Question

How do you multiply\[\left( 1+3i \right)\left( -4-7i \right)\left( 6-5i \right)\]?

Answer 1

Hint: In the given question, we have been asked to multiply an expression given. In order to solve the question, first we need to multiply any two of the brackets together and the result with the third bracket. To multiply we use the distributive property of multiplication over addition and subtraction i.e. \[a\times \left( b\pm c \right)=\left( a\times b \right)\pm \left( a\times c \right)\] to simplify the given expression. We will need to use the distributive property of multiplicative over summation or subtraction.

Formula used:
Distributive property of multiplication over addition and subtraction states that the product of a number say ‘a’ with product of two numbers, say ‘b’ and ‘c’ is equal to the addition or subtraction of products of the number ‘a’ multiplied separately with ‘b’ and ‘c’:
\[a\times \left( b\pm c \right)=ab\pm ac\]

Complete step by step solution:
We have given that,
\[\Rightarrow \left( 1+3i \right)\left( -4-7i \right)\left( 6-5i \right)\]
Taking,
\[\Rightarrow \left( 1+3i \right)\left( -4-7i \right)\]
Using the distributive property of multiplication over subtraction, i.e. when a number is multiplied by the subtraction of numbers.
\[\Rightarrow a\times \left( b-c \right)=\left( a\times b \right)-\left( a\times c \right)\]
Open the brackets in the given expression, we get
\[\Rightarrow \left( 1+3i \right)\left( -4-7i \right)=1\left( -4-7i \right)+3i\left( -4-7i \right)\]
Applying the distributive property in the above expression, we get
\[\Rightarrow \left( 1\times \left( -4 \right) \right)-\left( 1\times 7i \right)+\left( 3i\times \left( -4 \right) \right)-\left( 3i\times 7i \right)\]
Simplifying the above expression, we get
\[\Rightarrow -4-7i-12i-21{{i}^{2}}\]
Putting the value of \[{{i}^{2}}=-1\] in the above expression,
\[\Rightarrow -4-7i-12i-21\times \left( -1 \right)\]
\[\Rightarrow -4-7i-12i+21\]
Simplifying the numbers in the above expression, we get
\[\Rightarrow 17-7i-12i\]
Combining the like terms, we get
\[\Rightarrow 17-19i\]
Now, we multiply \[\left( 17-19i \right)\left( 6-5i \right)\]
Similarly,
\[\Rightarrow \left( 17-19i \right)\left( 6-5i \right)=17\left( 6-5i \right)-19i\left( 6-5i \right)\]
\[\Rightarrow \left( 17\times 6 \right)-\left( 17\times 5i \right)-\left( \left( 19i\times 6 \right)-\left( 19i\times 5i \right) \right)\]
Simplifying the above, we get
\[\Rightarrow 102-85i-114i+95{{i}^{2}}\]
Putting the value of \[{{i}^{2}}=-1\] in the above expression,
\[\Rightarrow 102-85i-114+95\times \left( -1 \right)\]
\[\Rightarrow 102-199i-95\]
Simplifying the numbers, we get
\[\Rightarrow 7-199i\]
Thus,
\[\Rightarrow \left( 1+3i \right)\left( -4-7i \right)\left( 6-5i \right)=7-199i\]

Therefore, the product of \[\left( 1+3i \right)\left( -4-7i \right)\left( 6-5i \right)\] using distributive property of multiplication is equal to \[7-199i\]. It is the required answer.

Note: To solve these types of questions, we need to know there are 4 properties for the arithmetic operation called closure, commutative, associative and distributive. The distributive property of multiplication requires one more operation either addition or subtraction. We know that the Distributive property of multiplication over addition and subtraction states that the when a number is multiplied by a summation or subtractions of numbers. We can solve this question by expanding the second bracket first, we will get the same answer.