Evaluate the following, using the distributive property.
(i) \[ - 39 \times 99\]
(ii) \[\left( { - 85} \right) \times 43 + 43 \times \left( { - 15} \right)\]
(iii) \[53 \times \left( { - 9} \right) - \left( { - 109} \right) \times 53\]
(iv) \[68 \times \left( { - 17} \right) + \left( { - 68} \right) \times 3\]

Hint: First we have to know the distributive property that gives us how to solve expressions in the form of \[a\left( {b + c} \right)\]. In this property, we need to remember to multiply first, before doing the addition. When a given expression contains large value numbers in product form, then each number can be expressed as the sum or difference of the multiple of \[10\]and the number less than \[10\].

Complete step by step solution:
(i): Given \[ - 39 \times 99\]--(1)
Then the equation (1) rewrite as
\[ - 39 \times 99 = - 39 \times \left( {100 - 1} \right)\]--(2)
Using the distributive property in the equation (2), we get
\[ - 39 \times 99 = - 3900 + 39 = - 3861\]
Hence, the value of \[ - 39 \times 99\] is \[ - 3861\].

(ii): Given \[\left( { - 85} \right) \times 43 + 43 \times \left( { - 15} \right)\]--(3)
Using the distributive property in the equation (3), we get
\[43 \times \left( { - 85 - 15} \right) = - 43 \times 100 = - 4300\]
Hence, the value of \[\left( { - 85} \right) \times 43 + 43 \times \left( { - 15} \right)\] is \[ - 4300\].

(iii): Given \[53 \times \left( { - 9} \right) - \left( { - 109} \right) \times 53\]--(4)
Then the equation (4) rewrite as
 \[53 \times \left( { - 9} \right) - \left( { - 100 - 9} \right) \times 53\]--(5)
Using the distributive property in the equation (5), we get
\[ - 53 \times 9 + 100 \times 53 + 9 \times 53 = 5300\]
Hence, the value of \[53 \times \left( { - 9} \right) - \left( { - 109} \right) \times 53\] is \[5300\].

(vi): Given \[68 \times \left( { - 17} \right) + \left( { - 68} \right) \times 3\]--(6)
Then the equation (6) rewrite as
 \[68 \times \left( { - 17} \right) + 68 \times \left( { - 3} \right)\]--(7)
Using the distributive property in the equation (7), we get
\[68 \times \left( { - 17 - 3} \right) = - 1360\]
Hence, the value of \[68 \times \left( { - 17} \right) + \left( { - 68} \right) \times 3\] is \[ - 1360\].

Note:
Note that the distributive property is sometimes called the distributive law of multiplication and division. We use the distributive property because the two terms inside the parentheses can’t be added because they’re not like terms. Make sure you apply the outside number to all of the terms inside the parentheses/brackets.