Simplify the equation \[\left( -90 \right)\times 52+\left( -52 \right)\times \left( -25 \right)\] using a suitable frequency.

Hint: We use different methods of doing binary operations. We use associative, distributive to get to the solution. In the given problem there are two parts. Each part has a common number. We use that to apply distributive law.

Complete step-by-step solution:
The given equation is \[\left( -90 \right)\times 52+\left( -52 \right)\times \left( -25 \right)\].
The theorem of associative tells us that for three numbers a, b, c, $a+\left( b+c \right)=\left( a+b \right)+c$.
Whereas the theorem of distributive tells us $a\left( b+c \right)=ab+ac$.
We know that $\left( -a \right)=-a$. So, we can write \[\left( -90 \right)\times 52=-\left( 90 \right)\times 52=-\left( 90\times 52 \right)\].
Similarly, \[\left( -52 \right)\times \left( -25 \right)=-\left( 52 \right)\times \left( -25 \right)=-\left( 52\times \left( -25 \right) \right)=52\times 25\].
Now the whole equation becomes
\[\begin{align}
  & \left( -90 \right)\times 52+\left( -52 \right)\times \left( -25 \right) \\
 & =-\left( 90\times 52 \right)+52\times 25 \\
\end{align}\]
Both equations have 52. So, we use distributive law which tells us $ab+ac=a\left( b+c \right)$.
The equation changes to
\[\begin{align}
  & -\left( 90\times 52 \right)+52\times 25 \\
 & =52\times \left( -90 \right)+52\times 25 \\
 & =52\times \left( 25-90 \right) \\
\end{align}\]
We solve the equation to get the answer using simple multiplication.
\[\begin{align}
  & 52\times \left( 25-90 \right) \\
 & =52\times \left( -65 \right) \\
 & =-\left( 52\times 65 \right) \\
 & =-3380 \\
\end{align}\]
So, the given equation solves to give answer of 3380.

Note: We can use the method of associative. It tells us $a+\left( b+c \right)=\left( a+b \right)+c$. The formula can be applied for multiplication also. We can also use the method of distribution to break the numbers into similar parts.