Question

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

Answer 1

We have the following complex number:

$11i^7-2i^5+5i-11$

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^1=i,$$i^2=(\sqrt[]{-1})^2=-1,$$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:

$i^7=i^3=-i,$

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.