How can you simplify $\dfrac{125{{a}^{15}}{{b}^{7}}}{-8{{a}^{3}}{{b}^{4}}}$?

Hint: Now consider the given expression $\dfrac{125{{a}^{15}}{{b}^{7}}}{-8{{a}^{3}}{{b}^{4}}}$ . Now to simplify the given expression we will first use the property of indices which states $\dfrac{{{a}^{m}}}{{{b}^{m}}}={{a}^{m-n}}$ . Using this we will remove the indices from the denominator and hence write the whole expression.

Complete step by step solution:
Now the given expression is a fraction with variables a and b.
Now first let us understand the concept of indices.
An index is nothing but a number or a variable raised to another number.
Now let us consider ${{a}^{2}}$ for example.
Here we read this as a raised to 2. Now in this expression 2 is called the power of a.
Now what does this mean? Power represents the number of time its base is multiplied to itself.
Hence if 2 is the power of a in ${{a}^{2}}$ then we have ${{a}^{2}}=a\times a$ .
Now that we have understood indices let us understand some basic properties of indices.
Now if we have ${{a}^{m}}\times {{a}^{n}}$ then we can simplify and write this as ${{a}^{m+n}}$ . Hence we have ${{a}^{m}}{{a}^{n}}={{a}^{m+n}}$ .
Now similarly let us say that the two terms are in division and we have $\dfrac{{{a}^{m}}}{{{a}^{n}}}$ then to simplify we have $\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}$ .
Now consider the given expression $\dfrac{125{{a}^{15}}{{b}^{7}}}{-8{{a}^{3}}{{b}^{4}}}$
Now using the division rule which states $\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}$ we get,
$\begin{align}
  & \Rightarrow \dfrac{125{{a}^{15-3}}{{b}^{7-4}}}{-8} \\
 & \Rightarrow \dfrac{125{{a}^{12}}{{b}^{3}}}{-8} \\
\end{align}$
 Hence the given expression can be written as $\dfrac{125{{a}^{12}}{{b}^{3}}}{-8}$ .

Note: Now note that we can only simplify the terms in multiplication and division for which the base of the index is same. Hence we cannot simplify ${{a}^{2}}{{b}^{3}}$. Also note that the power of a number can be positive number negative number, fraction or 0. If the power is 0 then the value of the number is 0 no matter what the base is.