Questionmplify the expression to a + bi form: $2 i^{5}-3 i^{7}+i^{76}+5 i^{17}$
Answer

Studdy Solution

STEP 1

What is this asking? Simplify a bunch of imaginary numbers raised to different powers and write the answer in the standard a + bi form. Watch out! Remember the powers of i cycle!
Don't forget to simplify each term completely before combining them.

STEP 2

1. Simplify each term
2. Combine like terms

STEP 3

Alright, let's break down this imaginary extravaganza!
We've got $2i^5$ .
Remember that $i^2 = -1$ .
So, $i^5 = i^2 \cdot i^2 \cdot i = (-1) \cdot (-1) \cdot i = i$ .
Therefore, $2i^5 = 2i$ .

STEP 4

Next up: $-3i^7$ .
Since $i^2 = -1$ , we have $i^7 = i^2 \cdot i^2 \cdot i^2 \cdot i = (-1) \cdot (-1) \cdot (-1) \cdot i = -i$ .
So, $-3i^7 = -3(-i) = 3i$ .

STEP 5

Now for $i^{76}$ .
Notice that 76 is a multiple of 4, since $76 = 4 \cdot 19$ .
This means $i^{76} = (i^4)^{19} = 1^{19} = 1$ , because $i^4 = i^2 \cdot i^2 = (-1) \cdot (-1) = 1$ .

STEP 6

Finally, let's tackle $5i^{17}$ .
We can write 17 as $17 = 4 \cdot 4 + 1$ .
So, $i^{17} = (i^4)^4 \cdot i = 1^4 \cdot i = i$ .
Therefore, $5i^{17} = 5i$ .

STEP 7

Let's put it all together!
We have $2i + 3i + 1 + 5i$ .
Combining the imaginary terms, we get $2i + 3i + 5i = 10i$ .
The real term is just 1.

STEP 8

So, our final answer in a + bi form is $1 + 10i$ .

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