1}{2}\cdot \frac{2}{3}\right)}^3={\left(\frac{15}{2}\cdot \frac{2}{3}\right)}^3=5^3=125$

${\left(\frac{2}{7}\right)}^8\cdot {3,5}^8={\left(\frac{2}{7}\cdot 3,5\right)}^8={\left(\frac{2}{7}\cdot 3\frac{1}{2}\right)}^8={\left(\frac{2}{7}\cdot \frac{7}{2}\right)}^8=1^8=1$

${0,75}^5\cdot {\left(2\frac{2}{3}\right)}^5={\left(0,75\cdot 2\frac{2}{3}\right)}^5={\left(\frac{3}{4}\cdot \frac{8}{3}\right)}^5=2^5=32$

${10}^3\cdot {0,01}^3={\left(10\cdot 0,01\right)}^3={\left(0,1\right)}^3=0,001$