Divide The Following Expression:$\[ \frac{8+i}{8-i} \\]Express Your Answer In The Form \[$ A + Bi \$\]. (Type Integers Or Fractions.)

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Introduction

Complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields, including algebra, geometry, and calculus. In this article, we will focus on dividing a complex expression, specifically the expression $\frac{8+i}{8-i}$. We will use the technique of multiplying by the conjugate to simplify the expression and express the result in the form $a + bi$.

Multiplying by the Conjugate

To divide a complex number by another complex number, we can multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number $a + bi$ is defined as $a - bi$. In this case, the conjugate of $8 - i$ is $8 + i$.

Step 1: Multiply the Numerator and Denominator by the Conjugate

We will multiply the numerator and denominator by the conjugate of the denominator, which is $8 + i$.

$\frac{8+i}{8-i} \cdot \frac{8+i}{8+i}$

Step 2: Simplify the Expression

Now, we will simplify the expression by multiplying the numerator and denominator.

$\frac{(8+i)(8+i)}{(8-i)(8+i)}$

Step 3: Expand the Numerator and Denominator

We will expand the numerator and denominator using the distributive property.

$\frac{64 + 16i + 8i + i^2}{64 - i^2}$

Step 4: Simplify the Expression Further

We will simplify the expression further by combining like terms and using the fact that $i^2 = -1$.

$\frac{64 + 24i - 1}{64 + 1}$

Step 5: Simplify the Expression Even Further

We will simplify the expression even further by combining like terms.

$\frac{63 + 24i}{65}$

Step 6: Express the Result in the Form $a + bi$

We will express the result in the form $a + bi$ by dividing the numerator by the denominator.

$\frac{63}{65} + \frac{24}{65}i$

Conclusion

In this article, we have divided the complex expression $\frac{8+i}{8-i}$ using the technique of multiplying by the conjugate. We have simplified the expression and expressed the result in the form $a + bi$. The final result is $\frac{63}{65} + \frac{24}{65}i$.

Final Answer

The final answer is $\boxed{\frac{63}{65} + \frac{24}{65}i}$.

Related Topics

• Complex Numbers
• Conjugate
• Multiplying by the Conjugate
• Simplifying Complex Expressions

References

• [1] "Complex Numbers" by Math Open Reference
• [2] "Conjugate" by Wolfram MathWorld
• [3] "Multiplying by the Conjugate" by Purplemath

Note: The references provided are for informational purposes only and are not necessarily endorsed by the author.

Introduction

In our previous article, we explored the process of dividing a complex expression using the technique of multiplying by the conjugate. We simplified the expression $\frac{8+i}{8-i}$ and expressed the result in the form $a + bi$. In this article, we will address some common questions and concerns related to complex expression division.

Q: What is the conjugate of a complex number?

A: The conjugate of a complex number $a + bi$ is defined as $a - bi$. For example, the conjugate of $3 + 4i$ is $3 - 4i$.

Q: Why do we multiply by the conjugate when dividing complex numbers?

A: Multiplying by the conjugate eliminates the imaginary part of the denominator, making it easier to simplify the expression. This technique is particularly useful when dividing complex numbers.

Q: Can we use the conjugate to simplify expressions with complex numbers in the numerator?

A: Yes, we can use the conjugate to simplify expressions with complex numbers in the numerator. However, we must be careful to multiply both the numerator and the denominator by the conjugate.

Q: How do we simplify complex expressions with multiple complex numbers?

A: To simplify complex expressions with multiple complex numbers, we can use the distributive property and combine like terms. We can also use the technique of multiplying by the conjugate to eliminate the imaginary part of the denominator.

Q: Can we use a calculator to simplify complex expressions?

A: Yes, we can use a calculator to simplify complex expressions. However, it's essential to understand the underlying mathematical concepts and techniques to ensure accurate results.

Q: What are some common mistakes to avoid when dividing complex numbers?

A: Some common mistakes to avoid when dividing complex numbers include:

• Not multiplying by the conjugate
• Not combining like terms
• Not using the distributive property
• Not checking for errors in the calculation

Q: How do we express the result of a complex expression division in the form $a + bi$?

A: To express the result of a complex expression division in the form $a + bi$, we can use the technique of multiplying by the conjugate and then simplifying the expression.

Q: Can we use complex expression division to solve real-world problems?

A: Yes, complex expression division can be used to solve real-world problems, such as:

• Electrical engineering: Complex numbers are used to represent AC circuits and analyze their behavior.
• Signal processing: Complex numbers are used to represent signals and analyze their properties.
• Control systems: Complex numbers are used to analyze the behavior of control systems.

Conclusion

In this article, we have addressed some common questions and concerns related to complex expression division. We have discussed the technique of multiplying by the conjugate, simplified complex expressions, and expressed the result in the form $a + bi$. We have also highlighted some common mistakes to avoid and provided examples of real-world applications of complex expression division.

Final Answer

The final answer is $\boxed{\frac{63}{65} + \frac{24}{65}i}$.

Related Topics

• Complex Numbers
• Conjugate
• Multiplying by the Conjugate
• Simplifying Complex Expressions
• Real-World Applications of Complex Numbers

References

• [1] "Complex Numbers" by Math Open Reference
• [2] "Conjugate" by Wolfram MathWorld
• [3] "Multiplying by the Conjugate" by Purplemath
• [4] "Real-World Applications of Complex Numbers" by IEEE Xplore