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

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

### Problem 1: Simplify powers of $ i $
#### Part a: $ i^{47} $
# Explanation:
## Step1: Recall $ i^n $ cycle: $ i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1 $, cycle length 4.
Divide exponent 47 by 4: $ 47 \div 4 = 11 $ remainder $ 3 $. So $ i^{47} = i^{4 \times 11 + 3} = (i^4)^{11} \times i^3 $.
## Step2: Substitute $ i^4 = 1 $, $ i^3 = -i $.
$ (1)^{11} \times (-i) = -i $.
# Answer: $ -i $

#### Part b: $ i^{102} $
# Explanation:
## Step1: Divide 102 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 $, $ i^2 = -1 $.
$ (1)^{25} \times (-1) = -1 $.
# Answer: $ -1 $

#### Part c: $ i^{133} $
# Explanation:
## Step1: Divide 133 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 $, $ i^1 = i $.
$ (1)^{33} \times i = i $.
# Answer: $ i $

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

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

### Problem 3: Compute complex number operations
#### Part a: $ (46 + 26i) + (-25 + 67i) $
# Explanation:
## Step1: Add real parts and imaginary parts separately.
Real: $ 46 + (-25) = 21 $. Imaginary: $ 26i + 67i = 93i $.
## Step2: Combine.
$ 21 + 93i $.
# Answer: $ 21 + 93i $

#### Part b: $ (23 + 24i) - (26 + 46i) $
# Explanation:
## Step1: Subtract real parts and imaginary parts separately.
Real: $ 23 - 26 = -3 $. Imaginary: $ 24i - 46i = -22i $.
## Step2: Combine.
$ -3 - 22i $.
# Answer: $ -3 - 22i $

#### Part c: $ (16 + 19i) + (-17 + 96i) - (75 - 46i) $
# Explanation:
## Step1: First, add the first two complex numbers: $ (16 - 17) + (19i + 96i) = -1 + 115i $.
## Step2: Subtract the third complex number: $ (-1 - 75) + (115i + 46i) $.
Real: $ -76 $. Imaginary: $ 161i $.
## Step3: Combine.
$ -76 + 161i $.
# Answer: $ -76 + 161i $

### Problem 4: Find the product of complex numbers
#### Part a: $ (-3 - 24i)(27 + 8i) $
# Explanation:
## Step1: Use distributive property (FOIL): $ -3 \times 27 + (-3) \times 8i + (-24i) \times 27 + (-24i) \times 8i $.
## Step2: Simplify each term: $ -81 - 24i - 648i - 192i^2 $.
## Step3: Recall $ i^2 = -1 $, so $ -192i^2 = -192(-1) = 192 $.
## Step4: Combine real and imaginary parts: $ (-81 + 192) + (-24i - 648i) = 111 - 672i $.
# Answer: $ 111 - 672i $

#### Part b: $ (-14 - 32i)(7 - 23i) $
# Explanation:
## Step1: Use distributive property: $ -14 \times 7 + (-14) \times (-23i) + (-32i) \times 7 + (-32i) \times (-23i) $.
## Step2: Simplify each term: $ -98 + 322i - 224i + 736i^2 $.
## Step3: Recall $ i^2 = -1 $, so $ 736i^2 = 736(-1) = -736 $.
## Step4: Combine real and imaginary parts: $ (-98 - 736) + (322i - 224i) = -834 + 98i $.
# Answer: $ -834 + 98i $

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QUESTION IMAGE

Question

Explanation:

Problem 1: Simplify powers of \( i \)

Part a: \( i^{47} \)

Step1: Recall \( i^n \) cycle: \( i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1 \), cycle length 4.

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

Step1: Divide 102 by 4: \( 102 \div 4 = 25 \) remainder \( 2 \). So \( i^{102} = i^{4 \times 25 + 2} = (i^4)^{25} \times i^2 \).

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

Step1: Divide 133 by 4: \( 133 \div 4 = 33 \) remainder \( 1 \). So \( i^{133} = i^{4 \times 33 + 1} = (i^4)^{33} \times i^1 \).

Step2: Substitute \( i^4 = 1 \), \( i^1 = i \).

Answer:

Part b: \( i^{102} \)