### Theory:

Basic properties of Exponents.
$\begin{array}{l}{a}^{n}\cdot {a}^{m}={a}^{n+m};\\ \\ {a}^{n}:{a}^{m}={a}^{n-m},\phantom{\rule{0.147em}{0ex}}n>m,\phantom{\rule{0.147em}{0ex}}a\ne 0;\\ \\ {\left({a}^{n}\right)}^{m}={a}^{n\cdot m}\end{array}$
Where n and $$m$$ are integers.
In practice, several properties are often applied simultaneously.
Example:
Calculate$\frac{{\left({5}^{4}\cdot 5\right)}^{3}}{{5}^{7}\cdot {5}^{6}}$

Perform actions in the numerator:

1.${5}^{4}\cdot 5={5}^{4}\cdot {5}^{1}={5}^{4+1}={5}^{5}$ — applied property ${a}^{n}\cdot {a}^{m}={a}^{n+m}$.

2. ${\left({5}^{5}\right)}^{3}={5}^{5\cdot 3}={5}^{15}$ — applied property ${\left({a}^{n}\right)}^{m}={a}^{n\cdot m}$.

Perform the multiplication in the denominator:

3. ${5}^{7}\cdot {5}^{6}={5}^{7+6}={5}^{13}$ — applied property ${a}^{n}\cdot {a}^{m}={a}^{n+m}$.

Replace the fraction line with division:

4.${5}^{15}:{5}^{13}={5}^{15-13}={5}^{2}$ — applied property ${a}^{n}:{a}^{m}={a}^{n-m}$.

5.  ${5}^{2}=25$.

Answer: $$25$$.