find the product of the given complex number and its conjugate. write the answer in the form a + bi.

(4 + 5i)

the product of (4 + 5i) and its conjugate is .
(simplify your answer. use integers or fractions for any numbers in the expression. type your answer in the form a + bi.)

Question

find the product of the given complex number and its conjugate. write the answer in the form a + bi.

(4 + 5i)

the product of (4 + 5i) and its conjugate is  .
(simplify your answer. use integers or fractions for any numbers in the expression. type your answer in the form a + bi.)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Complex Conjugates", "Product of Complex Conjugates" ], "new_concepts": [], "current_concepts": [ "Complex Numbers", "Complex Conjugates", "Product of Complex Conjugates" ] } ## Identify the complex conjugate Using the *Complex Conjugates* knowledge point $$ \begin{aligned} &z = 4 + 5i \ &\bar{z} = 4 - 5i \end{aligned} $$ ## Calculate the product of the complex conjugates Using the *Product of Complex Conjugates* knowledge point $$ \begin{aligned} z \cdot \bar{z} &= (4 + 5i)(4 - 5i) \ &= 4^2 - (5i)^2 \ &= 16 - 25i^2 \ &= 16 - 25(-1) \ &= 16 + 25 \ &= 41 \end{aligned} $$ ## Write the final answer in the requested form Using the *Complex Numbers* knowledge point $$ 41 = 41 + 0i $$ The product of $4 + 5i$ and its conjugate is $41 + 0i$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Complex Numbers" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the complex conjugate

Calculate the product of the complex conjugates

Write the final answer in the requested form

Answer: