These Roots And Logs Could Make A Forest

$\sqrt[3]{125^{\log_{5} 4+\log_{5} 3}}=\color{#D61F06}A$ $\rightarrow\sqrt[3]{125^{\log_{5} 12}}$ $\rightarrow\sqrt[3]{5^{3\log_{5} 12}}$ $\rightarrow\sqrt[3]{5^{\log_{5} 12^3}}$ $\rightarrow\sqrt[3]{12^{3}}$ $\rightarrow{\boxed{12=\color{#D61F06}A}}$

$\displaystyle{=\sqrt[3]{125^{\log_54+\log_53} }}$

$\displaystyle{ =\sqrt[3]{125^{\log_512}}} \quad \quad \left( \textbf{log Property:} \quad \log_ab+\log_ac=\log_abc\right)$

$\displaystyle{=\sqrt[3]{5^{3\log_512}}}$

$\displaystyle{=\sqrt[3]{5^{\log_512^3}}} \quad \quad \quad \left( \textbf{log Property:} \quad p\log_ab=\log_ab^p\right)$

$\displaystyle{=\sqrt[3]{12^3}} \quad \quad \quad \quad \quad \left( \textbf{log Property:}\quad a^{\log_ab}=b\right)$

$\displaystyle{=\boxed{12}}$

See: $\displaystyle{ \sqrt{b^{ \text{LOG}_{b}{x}+ \text{LOG}_{b}{y}}} = \sqrt{ xy} }$