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Evaluate $(1+2i)(3+4i)$

The product of one plus two times imaginary unit and three plus four times imaginary unit is the given mathematical expression in this math problem.

multiplying complex numbers


The product of $1$ plus $2i$ and $3$ plus $4i$ is a mathematical representation for the multiplication of them. Therefore, the above mathematical expression can be written as follows.

$=\,\,\,$ $(1+2i) \times (3+4i)$

Trick to avoid confusion in multiplication

Multiplying the terms in the complex number $3$ plus $4i$ by another complex number $1$ plus $2i$ confuses some learners. So, let’s denote the complex number $1$ plus $2i$ by a variable. In this problem, the complex number $1+2i$ is denoted by a variable $z$.

$=\,\,\,$ $z \times (3+4i)$

Multiply the terms by their coefficient

The complex $3+4i$ is a binomial and its terms are multiplied by a variable $z$. So, the terms $3$ and $4i$ can be multiplied by the variable $z$ as per the distributive property of multiplication over the addition.

$=\,\,\,$ $z \times 3$ $+$ $z \times 4i$

The use of variable $z$ is over. So, replace the variable $z$ by its actual value in the above mathematical expression.

$=\,\,\,$ $(1+2i) \times 3$ $+$ $(1+2i) \times 4i$

According to the commutative property of multiplication, the positions of the factors can be changed in each term.

$=\,\,\,$ $3 \times (1+2i)$ $+$ $4i \times (1+2i)$

Now, use the distributive property one more time in each term to distribute the coefficient over the addition of the terms.

$=\,\,\,$ $3 \times 1$ $+$ $3 \times 2i$ $+$ $4i \times 1$ $+$ $4i \times 2i$

Simplify the Mathematical expression

The complex number $1$ plus $2i$ is multiplied by another complex number $3$ plus $4i$. The multiplication of them formed a mathematical expression. Now, it is time to simplify the mathematical expression to find the product of the given complex numbers $1$ plus $2i$ and $3$ plus $4i$.

$=\,\,\,$ $3$ $+$ $6i$ $+$ $4i$ $+$ $8i^2$

Second and third terms are like terms in the above mathematical expression. Therefore, add the like terms $6i$ and $4i$ to find the sum of them.

$=\,\,\,$ $3$ $+$ $10i$ $+$ $8i^2$

$=\,\,\,$ $3$ $+$ $10i$ $+$ $8 \times i^2$

According to the complex numbers, the square of imaginary unit is negative one.

$=\,\,\,$ $3$ $+$ $10i$ $+$ $8 \times (-1)$

$=\,\,\,$ $3$ $+$ $10i$ $-$ $8$

Now, use the commutative property to write the terms in an order for our convenience.

$=\,\,\,$ $3$ $-$ $8$ $+$ $10i$

Look at the first and second terms. They are numbers. So, subtract the number $8$ from number $3$ to find difference of them.

$=\,\,\,$ $-5+10i$

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