Question

Add the values $ab - bc, bc - ca, ca - ab$.

Answer 1

Hint: In order to add the three values, we need to know how to add three operands. Either we can take two operands at a time, adding them and the result will be added to the third operand. Or we can add the three operands directly. Try solving both the methods for better understanding.

Complete step-by-step solution:
We are given three operands that are $ab - bc$, $bc - ca$ and $ca - ab$.
We need to be adding the three operands. For that we are adding the first two operands that are $ab - bc$ and $bc - ca$ at a time. So, numerically we can write it as:
$= ab - bc + bc - ca$
Since, we can see in the above equation the 2nd and 3rd operand are the same and also have different signs, so they can be cancelled. So, now we get:
$ \Rightarrow ab - bc + bc - ca \\
   = ab - ca $
Now, adding the third operand $ca - ab$ in the above equation in order to get the sum of the three operands.
$= ab - bc + bc - ca \\
   = ab - ca \\
= ab - ca + ca - ab $
We can see that the 2nd and 3rd operands are same and also have opposite signs so, they are cancelled, and we get:
$ \Rightarrow ab - ca + ca - ab \\
   = - ca + ca $
Since, the remaining operands are same and have opposite signs so, they are also cancelled, and we get:
$ \Rightarrow ab - ca + ca - ab \\
= - ca + ca \\
   = 0 $
So, we get the sum as:
$ \Rightarrow ab - bc + bc - ca + ca - ab \\
   = 0 $
Therefore, the sum of the operands $ab - bc, bc - ca, ca - ab$ is $0$.

Note: We can also add the three operands directly as: $ab - bc + bc - ca + ca - ab$ then cancelling out the same values with opposite signs and we get: $ab - bc + bc - ca + ca - ab = 0$. Operands are the values that are before and after any operator, for example in $a + b$, a and b are the operands and + is an operator.