Question

How do you multiply \[\left( 2-4i \right)\left( 3+5i \right)\]?

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 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. Here, we will need to use the distributive property of multiplication over addition or subtraction both as a number is multiplied by the subtraction of two numbers and the summation of two numbers.

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,
\[\Rightarrow \left( 2-4i \right)\left( 3+5i \right)\]
Using the distributive property of multiplication over addition, i.e. when a number is multiplied by the summation of numbers.
\[\Rightarrow a\times \left( b+c \right)=\left( a\times b \right)+\left( a\times c \right)\]
Open the brackets in the above expression, we obtain
\[\Rightarrow \left( 2-4i \right)\left( 3+5i \right)=2\left( 3+5i \right)-4i\left( 3+5i \right)\]
Applying the distributive property in the above expression, we get
\[\Rightarrow \left( 2\times 3 \right)+\left( 2\times 5i \right)-\left( \left( 4i\times 3 \right)+\left( 4i\times 5i \right) \right)\]
Simplifying the above expression, we get
\[\Rightarrow 6+10i-12i-20{{i}^{2}}\]
Combining the like terms in the expression, we get
\[\Rightarrow 6-2i-20{{i}^{2}}\]
As we know that the value of \[{{i}^{2}}\]= -1,
Substitute the value of \[{{i}^{2}}\]= -1 in the above expression, we get
\[\Rightarrow 6-2i-20\times \left( -1 \right)\]
\[\Rightarrow 6-2i+20\]
Simplifying the numbers in the above expression, we get
\[\Rightarrow 26-2i\]
Thus,
\[\Rightarrow \left( 2-4i \right)\left( 3+5i \right)=26-2i\]

Therefore, the product of \[\left( 2-4i \right)\left( 3+5i \right)\] using the distributive property of multiplication is equal to \[26-2i\]. 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.