Question

Verify:$\left( -82 \right)\times \left\{ \left( -4 \right)+19 \right\}=\left( -82 \right)\times \left( -4 \right)+\left( -82 \right)\times 19$

Answer 1

Hint: As we need to verify the above equation means, we need to give out the result that the LHS (Left Hand Side) of the equation is equal to the RHS (right hand side) of the equation. Since there are multiple mathematical operations used on both sides which makes the equation seem complicated, we will just simplify both sides of the equation using the rule of BODMAS, which will help us to find that LHS is equal to RHS.

Complete step by step answer:
Moving ahead with the question in step wise manner;
According to question we had a equation$\left( -82 \right)\times \left\{ \left( -4 \right)+19 \right\}=\left( -82 \right)\times \left( -4 \right)+\left( -82 \right)\times 19$, which we need to verify, so in short we had to prove that what is written on LHS side and RHS side of equation both are correct. So we will just simplify both sides of the equation separately and if both side results come out to be the same then it means the equation written is correct, means we had verified the equation.
Now to solve the above equation, it seems to be complicated as multiple mathematical operations$\left( +,-,\times \right)$which make it complicated to solve the equation, so we will use the rule of BODMAS to solve it.
Which states that when we had multiply operation in single equation or expression then rule sets the priority to solve it i.e. BODMAS. Its priority order is Bracket i.e. first solve the expression which is in bracket, then order or indices, then further we will solve division which is further followed by multiplication then addition and finally by subtraction.
Now applying the same rule in our question, let us first solve LHS side of the given equation i.e. $\left( -82 \right)\times \left\{ \left( -4 \right)+19 \right\}=\left( -82 \right)\times \left( -4 \right)+\left( -82 \right)\times 19$
As in this LHS side expression, according to BODMAS rule we had to solve first the values with are inside the brackets, so as in LHS side we have bracket in$\left( -4 \right)+19$so we will solve it first, and RHS side will remain as it is, which will give us as;
\[\begin{align}
  & \left( -82 \right)\times \left\{ \left( -4 \right)+19 \right\}=\left( -82 \right)\times \left( -4 \right)+\left( -82 \right)\times 19 \\
 & \left( -82 \right)\times \left\{ -4+19 \right\}=\left( -82 \right)\times \left( -4 \right)+\left( -82 \right)\times 19 \\
 & \left( -82 \right)\times \left\{ 15 \right\}=\left( -82 \right)\times \left( -4 \right)+\left( -82 \right)\times 19 \\
\end{align}\]
Now in the LHS side we had only simple multiplication, so we can now solve it easily, so we will get;
\[\begin{align}
  & \left( -82 \right)\times \left\{ 15 \right\}=\left( -82 \right)\times \left( -4 \right)+\left( -82 \right)\times 19 \\
 & -1230=\left( -82 \right)\times \left( -4 \right)+\left( -82 \right)\times 19 \\
\end{align}\]
So we got the LHS side equal to\[-1230\]. Now let us solve in RHS side using the same rule of BODMAS, so we will get it as;
On the RHS side we do not have any expression between the brackets, we had only numbers, so we had only multiplication which comes after division according to rule, as we do not have division indices operation in the expression so we will do multiplication first. So we will get the RHS expression as;
\[\begin{align}
  & -1230=\left( -82 \right)\times \left( -4 \right)+\left( -82 \right)\times 19 \\
 & -1230=328+(-1558) \\
\end{align}\]
Now we will open the bracket of 1558, which will give us;
\[-1230=328-1558\]
Now we had simple subtraction in RHS side which on solving give us;
\[-1230=-1230\]
It gave us RHS equal to LHS, meaning we had verified the equation.
Hence verified.

Note: BODMAS is the only rule to solve such multiple operations containing equations or expressions. Rule is named after its priority order i.e. ‘B’ for brackets, ‘O’ for order, ‘D’ for division, ‘M’ for multiplication, ‘A’ for addition and at last ‘S’ for subtraction.