23. Express the following in the form a+bi. 11^7 − 2^5 + 5 − 11a. 11+8i b. -11+8i c. -11-8i d. -8+11i e. 8+11i

Question

asked May 18, 2023 53.6k views

1 Answer

Jsight · Answer 1 · 2023-05-23T03:09:00+0000

We have the following complex number:

Let's begin by noting how the powers of i work. To begin with, let's remember that

$i=\sqrt[]{-1}\text{.}$

Knowing that:

$i^2=(\sqrt[]{-1})^2=-1,$
$i^3=i^2\cdot i=-1\cdot i=-i,$
$i^4=i^2\cdot i^2=(-1)(-1)=1.$

Now, notice that

$i^5=i^4\cdot i=1\cdot i=i,$

so the powers of i actually repeat after four integers. In other words:

$i^1=i^5=i^9=i^(13)=\ldots$
$i^2=i^6=i^(10)=i^(14)=\ldots$
$i^3=i^7=i^(11)=i^(15)=\ldots$
$i^4=i^8=i^(12)=i^(16)=\ldots$

This also works the same way for negative powers. Now that we know this, let's focus on the powers of i on the number we were given:

so

$11i^7=-11i\text{.}$
i^5=i,

so

$-2i^5=-2i\text{.}$

Putting all of them together:

$11i^7-2i^5+5i-11=-11i-2i+5i-11=-8i-11=-11-8i\text{.}$

So, the correct answer is option c.