$\text{I.}$

$5^{24}={\left(5^4\right)}^6={\left(5^8\right)}^3={\left(5^{24}\right)}^1$

 

$\text{II.}$

${\left(-8\right)}^{32}={\left({\left(-8\right)}^4\right)}^8={\left({\left(-8\right)}^{16}\right)}^2={\left({\left({\left(-8\right)}^1\right)}^2\right)}^{16}$

 

$\text{III.}$

${\text{a}}^{100}={\left({\text{a}}^{10}\right)}^{10}={\left({\text{a}}^2\right)}^{50}={\left({\text{a}}^{50}\right)}^2$

 

$\text{IV.}$

${\text{b}}^{120}={\left({\text{b}}^2\right)}^{60}={\left({\text{b}}^3\right)}^{40}={\left({\text{b}}^5\right)}^6\cdot {\left({\text{b}}^3\right)}^{30}$