1. simplify the following:
a) $i^{47}$
$=i^{4×11 + 3}$
$=(i^4)^{11}×i^3$
$=-i$
b) $i^{102}$
=
c) $i^{133}$
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) $i^{47}$
      $=i^{4×11 + 3}$
      $=(i^4)^{11}×i^3$
      $=-i$
   b) $i^{102}$
      =
   c) $i^{133}$
   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
#### Part a: Simplify $ i^{47} $
# Explanation:
## Step 1: Recall the cycle of $ i $
The powers of $ i $ cycle every 4: $ i^1 = i $, $ i^2 = -1 $, $ i^3 = -i $, $ i^4 = 1 $, and then repeat. So we divide the exponent by 4 and find the remainder.
$ 47 \div 4 = 11 $ with a remainder of $ 3 $. So $ i^{47} = i^{4 \times 11 + 3} = (i^4)^{11} \times i^3 $
## Step 2: Simplify using the cycle
Since $ i^4 = 1 $ and $ i^3 = -i $, we have $ (1)^{11} \times (-i) = -i $
# Answer: $ -i $

#### Part b: Simplify $ i^{102} $
# Explanation:
## Step 1: Find the remainder when 102 is divided by 4
$ 102 \div 4 = 25 $ with a remainder of $ 2 $. So $ i^{102} = i^{4 \times 25 + 2} = (i^4)^{25} \times i^2 $
## Step 2: Simplify using the cycle
Since $ i^4 = 1 $ and $ i^2 = -1 $, we have $ (1)^{25} \times (-1) = -1 $
# Answer: $ -1 $

#### Part c: Simplify $ i^{133} $
# Explanation:
## Step 1: Find the remainder when 133 is divided by 4
$ 133 \div 4 = 33 $ with a remainder of $ 1 $. So $ i^{133} = i^{4 \times 33 + 1} = (i^4)^{33} \times i^1 $
## Step 2: Simplify using the cycle
Since $ i^4 = 1 $ and $ i^1 = i $, we have $ (1)^{33} \times i = i $
# Answer: $ i $

#### Part d: Simplify $ i^{1212} $
# Explanation:
## Step 1: Find the remainder when 1212 is divided by 4
$ 1212 \div 4 = 303 $ with a remainder of $ 0 $. When the remainder is 0, the exponent is a multiple of 4, so $ i^{1212} = (i^4)^{303} $
## Step 2: Simplify using the cycle
Since $ i^4 = 1 $, we have $ (1)^{303} = 1 $
# Answer: $ 1 $

### Problem 2: Simplify $ -21 + 26i + 45i^2 + 28i^5 $
# Explanation:
## Step 1: Simplify the powers of $ i $
Recall that $ i^2 = -1 $, $ i^3 = -i $, $ i^4 = 1 $, $ i^5 = i^4 \times i = 1 \times i = i $
So $ 45i^2 = 45(-1) = -45 $ and $ 28i^5 = 28i $
## Step 2: Combine like terms
Real parts: $ -21 - 45 = -66 $
Imaginary parts: $ 26i + 28i = 54i $
So the simplified expression is $ -66 + 54i $
# Answer: $ -66 + 54i $

### Problem 3
#### Part a: $ (46 + 26i) + (-25 + 67i) $
# Explanation:
## Step 1: Combine real parts and imaginary parts separately
Real parts: $ 46 + (-25) = 21 $
Imaginary parts: $ 26i + 67i = 93i $
# Answer: $ 21 + 93i $

#### Part b: $ (23 + 24i) - (26 + 46i) $
# Explanation:
## Step 1: Distribute the negative sign
$ 23 + 24i - 26 - 46i $
## Step 2: Combine real parts and imaginary parts separately
Real parts: $ 23 - 26 = -3 $
Imaginary parts: $ 24i - 46i = -22i $
# Answer: $ -3 - 22i $

#### Part c: $ (16 + 19i) + (-17 + 96i) - (75 - 46i) $
# Explanation:
## Step 1: Distribute the negative sign
$ 16 + 19i - 17 + 96i - 75 + 46i $
## Step 2: Combine real parts and imaginary parts separately
Real parts: $ 16 - 17 - 75 = -76 $
Imaginary parts: $ 19i + 96i + 46i = 161i $
# Answer: $ -76 + 161i $

### Problem 4
#### Part a: $ (-3 - 24i)(27 + 8i) $
# Explanation:
## Step 1: Use the distributive property (FOIL method)
$ -3 \times 27 + (-3) \times 8i + (-24i) \times 27 + (-24i) \times 8i $
$ = -81 - 24i - 648i - 192i^2 $
## Step 2: Simplify $ i^2 $ and combine like terms
Since $ i^2 = -1 $, $ -192i^2 = -192(-1) = 192 $
Real parts: $ -81 + 192 = 111 $
Imaginary parts: $ -24i - 648i = -672i $
# Answer: $ 111 - 672i $

#### Part b: $ (-14 - 32i)(7 - 23i) $
# Explanation:
## Step 1: Use the distributive property (FOIL method)
$ -14 \times 7 + (-14) \times (-23i) + (-32i) \times 7 + (-32i) \times (-23i) $
$ = -98 + 322i - 224i + 736i^2 $
## Step 2: Simplify $ i^2 $ and combine like terms
Since $ i^2 = -1 $, $ 736i^2 = 736(-1) = -736 $
Real parts: $ -98 - 736 = -834 $
Imaginary parts: $ 322i - 224i = 98i $
# Answer: $ -834 + 98i $

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 1

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

Step 1: Recall the cycle of \( i \)

Step 2: Simplify using the cycle

Step 1: Find the remainder when 102 is divided by 4

Step 2: Simplify using the cycle

Step 1: Find the remainder when 133 is divided by 4

Step 2: Simplify using the cycle

Answer:

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