Perform The Indicated Operation And Simplify. Express The Answer In Terms Of $i$ (as A Complex Number).$(8+6i) \cdot \overline{(8+6i)} = \square$
Introduction
In mathematics, complex numbers are a fundamental concept that extends the real number system to include numbers with both real and imaginary parts. The imaginary unit, denoted by $i$, is defined as the square root of $-1$. Complex numbers are used to represent points in a two-dimensional plane, and they have numerous applications in various fields, including algebra, geometry, trigonometry, and engineering.
Complex Number Notation
A complex number is typically denoted by $z = a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit. The real part of the complex number is $a$, and the imaginary part is $b$. The conjugate of a complex number $z = a + bi$ is denoted by $\overline{z} = a - bi$.
Multiplying a Complex Number by its Conjugate
The problem requires us to multiply a complex number by its conjugate. To do this, we need to understand the properties of complex number multiplication. When we multiply two complex numbers, we can use the distributive property to expand the product.
Step 1: Multiply the Complex Number by its Conjugate
Let's start by multiplying the complex number $8 + 6i$ by its conjugate $\overline{8 + 6i} = 8 - 6i$.
Step 2: Expand the Product
Using the distributive property, we can expand the product as follows:
Step 3: Simplify the Expression
Now, let's simplify the expression by combining like terms:
Step 4: Simplify the Imaginary Terms
Since $i^2 = -1$, we can simplify the expression further:
Step 5: Simplify the Expression
Now, let's simplify the expression by evaluating the expression:
Conclusion
In conclusion, when we multiply a complex number by its conjugate, we get a real number. In this case, the product of $8 + 6i$ and its conjugate $\overline{8 + 6i} = 8 - 6i$ is $100$.
Applications of Complex Number Operations
Complex number operations have numerous applications in various fields, including:
- Algebra: Complex numbers are used to solve polynomial equations and to find the roots of quadratic equations.
- Geometry: Complex numbers are used to represent points in a two-dimensional plane and to perform geometric transformations.
- Trigonometry: Complex numbers are used to represent trigonometric functions and to solve trigonometric equations.
- Engineering: Complex numbers are used to represent electrical circuits and to analyze their behavior.
Conclusion
In conclusion, complex number operations are a fundamental concept in mathematics that has numerous applications in various fields. By understanding the properties of complex number multiplication, we can solve problems involving complex numbers and apply them to real-world situations.
Final Answer
The final answer is:
Introduction
In our previous article, we discussed the concept of complex number operations, specifically multiplying a complex number by its conjugate. We also provided a step-by-step solution to the problem. In this article, we will answer some frequently asked questions (FAQs) related to complex number operations.
Q: What is a complex number?
A: A complex number is a number that can be expressed in the form $z = a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit.
Q: What is the conjugate of a complex number?
A: The conjugate of a complex number $z = a + bi$ is denoted by $\overline{z} = a - bi$.
Q: Why do we multiply a complex number by its conjugate?
A: We multiply a complex number by its conjugate to eliminate the imaginary part and obtain a real number. This is useful in various applications, such as solving polynomial equations and finding the roots of quadratic equations.
Q: How do we multiply a complex number by its conjugate?
A: To multiply a complex number by its conjugate, we use the distributive property to expand the product. We then simplify the expression by combining like terms and using the fact that $i^2 = -1$.
Q: What is the result of multiplying a complex number by its conjugate?
A: The result of multiplying a complex number by its conjugate is a real number. In the case of the problem we solved earlier, the product of $8 + 6i$ and its conjugate $\overline{8 + 6i} = 8 - 6i$ is $100$.
Q: What are some applications of complex number operations?
A: Complex number operations have numerous applications in various fields, including algebra, geometry, trigonometry, and engineering. Some specific applications include:
- Algebra: Complex numbers are used to solve polynomial equations and to find the roots of quadratic equations.
- Geometry: Complex numbers are used to represent points in a two-dimensional plane and to perform geometric transformations.
- Trigonometry: Complex numbers are used to represent trigonometric functions and to solve trigonometric equations.
- Engineering: Complex numbers are used to represent electrical circuits and to analyze their behavior.
Q: How do I simplify complex number expressions?
A: To simplify complex number expressions, you can use the following steps:
- Combine like terms: Combine the real and imaginary parts of the expression separately.
- Use the fact that $i^2 = -1$: Replace $i^2$ with $-1$ in the expression.
- Simplify the expression: Simplify the expression by combining like terms and using the fact that $i^2 = -1$.
Conclusion
In conclusion, complex number operations are a fundamental concept in mathematics that has numerous applications in various fields. By understanding the properties of complex number multiplication and simplifying complex number expressions, we can solve problems involving complex numbers and apply them to real-world situations.
Final Answer
The final answer is: