$\text{I.}{5}^{10}\cdot 5^7=5^{17}=5^{12}\cdot 5^5$

$\text{B, C}$

 

$\text{II.}{\left(1\frac{1}{4}\right)}^2\cdot {\left(1\frac{1}{4}\right)}^8={\left(1\frac{1}{4}\right)}^{2+8}={\left(1\frac{1}{4}\right)}^{10}$

$\text{D}$

 

$\text{III.}{\left(1,4\right)}^9={\left(1,4\right)}^7\cdot {\left(1,4\right)}^2={\left(1,4\right)}^4\cdot {\left(1,4\right)}^2\cdot {\left(1,4\right)}^3={\left(1,4\right)}^1\cdot {\left(1,4\right)}^8={\left(1,4\right)}^9$

$\text{A, B, C}$

 

 

$\text{IV.}{x}^{24}\cdot x^{36}=x^{24+36}=x^{60}$

$\text{B, D}$

 

$\text{V. B}$