How do you simplify \[{{\left( -5{{a}^{2}}{{b}^{7}} \right)}^{7}}\]?

Hint: In this problem, we have to simplify the given exponential expression. We know that, since the whole thing is a multiplication, we can distribute the power to each term in the bracket. We can also see that there is a negative, and the power is an odd number therefore, the result should be a negative. We can power the terms individually and combine them, to get the simplified form.

Complete step by step solution:
We know that the given expression to be simplified is,
 \[{{\left( -5{{a}^{2}}{{b}^{7}} \right)}^{7}}\]
We know that, since the whole thing is a multiplication, we can distribute the power to each term in the bracket,
\[\Rightarrow \left[ {{\left( - \right)}^{7}}{{\left( 5 \right)}^{7}}{{\left( {{a}^{2}} \right)}^{7}}{{\left( {{b}^{7}} \right)}^{7}} \right]\]
We know that there is a negative, and the power is an odd number therefore, the result should be a negative.
Now we can simplify the above step one by one, we get
\[\Rightarrow \left[ -{{\left( 5 \right)}^{7}}{{\left( a \right)}^{2\times 7}}{{\left( b \right)}^{7\times 7}} \right]\]
We also know that,
\[\Rightarrow {{5}^{7}}=5\times 5\times 5\times 5\times 5\times 5\times 5=3125\]
\[\begin{align}
  & \Rightarrow {{\left( a \right)}^{2\times 7}}={{a}^{14}} \\
 & \Rightarrow {{\left( b \right)}^{7\times 7}}={{b}^{49}} \\
\end{align}\]
We can now substitute the above values combine these terms in the above step, we get
\[\Rightarrow -3125{{a}^{14}}{{b}^{49}}\]

Therefore, the simplified form of the given expression \[{{\left( -5{{a}^{2}}{{b}^{7}} \right)}^{7}}\] is \[-3125{{a}^{14}}{{b}^{49}}\].

Note: Students make mistakes while multiplying the power terms. We should always remember that the negative sign with odd power plays a significant role, as the negative terms with odd power gives a negative solution and the terms inside the brackets are in multiplication, so we can individually take power for each term. We have to concentrate while multiplying terms with the highest power as it will take time.