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.