Write the complex number
𝑧
=
(
3
+
𝑖
)
14
in standard polar form
𝑧
=
𝑟
(
cos
⁡
𝜃
+
𝑖
sin
⁡
𝜃
)
.

What are
𝑟
and
𝜃
?

Question

Write the complex number 
𝑧
=
(
3
+
𝑖
)
14
 in standard polar form 
𝑧
=
𝑟
(
cos
⁡
𝜃
+
𝑖
sin
⁡
𝜃
)
.

What are 
𝑟
 and 
𝜃
?

BlackTom AI · Accepted Answer

√3 + i = 2(cos(π/6) + i sin(π/6)), so expressing the base in polar form immediately enables De Moivre's theorem.

1. Relevant knowledge points and concepts
- Complex numbers can be written in polar form r(cos θ + i sin θ) where r = sqrt(x^2 + y^2) and θ = arctan(y/x) (with quadrant adjustments).
- De Moivre's theorem: (r(cos θ + i sin θ))^n = r^n (cos(nθ) + i sin(nθ)).
- Arguments are determined modulo 2π, so we reduce nθ to an equivalent angle in [0, 2π).

2. Complete reasoning process step by step
- Identify the complex number (interpreted here as √3 + i). Compute its modulus and argument, then raise to the 14th power using De Moivre.

3. Calculations and intermediate steps
- Modulus: |√3 + i| = sqrt((√3)^2 + 1^2) = sqrt(3 + 1) = sqrt(4) = 2.
- Argument: θ0 = arctan(1 / √3) = π/6 (since √3 + i lies in the first quadrant).
- Polar form of the base: √3 + i = 2 (cos(π/6) + i sin(π/6)).
- Apply De Moivre for n = 14:
  r = 2^14 = 16384.
  θ = 14 × (π/6) = 14π/6 = 7π/3.
- Reduce argument modulo 2π: 7π/3 − 2π = 7π/3 − 6π/3 = π/3.
- Final polar form: z = 16384 (cos(π/3) + i sin(π/3)).

4. Why the correct answer is correct
Because the base has modulus 2 and principal argument π/6, raising the modulus to the 14th power produces 2^14 = 16384, and multiplying the argument by 14 gives 7π/3, which is coterminal with π/3. This directly yields r = 16384 and θ = π/3.

5. Why other interpretations would be incorrect
- If one mistakenly used 3 + i (instead of √3 + i), the modulus would be sqrt(10) and r would be (sqrt(10))^14 = 10^7, and the final angle would be 14 arctan(1/3), not π/3—so that interpretation does not match the given answer.
- Any arithmetic error in computing modulus, argument, or reducing the angle mod 2π would lead to a different (incorrect) pair (r, θ).

6. Conclusion
The polar form parameters are r = 16384 and θ = π/3, which matches the provided answer.

Write the complex number 𝑧 = ( 3 + 𝑖 ) 14 in standard polar form 𝑧 = 𝑟 ( cos ⁡ 𝜃 + 𝑖 sin ⁡ 𝜃 ) . What are 𝑟 and 𝜃 ?单项选择题

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