Question

Explain the given complex number in the form $a + ib = 3(7 + 7i) + i(7 + i7)$ .

Answer 1

In this que4stion we have to write the given complex number in the form of $a + ib$ . We know that a complex number is a number that can be expressed in the form of $a + ib$ , where $a$ and $b$ are the real numbers and the other symbol is known as iota i.e. $i$ . It represents the imaginary unit. So here we will simplify and multiply the given equation and after multiplying we will arrange the equation and then simplify it.

Complete step-by-step answer:
Here we have been given the equation:
$3(7 + 7i) + i(7 + i7)$ .
We will now multiply the given equation and it gives us:
$= 3 \times 7 + 3 \times 7i + i \times 7 + i \times 7i$
$= 21 + 21i + 7i + 7{i^2}$
We can add the similar terms of the expression and we have:
$= 21 + 28i + 7{i^2}$
We will now substitute the value of the square of iota i.e.
${i^2} = - 1$
By substituting this value we can write the expression as:
$= 21 + 28i + 7 \times ( - 1)$
$= 21 + 28i - 7$
We can subtract similar value terms and it gives us :
$14 + 28i$
This expression cannot be solved any further and we can see that this is in the form of complex numbers i.e. $a + ib$
Where we have
$a = 14,b = 28$ .
Hence the required answer is
$14 + 28i$ .

Note: We should note the original value of iota is
$i = \sqrt { - 1}$ .
So when we will find the square of iota we have to square the other side too, it gives us
${i^2} = {\left( {\sqrt { - 1} } \right)^2}$
Therefore we have ${i^2} = - 1$ . We should know that if the real part of the complex number is zero, then we are only left with an imaginary number. For example, we have $2i$ . Here the real part is zero, we can also express this number in the form of $a + ib$ .
It can be represented as
$0 + 2i$ .