Question

Verify the following:
a)\[18 \times \left[ {7 + \left( { - 3} \right)} \right] = \left[ {18 \times 7} \right] + \left[ {18 \times \left( { - 3} \right)} \right]\]
b)\[\left( { - 21} \right) \times \left[ {\left( { - 4} \right) + \left( { - 6} \right)} \right] = \left[ {\left( { - 21} \right) \times \left( { - 4} \right)} \right] + \left[ {\left( { - 21} \right) \times \left( { - 6} \right)} \right]\]

Answer 1

Hint: For solving these types of questions, always remember to first solve the brackets. In these questions, we will be first applying the distributive property to verify the question by using it on the expression given on the left-hand side and then mold it into the one given on the other side, the right-hand side. Then we will be actually verifying the expression by calculating the left-hand side, noting the value, then calculating the right-hand side, again noting its value and then comparing the two to check if they come out to be equal.

Formula Used:
We will be using the fact that the operation of multiplication follows the distributive property.
A\[ \times \](B + C) = (A\[ \times \]B) + (A\[ \times \]C)

Complete step-by-step answer:
Solving the part a) first.
The expression given to be checked is \[18 \times \left[ {7 + \left( { - 3} \right)} \right] = \left[ {18 \times 7} \right] + \left[ {18 \times \left( { - 3} \right)} \right]\]
Solving through the left-hand side first, we have,
\[18 \times [7 + ( - 3)]\]
Now, inside the bracket we have \[[7 + \left( { - 3} \right)]\]. We know plus and minus combine to make a minus. So,
\[\left[ {7 + \left( { - 3} \right)} \right] = \left[ {7 - 3} \right] = 4\]
Hence, \[18 \times \left[ {7 + \left( { - 3} \right)} \right] = 18 \times 4 = 72\]
So, solving the left-hand side we get the value as \[72\].
Now, we shall solve the right-hand side, and we have,
\[\left[ {18 \times 7} \right] + \left[ {18 \times \left( { - 3} \right)} \right]\]
Now, first we have \[\left[ {18 \times 7} \right] = 126\] and then we have \[\left[ {18 \times \left( { - 3} \right)} \right] = - 54\], so,
\[\left[ {18 \times 7} \right] + \left[ {18 \times \left( { - 3} \right)} \right]\]
\[ = 126 + \left( { - 54} \right)\]
Again, plus and minus give a minus, so
\[ = 126 - 54 = 72\]
Hence, the value of the right-hand side is 72.
Clearly, the value of the left and right-hand side is equal, and thus, the expression is verified.

The expression given to be checked is \[ - 21 \times \left[ {\left( { - 4} \right) + \left( { - 6} \right)} \right] = \left[ {\left( { - 21} \right) \times \left( { - 4} \right)} \right] + \left[ {\left( { - 21} \right) \times \left( { - 6} \right)} \right]\]
Solving through the left-hand side first, we have,
\[ - 21 \times \left[ {\left( { - 4} \right) + \left( { - 6} \right)} \right]\]
Now, inside the bracket we have \[\left[ {\left( { - 4} \right) + \left( { - 6} \right)} \right]\]. We know plus and minus combine to make a minus. So,
\[\left[ {\left( { - 4} \right) + \left( { - 6} \right)} \right] = \left[ { - 4 - 6} \right] = - 10\]
Hence, \[ - 21 \times \left[ {\left( { - 4} \right) + \left( { - 6} \right)} \right] = - 21 \times \left[ { - 10} \right] = 210\]
So, solving the left-hand side we get the value as \[210\].
Now, we shall solve the right-hand side, and we have,
\[\left[ {\left( { - 21} \right) \times \left( { - 4} \right)} \right] + \left[ {\left( { - 21} \right) \times \left( { - 6} \right)} \right]\]
Now, first we have \[\left[ {\left( { - 21} \right) \times \left( { - 4} \right)} \right] = 84\] and then we have \[\left[ {\left( { - 21} \right) \times \left( { - 6} \right)} \right] = 126\], so,
\[\left[ {\left( { - 21} \right) \times \left( { - 4} \right)} \right] + \left[ {\left( { - 21} \right) \times \left( { - 6} \right)} \right]\]
\[ = 84 + 126 = 210\]
Hence, the value of the right-hand side is \[210\].
Clearly, the value of the left and right-hand side is equal, and thus, the expression is verified.

Note: In these questions, make sure to get the operations with the signs correct or else, the answer comes out wrong. These questions are easiest if they are solved by first solving the left-hand side, noting the result, then solving the right-hand side, noting the result again and then comparing the two values.