**Problem 9: Simplification of Complex Expressions** Simplify the expression \((1 + 2xi)^2 + (4 + 3xi)\), where \(i\) is the imaginary unit. **Explanation:** To solve this problem, we need to simplify a mathematical expression that involves complex numbers. Complex numbers are numbers that consist of a real part and an imaginary part, typically expressed in the form \(a + bi\), where \(i\) is the imaginary unit with the property that \(i^2 = -1\). **Steps:** 1. **Expand the Square Term:** \((1 + 2xi)^2\) needs to be expanded. Use the formula \((a + b)^2 = a^2 + 2ab + b^2\). 2. **Combine Like Terms:** After expanding, combine any like terms from the two parts of the expression \((1 + 2xi)^2\) and \((4 + 3xi)\). 3. **Simplify:** Simplify the expression by combining real parts (constant terms) and imaginary parts (coefficients of \(i\)) separately. This problem requires a foundational understanding of algebraic manipulations and properties of imaginary numbers to arrive at the simplest form of the expression.
**Problem 9: Simplification of Complex Expressions** Simplify the expression \((1 + 2xi)^2 + (4 + 3xi)\), where \(i\) is the imaginary unit. **Explanation:** To solve this problem, we need to simplify a mathematical expression that involves complex numbers. Complex numbers are numbers that consist of a real part and an imaginary part, typically expressed in the form \(a + bi\), where \(i\) is the imaginary unit with the property that \(i^2 = -1\). **Steps:** 1. **Expand the Square Term:** \((1 + 2xi)^2\) needs to be expanded. Use the formula \((a + b)^2 = a^2 + 2ab + b^2\). 2. **Combine Like Terms:** After expanding, combine any like terms from the two parts of the expression \((1 + 2xi)^2\) and \((4 + 3xi)\). 3. **Simplify:** Simplify the expression by combining real parts (constant terms) and imaginary parts (coefficients of \(i\)) separately. This problem requires a foundational understanding of algebraic manipulations and properties of imaginary numbers to arrive at the simplest form of the expression.
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Transcribed Image Text:**Problem 9: Simplification of Complex Expressions**
Simplify the expression \((1 + 2xi)^2 + (4 + 3xi)\), where \(i\) is the imaginary unit.
**Explanation:**
To solve this problem, we need to simplify a mathematical expression that involves complex numbers. Complex numbers are numbers that consist of a real part and an imaginary part, typically expressed in the form \(a + bi\), where \(i\) is the imaginary unit with the property that \(i^2 = -1\).
**Steps:**
1. **Expand the Square Term:**
\((1 + 2xi)^2\) needs to be expanded. Use the formula \((a + b)^2 = a^2 + 2ab + b^2\).
2. **Combine Like Terms:**
After expanding, combine any like terms from the two parts of the expression \((1 + 2xi)^2\) and \((4 + 3xi)\).
3. **Simplify:**
Simplify the expression by combining real parts (constant terms) and imaginary parts (coefficients of \(i\)) separately.
This problem requires a foundational understanding of algebraic manipulations and properties of imaginary numbers to arrive at the simplest form of the expression.
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