Math Problem Statement
Simplify (2+3i)⋅(1−4i)(2 + 3i) \cdot (1 - 4i)(2+3i)⋅(1−4i).
Solution
We are tasked with simplifying the expression . Let's go step by step.
Step 1: Use the distributive property (FOIL method)
We distribute the terms: Now simplify each term:
Step 2: Simplify using
Recall that , so:
Step 3: Combine all terms
Now substitute back into the expression: Combine the real terms and the imaginary terms:
Final Answer:
Would you like more details on this process or clarification on any step?
Here are 5 related questions you could explore:
- What happens when you multiply two complex numbers geometrically?
- How do you find the conjugate of a complex number?
- Can you simplify ?
- What is the modulus of the complex number ?
- How do you divide two complex numbers?
Tip: When multiplying complex numbers, always remember that , which simplifies the terms with imaginary units.
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Math Problem Analysis
Mathematical Concepts
Complex Numbers
Multiplication of Complex Numbers
Formulas
(a + bi)(c + di) = (ac - bd) + (ad + bc)i
Theorems
i^2 = -1
Suitable Grade Level
Grades 9-12