mcr3u independent study unit – complex numbers assignment
1. simplify the following:
a) $i^{47}$
c) $i^{133}$
b) $i^{102}$
d) $i^{1212}$

2. simplify the expression: $-21 + 26i + 45i^2 + 28i^5$. your answer must have a real part and an imaginary part.

3. compute the following expressions.
a) $(46 + 26i) + (-25 + 67i)$
b) $(23 + 24i) - (26 + 46i)$
c) $(16 + 19i) + (-17 + 96i) - (75 - 46i)$

4. find the product.
a) $(-3 - 24i)(27 + 8i)$
b) $(-14 - 32i)(7 - 23i)$

Question

mcr3u independent study unit – complex numbers assignment
1. simplify the following:
   a) $i^{47}$
   c) $i^{133}$
   b) $i^{102}$
   d) $i^{1212}$

2. simplify the expression: $-21 + 26i + 45i^2 + 28i^5$. your answer must have a real part and an imaginary part.

3. compute the following expressions.
   a) $(46 + 26i) + (-25 + 67i)$
   b) $(23 + 24i) - (26 + 46i)$
   c) $(16 + 19i) + (-17 + 96i) - (75 - 46i)$

4. find the product.
   a) $(-3 - 24i)(27 + 8i)$
   b) $(-14 - 32i)(7 - 23i)$

Accepted Answer

### Question 1a: Simplify $ i^{47} $
# Explanation:
## Step1: Recall $ i $ powers cycle
The powers of $ i $ cycle every 4: $ i^1 = i $, $ i^2 = -1 $, $ i^3 = -i $, $ i^4 = 1 $, then repeat.
Divide 47 by 4: $ 47 \div 4 = 11 $ remainder $ 3 $.
## Step2: Simplify using cycle
So $ i^{47} = i^{4 \times 11 + 3} = (i^4)^{11} \times i^3 $.
Since $ i^4 = 1 $, this becomes $ 1^{11} \times i^3 = i^3 $.
And $ i^3 = -i $.
# Answer: $ -i $

### Question 1b: Simplify $ i^{102} $
# Explanation:
## Step1: Use $ i $ cycle
Divide 102 by 4: $ 102 \div 4 = 25 $ remainder $ 2 $.
## Step2: Simplify
$ i^{102} = i^{4 \times 25 + 2} = (i^4)^{25} \times i^2 $.
$ i^4 = 1 $, so $ 1^{25} \times i^2 = i^2 $.
$ i^2 = -1 $.
# Answer: $ -1 $

### Question 1c: Simplify $ i^{133} $
# Explanation:
## Step1: Divide exponent by 4
$ 133 \div 4 = 33 $ remainder $ 1 $.
## Step2: Simplify using cycle
$ i^{133} = i^{4 \times 33 + 1} = (i^4)^{33} \times i^1 $.
$ i^4 = 1 $, so $ 1^{33} \times i = i $.
# Answer: $ i $

### Question 1d: Simplify $ i^{1212} $
# Explanation:
## Step1: Divide by 4
$ 1212 \div 4 = 303 $ with remainder 0.
## Step2: Simplify
$ i^{1212} = i^{4 \times 303} = (i^4)^{303} $.
$ i^4 = 1 $, so $ 1^{303} = 1 $.
# Answer: $ 1 $

### Question 2: Simplify $ -21 + 26i + 45i^2 + 28i^5 $
# Explanation:
## Step1: Replace $ i^2, i^5 $
$ i^2 = -1 $, $ i^5 = i^{4 + 1} = i^4 \times i = 1 \times i = i $.
Substitute: $ -21 + 26i + 45(-1) + 28i $.
## Step2: Combine like terms
Real parts: $ -21 - 45 = -66 $.
Imaginary parts: $ 26i + 28i = 54i $.
So the expression becomes $ -66 + 54i $.
# Answer: $ -66 + 54i $

### Question 3a: Compute $ (46 + 26i) + (-25 + 67i) $
# Explanation:
## Step1: Combine real and imaginary parts
Real: $ 46 - 25 = 21 $.
Imaginary: $ 26i + 67i = 93i $.
## Step2: Write result
$ 21 + 93i $.
# Answer: $ 21 + 93i $

### Question 3b: Compute $ (23 + 24i) - (26 + 46i) $
# Explanation:
## Step1: Distribute the subtraction
$ 23 + 24i - 26 - 46i $.
## Step2: Combine like terms
Real: $ 23 - 26 = -3 $.
Imaginary: $ 24i - 46i = -22i $.
## Step3: Write result
$ -3 - 22i $.
# Answer: $ -3 - 22i $

### Question 3c: Compute $ (16 + 19i) + (-17 + 96i) - (75 - 46i) $
# Explanation:
## Step1: Distribute subtraction
$ 16 + 19i - 17 + 96i - 75 + 46i $.
## Step2: Combine real parts
$ 16 - 17 - 75 = -76 $.
## Step3: Combine imaginary parts
$ 19i + 96i + 46i = 161i $.
## Step4: Write result
$ -76 + 161i $.
# Answer: $ -76 + 161i $

### Question 4a: Find the product $ (-3 - 24i)(27 + 8i) $
# Explanation:
## Step1: Use distributive property (FOIL)
$ -3 \times 27 + (-3) \times 8i + (-24i) \times 27 + (-24i) \times 8i $.
## Step2: Compute each term
$ -81 - 24i - 648i - 192i^2 $.
## Step3: Replace $ i^2 $
$ i^2 = -1 $, so $ -192(-1) = 192 $.
## Step4: Combine like terms
Real: $ -81 + 192 = 111 $.
Imaginary: $ -24i - 648i = -672i $.
## Step5: Write result
$ 111 - 672i $.
# Answer: $ 111 - 672i $

### Question 4b: Find the product $ (-14 - 32i)(7 - 23i) $
# Explanation:
## Step1: Distribute (FOIL)
$ -14 \times 7 + (-14) \times (-23i) + (-32i) \times 7 + (-32i) \times (-23i) $.
## Step2: Compute each term
$ -98 + 322i - 224i + 736i^2 $.
## Step3: Replace $ i^2 $
$ i^2 = -1 $, so $ 736(-1) = -736 $.
## Step4: Combine like terms
Real: $ -98 - 736 = -834 $.
Imaginary: $ 322i - 224i = 98i $.
## Step5: Write result
$ -834 + 98i $.
# Answer: $ -834 + 98i $

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Question 1a: Simplify \( i^{47} \)

Step1: Recall \( i \) powers cycle

Step2: Simplify using cycle

Step1: Use \( i \) cycle

Step2: Simplify

Step1: Divide exponent by 4

Step2: Simplify using cycle

Answer:

Question 1b: Simplify \( i^{102} \)