(
6
−
6
𝑖
−
9
3
−
27
𝑖
)
256
=
_
_
_
_
_
_
_
_
_
_

Hints:

Convert the complex numbers to polar form.
If
𝑧
=
𝑟
(
cos
⁡
𝜃
+
𝑖
sin
⁡
𝜃
)
then
𝑧
𝑛
=
𝑟
𝑛
(
cos
⁡
𝑛
𝜃
+
𝑖
sin
⁡
𝑛
𝜃
)
.
If
𝑧
1
=
𝑟
1
(
cos
⁡
𝜃
1
+
𝑖
sin
⁡
𝜃
1
)
and
𝑧
2
=
𝑟
2
(
cos
⁡
𝜃
2
+
𝑖
sin
⁡
𝜃
2
)
then:

𝑧
1
𝑧
2
=
𝑟
1
𝑟
2
[
cos
⁡
(
𝜃
1
+
𝜃
2
)
+
𝑖
sin
⁡
(
𝜃
1
+
𝜃
2
)
]
and
𝑧
1
𝑧
2
=
𝑟
1
𝑟
2
[
cos
⁡
(
𝜃
1
−
𝜃
2
)
+
𝑖
sin
⁡
(
𝜃
1
−
𝜃
2
)
]
.

Question

(
6
−
6
𝑖
−
9
3
−
27
𝑖
)
256
=
_
_
_
_
_
_
_
_
_
_

Hints:

Convert the complex numbers to polar form.
If   
𝑧
=
𝑟
(
cos
⁡
𝜃
+
𝑖
sin
⁡
𝜃
)
 then  
𝑧
𝑛
=
𝑟
𝑛
(
cos
⁡
𝑛
𝜃
+
𝑖
sin
⁡
𝑛
𝜃
)
.
If 
𝑧
1
=
𝑟
1
(
cos
⁡
𝜃
1
+
𝑖
sin
⁡
𝜃
1
)
  and  
𝑧
2
=
𝑟
2
(
cos
⁡
𝜃
2
+
𝑖
sin
⁡
𝜃
2
)
 then:

𝑧
1
𝑧
2
=
𝑟
1
𝑟
2
[
cos
⁡
(
𝜃
1
+
𝜃
2
)
+
𝑖
sin
⁡
(
𝜃
1
+
𝜃
2
)
]
  and  
𝑧
1
𝑧
2
=
𝑟
1
𝑟
2
[
cos
⁡
(
𝜃
1
−
𝜃
2
)
+
𝑖
sin
⁡
(
𝜃
1
−
𝜃
2
)
]
.

BlackTom AI · Accepted Answer

The prompt presents a complex-number exponentiation task using polar form, but the provided data for answer choices is incomplete. I will first restate what is visible and then outline the full solution approach, while also explaining what is missing and how to proceed if the options were present.

Re-stated question elements:
- We are given a complex number expressed in a fragmented form: (6 − 6i − 9^3 − 27i) and then the operation z^256 on that quantity, i.e., (6 − 6i − 9^3 − 27i)^256, with hints about converting to polar form and using De Moivre’s formula: if z = r(cos θ + i sin θ) then z^n = r^n (cos(nθ) + i sin(nθ)). The hints also show how to multiply arguments when multiplying complex numbers, and how to raise a single complex number to a power.
- The target is to fill in blanks corresponding to the resulting expression after raising to the 256th power, presumably in polar form: r^256 [cos(256θ) + i sin(256θ)].
- However, the provided answer options array is empty, i.e., there are no listed choices to select from.

What this implies:
- Without actual answer options, we cannot identify a correct choice or explain why other choices are incorrect. The analysis below will still guide you through the expected reasoning steps so you can apply them once options are available.

Step-by-step reasoning you would apply to this problem (and what to check in the options):
1) Parse the complex number accurately.
   - If the expression is Z = 6 − 6i − 9^3 − 27i, compute the real and imaginary parts:
     a = 6 − 9^3, b = −6 − 27. Be careful with signs and the exponent; 9^3 equals 729, so a = 6 − 729 = −723, b = −33.
   - Thus Z = −723 − 33i. If the expression was intended differently (for example, a product or a sum of separate z1 and z2 terms), you would need to group accordingly. The key is to identify the actual a and b for the form Z = a + bi.
2) Convert Z to polar form.
   - Compute the modulus r = sqrt(a^2 + b^2).
   - Compute the argument θ = atan2(b, a), choosing the correct quadrant for (a, b).
   - Then Z = r (cos θ + i sin θ).
3) Apply De Moivre to raise to the 256th power.
   - Z^256 = r^256 [cos(256 θ) + i sin(256 θ)].
   - In many problems, it’s useful to reduce the angle 256θ modulo 2π to identify the principal point on the unit circle if you’re comparing to standard angles (e.g., multiples of π/3, π/6, etc.).
4) Compare with the given answer options.
   - Each option will typically present a form like r^256 (cos(256θ) + i sin(256θ)) with specific r and θ values, or a rectangular form a + bi after expanding the trigonometric expressions.
   - Check for simplifications: r^256 could be a very large magnitude; however, sometimes problems are crafted so that r is 1, or the angle makes the trigonometric terms collapse to simple values (0, ±1, or ±i).
5) If any option is in rectangular form, verify by reconstructing the polar form from the option and ensuring it matches Z^256 as computed from a and b.

Common misconceptions or pitfalls to avoid:
- Miscomputing the original a and b by mishandling the sign or the exponent (e.g., confusing 9^3 with 93 or forgetting the negative signs).
- Forgetting to use atan2(b, a) to place θ in the correct quadrant, which leads to an incorrect angle and thus an incorrect final result.
- Not reducing the angle 256θ modulo 2π, which can make it hard to recognize a standard angle pattern in the options.
- Treating r^256 as if it were simply 256 or ignoring the dramatic growth of the magnitude when r ≠ 1.

What to do when answer options are provided:
- For each option, express it in the polar form r'(cos φ + i sin φ) or in rectangular a + bi form, and verify consistency with the computed Z^256.
- If multiple options present plausible polar representations, check whether φ matches 256θ modulo 2π and whether r' equals r^256.
- If an option presents a unit-circle angle (e.g., multiples of π/3, π/2, π/6) after reducing φ, you can check by evaluating cos φ and sin φ to confirm the exact real and imaginary parts.

Bottom line given the current data:
- The core method and steps are clear, but since the answer options are missing, we cannot identify which choice is correct or explain why others are wrong in this specific instance.
- Once the answer options are supplied, you can apply the outlined steps to determine which option matches Z^256 and then craft precise explanations for why each incorrect option fails, using the same polar-de Moivre reasoning demonstrated here.

SMAT011 Weekly Quiz 3 |LA003

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Similar Questions

The polar form of 𝑧 = − 6 7 + 3 2 7 𝑖 is: Hint: 𝑧 = 𝑎 + 𝑏 𝑖 = 𝑟 ( cos ⁡ 𝜃 + 𝑖 sin ⁡ 𝜃 ) where 𝑟 = 𝑎 2 + 𝑏 2 and 𝜃 = tan − 1 ⁡ 𝑏 𝑎 . Also, don't forget to plot the complex number on the Argand diagram.

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