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**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)
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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**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.
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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