How do you simplify the algebraic expression $\left( {{p}^{5}}{{r}^{2}} \right){{\left( -7{{p}^{3}}{{r}^{4}} \right)}^{2}}\left( 6p{{r}^{3}} \right)$?

Hint: Now to simplify the given expression we will first solve the index by using the property ${{\left( ab \right)}^{m}}=\left( {{a}^{m}}\times {{b}^{m}} \right)$ . Now we will use ${{\left( {{a}^{n}} \right)}^{m}}={{a}^{m\times n}}$ to further simplify the expression. Next we will separate the variables p and r. Now using the law of indices which states ${{a}^{m}}{{a}^{n}}={{a}^{m+n}}$ we will rewrite the expression. Hence the given expression is simplified

Complete step by step solution:
Now consider the given expression $\left( {{p}^{5}}{{r}^{2}} \right){{\left( -7{{p}^{3}}{{r}^{4}} \right)}^{2}}\left( 6p{{r}^{3}} \right)$. Now the expression has 2 variables p and r.
Now to simplify the expression we first separate the terms with p and r. Now let us first try to bring all the same variables together.
Now using BODMAS we will first solve the order which is the index. Hence first let us consider the term ${{\left( -7{{p}^{3}}{{r}^{4}} \right)}^{2}}$
Now we know that ${{\left( a\times b \right)}^{n}}={{a}^{n}}\times {{b}^{n}}$ . Hence using this we get,
$\Rightarrow {{\left( -7{{p}^{3}}{{r}^{4}} \right)}^{2}}=\left( 49 \right){{\left( {{p}^{3}} \right)}^{2}}{{\left( {{r}^{4}} \right)}^{2}}$
Now again we know that ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}$ . Hence using this property of indices we get,
$\Rightarrow \left( -7{{p}^{3}}{{r}^{4}} \right)=49{{p}^{6}}{{r}^{8}}$
Now substituting this value in the given expression we get,
$\begin{align}
  & \Rightarrow \left( {{p}^{5}}{{r}^{2}} \right)\left( 49{{p}^{6}}{{r}^{8}} \right)\left( 6p{{r}^{3}} \right) \\
 & \Rightarrow 294\left( {{p}^{5}}{{r}^{2}}{{p}^{6}}{{r}^{8}}p{{r}^{3}} \right) \\
\end{align}$
Now let us write all the p terms together and all the r terms together by using $a\times b=b\times a$ . Hence we get,
$\Rightarrow 294\left( {{p}^{5}}{{p}^{6}}p \right)\left( {{r}^{2}}{{r}^{8}}{{r}^{3}} \right)$
Now we by the law of indices we have ${{a}^{m}}{{a}^{n}}={{a}^{m+n}}$ . Hence using this we get,
$\begin{align}
  & \Rightarrow 294{{p}^{5+6+1}}{{r}^{2+8+3}} \\
 & \Rightarrow 294{{p}^{12}}{{r}^{13}} \\
\end{align}$
Hence we get $\left( {{p}^{5}}{{r}^{2}} \right){{\left( -7{{p}^{3}}{{r}^{4}} \right)}^{2}}\left( 6p{{r}^{3}} \right)=294{{p}^{12}}{{r}^{13}}$

Note: Now note that we can just use the law of indices for the same variables. Hence we cannot multiply or divide the terms with different variables. Also note that if the power of any term is 0 then its value is 1.