Given four complex numbers
𝑧
1
=
2
+
3
𝑖
,
𝑧
2
=
−
7
−
5
𝑖
,
𝑧
3
=
−
9
+
7
𝑖
,
𝑧
4
=
2
+
5
𝑖
.

Calculate
|
𝑧
3
¯
𝑧
1
+
𝑧
2
𝑧
4
|
.

Formulae:

If
𝑧
=
𝑎
+
𝑏
𝑖
then
𝑧
¯
=
𝑎
−
𝑏
𝑖
and
|
𝑧
|
=
𝑎
2
+
𝑏
2
.
𝑎
+
𝑏
𝑖
𝑐
+
𝑑
𝑖
=
(
𝑎
+
𝑏
𝑖
)
(
𝑐
−
𝑑
𝑖
)
(
𝑐
+
𝑑
𝑖
)
(
𝑐
−
𝑑
𝑖
)
=
⋯

Question

Given four complex numbers  
𝑧
1
=
2
+
3
𝑖
,
𝑧
2
=
−
7
−
5
𝑖
,
𝑧
3
=
−
9
+
7
𝑖
,
𝑧
4
=
2
+
5
𝑖
 .

Calculate   
|
𝑧
3
¯
𝑧
1
+
𝑧
2
𝑧
4
|
.

Formulae:

If  
𝑧
=
𝑎
+
𝑏
𝑖
  then  
𝑧
¯
=
𝑎
−
𝑏
𝑖
  and  
|
𝑧
|
=
𝑎
2
+
𝑏
2
.
𝑎
+
𝑏
𝑖
𝑐
+
𝑑
𝑖
=
(
𝑎
+
𝑏
𝑖
)
(
𝑐
−
𝑑
𝑖
)
(
𝑐
+
𝑑
𝑖
)
(
𝑐
−
𝑑
𝑖
)
=
⋯

BlackTom AI · Accepted Answer

The given data presents a complex-number arithmetic problem, but there are no answer choices provided to evaluate. I will first outline what is being asked and then proceed with the computation step by step so you can see how the quantity is formed, even in the absence of multiple-choice options.

What is being asked:
- We are given four complex numbers:
  z1 = 2 + 3i, z2 = -7 - 5i, z3 = -9 + 7i, z4 = 2 + 5i.
- We need to compute the absolute value (modulus) of the complex number formed by taking the conjugate of z3, multiplying by z1, and adding the product of z2 and z4:
  | (conjugate(z3) * z1) + (z2 * z4) |.
- The conjugate of z3 is 
  ̅z3 = -9 - 7i (since z3 = -9 + 7i).

Step-by-step calculation of each product:
1) Compute ̅z3 * z1:
   ̅z3 = -9 - 7i and z1 = 2 + 3i.
   Multiply:
   (-9 - 7i) * (2 + 3i) = (-9*2) + (-9*3i) + (-7i*2) + (-7i*3i)
                      = -18 - 27i - 14i - 21(i^2).
   Since i^2 = -1, -21(i^2) = -21(-1) = +21.
   Real part: -18 + 21 = 3.
   Imaginary part: -27i - 14i = -41i.
   Therefore, ̅z3 * z1 = 3 - 41i.

2) Compute z2 * z4:
   z2 = -7 - 5i and z4 = 2 + 5i.
   Multiply:
   (-7 - 5i) * (2 + 5i) = (-7*2) + (-7*5i) + (-5i*2) + (-5i*5i)
                          = -14 - 35i - 10i - 25(i^2).
   With i^2 = -1, -25(i^2) = -25(-1) = +25.
   Real part: -14 + 25 = 11.
   Imaginary part: -35i - 10i = -45i.
   Therefore, z2 * z4 = 11 - 45i.

Step 3) Add the two results:
   (3 - 41i) + (11 - 45i) = (3 + 11) + (-41i - 45i) = 14 - 86i.

Step 4) Compute the modulus of 14 - 86i:
   |14 - 86i| = sqrt(14^2 + (-86)^2) = sqrt(196 + 7396) = sqrt(7592).

Result interpretation:
- The exact value is sqrt(7592).
- If you approximate, sqrt(7592) is between 87 and 88, closer to 87.2 (since 87^2 = 7569 and 88^2 = 7744).

Notes on the missing options:
- Because the provided data shows an answer field of "20
5" but no answer_options array, there are no explicit choices to evaluate as correct or incorrect.
- In a standard multiple-choice setup, we would compare the computed value (sqrt(7592)) to the given options and select the one that matches. Since the options are absent, the computed result stands as the correct mathematical outcome of the expression given.

Similar Questions

Let 𝑢 = 1 − 3 𝑖 and 𝑣 = 1 + 𝑖 . Which of the following is the Cartesian form of the complex number 𝑢 + 𝑢 𝑣 ?

设和 𝑢 = 1 − 3 𝑖 和 𝑣 = 4 + 𝑖 . 以下哪一种是复数 \（u + uv\）的笛卡尔形式？

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Let 𝑢 = 1 − 3 𝑖 and 𝑣 = 5 + 𝑖 . Which of the following is the Cartesian form of the complex number 𝑢 + 𝑢 𝑣 ?

Let 𝑢 = 4 − 3 𝑖 and 𝑣 = 3 + 𝑖 . Which of the following is the Cartesian form of the complex number 𝑢 + 𝑢 𝑣 ?

What is the result of the complex division 2 + 𝑖 4 − 𝑖 , when written in the form a + b 𝑖 ?

Which of the following options lists all of the solutions to the following equation? 𝑧 3 = 2 + 2 𝑖 .

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