mc3u independent study unit – complex numbers assignment
1. simplify the following:
a) $i^{87}$ 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

mc3u independent study unit – complex numbers assignment
1. simplify the following:
   a) $i^{87}$   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

### Problem 1a: Simplify $ i^{27} $
# Explanation:
## Step1: Recall $ i^n $ cycle (period 4)
The powers of $ i $ repeat every 4: $ i^1 = i $, $ i^2 = -1 $, $ i^3 = -i $, $ i^4 = 1 $. Divide exponent by 4: $ 27 \div 4 = 6 $ remainder $ 3 $. So $ i^{27} = i^{4 \times 6 + 3} = (i^4)^6 \times i^3 $.
## Step2: Substitute $ i^4 = 1 $ and $ i^3 = -i $
$ (1)^6 \times (-i) = -i $.
# Answer: $ -i $

### Problem 1b: Simplify $ i^{102} $
# Explanation:
## Step1: Divide exponent by 4
$ 102 \div 4 = 25 $ remainder $ 2 $. So $ i^{102} = i^{4 \times 25 + 2} = (i^4)^{25} \times i^2 $.
## Step2: Substitute $ i^4 = 1 $ and $ i^2 = -1 $
$ (1)^{25} \times (-1) = -1 $.
# Answer: $ -1 $

### Problem 1c: Simplify $ i^{133} $
# Explanation:
## Step1: Divide exponent by 4
$ 133 \div 4 = 33 $ remainder $ 1 $. So $ i^{133} = i^{4 \times 33 + 1} = (i^4)^{33} \times i^1 $.
## Step2: Substitute $ i^4 = 1 $ and $ i^1 = i $
$ (1)^{33} \times i = i $.
# Answer: $ i $

### Problem 1d: Simplify $ i^{1212} $
# Explanation:
## Step1: Divide exponent by 4
$ 1212 \div 4 = 303 $ remainder $ 0 $. So $ i^{1212} = i^{4 \times 303 + 0} = (i^4)^{303} \times i^0 $.
## Step2: Substitute $ i^4 = 1 $ and $ i^0 = 1 $
$ (1)^{303} \times 1 = 1 $.
# Answer: $ 1 $

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

### Problem 3a: Compute $ (46 + 26i) + (-25 + 67i) $
# Explanation:
## Step1: Add real parts and imaginary parts separately
Real: $ 46 + (-25) = 21 $. Imaginary: $ 26i + 67i = 93i $.
# Answer: $ 21 + 93i $

### Problem 3b: Compute $ (23 + 24i) - (26 + 46i) $
# Explanation:
## Step1: Subtract real parts and imaginary parts separately
Real: $ 23 - 26 = -3 $. Imaginary: $ 24i - 46i = -22i $.
# Answer: $ -3 - 22i $

### Problem 3c: Compute $ (16 + 19i) + (-17 + 96i) - (75 - 46i) $
# Explanation:
## Step1: Remove parentheses (distribute negative)
$ 16 + 19i - 17 + 96i - 75 + 46i $.
## Step2: Combine real parts and imaginary parts
Real: $ 16 - 17 - 75 = -76 $. Imaginary: $ 19i + 96i + 46i = 161i $.
# Answer: $ -76 + 161i $

### Problem 4a: Find the product $ (-3 - 24i)(27 + 8i) $
# Explanation:
## Step1: Use distributive property (FOIL)
$ -3(27) + (-3)(8i) + (-24i)(27) + (-24i)(8i) $.
## Step2: Simplify each term
$ -81 - 24i - 648i - 192i^2 $. Since $ i^2 = -1 $, substitute: $ -81 - 672i - 192(-1) $.
## Step3: Combine real parts
$ -81 + 192 - 672i = 111 - 672i $.
# Answer: $ 111 - 672i $

### Problem 4b: Find the product $ (-14 - 32i)(7 - 23i) $
# Explanation:
## Step1: Use distributive property (FOIL)
$ -14(7) + (-14)(-23i) + (-32i)(7) + (-32i)(-23i) $.
## Step2: Simplify each term
$ -98 + 322i - 224i + 736i^2 $. Since $ i^2 = -1 $, substitute: $ -98 + 98i + 736(-1) $.
## Step3: Combine real parts
$ -98 - 736 + 98i = -834 + 98i $.
# Answer: $ -834 + 98i $

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QUESTION IMAGE

Question

Explanation:

Problem 1a: Simplify \( i^{27} \)

Step1: Recall \( i^n \) cycle (period 4)

Step2: Substitute \( i^4 = 1 \) and \( i^3 = -i \)

Step1: Divide exponent by 4

Step2: Substitute \( i^4 = 1 \) and \( i^2 = -1 \)

Step1: Divide exponent by 4

Step2: Substitute \( i^4 = 1 \) and \( i^1 = i \)

Answer:

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