What Is The Value Of 15 I 2 + I \frac{15i}{2+i} 2 + I 15 I ​ ?A. − 3 + 6 I -3 + 6i − 3 + 6 I B. 3 + 6 I 3 + 6i 3 + 6 I C. 5 + 5 I 5 + 5i 5 + 5 I D. 5 − 5 I 5 - 5i 5 − 5 I

$$

Introduction

In mathematics, complex numbers are a fundamental concept that plays a crucial role in various fields, including algebra, geometry, and analysis. A complex number is a number that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, which satisfies the equation $i^2 = -1$. In this article, we will explore the value of the complex fraction $\frac{15i}{2+i}$.

Understanding Complex Fractions

To evaluate the complex fraction $\frac{15i}{2+i}$, we need to understand the concept of complex fractions. A complex fraction is a fraction that contains complex numbers in the numerator or denominator. In this case, the numerator is $15i$, and the denominator is $2+i$. To simplify this fraction, we can use the technique of multiplying both the numerator and denominator by the conjugate of the denominator.

Multiplying by the Conjugate

The conjugate of a complex number $a + bi$ is defined as $a - bi$. In this case, the conjugate of $2+i$ is $2-i$. To simplify the fraction, we can multiply both the numerator and denominator by the conjugate of the denominator:

$$\frac{15i}{2+i} \cdot \frac{2-i}{2-i}$$

Simplifying the Fraction

Now, let's simplify the fraction by multiplying the numerator and denominator:

$$\frac{15i(2-i)}{(2+i)(2-i)}$$

Expanding the numerator and denominator, we get:

$$\frac{30i - 15i^2}{4 - i^2}$$

Since $i^2 = -1$, we can substitute this value into the expression:

$$\frac{30i + 15}{4 + 1}$$

Simplifying further, we get:

$$\frac{30i + 15}{5}$$

Final Simplification

Now, let's simplify the fraction by dividing the numerator by the denominator:

$$\frac{30i + 15}{5} = 6i + 3$$

Therefore, the value of the complex fraction $\frac{15i}{2+i}$ is $6i + 3$.

Conclusion

In this article, we have explored the value of the complex fraction $\frac{15i}{2+i}$. We have used the technique of multiplying both the numerator and denominator by the conjugate of the denominator to simplify the fraction. The final value of the fraction is $6i + 3$. This result can be verified by checking the answer choices provided.

Answer

The correct answer is B. $3 + 6i$.

Discussion

The discussion of this problem involves understanding the concept of complex fractions and the technique of multiplying by the conjugate. This technique is essential in simplifying complex fractions and is used extensively in various mathematical applications.

Related Topics

• Complex numbers
• Conjugate of a complex number
• Multiplying complex fractions
• Simplifying complex fractions

References

• [1] "Complex Numbers" by Math Open Reference
• [2] "Conjugate of a Complex Number" by Math Is Fun
• [3] "Multiplying Complex Fractions" by Purplemath

Keywords

• Complex fractions
• Conjugate of a complex number
• Multiplying complex fractions
• Simplifying complex fractions
• Complex numbers
• Imaginary unit
• Algebra
• Geometry
• Analysis
  

Introduction

In our previous article, we explored the value of the complex fraction $\frac{15i}{2+i}$. We used the technique of multiplying both the numerator and denominator by the conjugate of the denominator to simplify the fraction. In this article, we will answer some frequently asked questions related to complex fractions and conjugates.

Q&A

Q1: What is a complex fraction?

A1: A complex fraction is a fraction that contains complex numbers in the numerator or denominator. Complex fractions can be simplified using various techniques, including multiplying by the conjugate.

Q2: What is the conjugate of a complex number?

A2: The conjugate of a complex number $a + bi$ is defined as $a - bi$. The conjugate of a complex number is used to simplify complex fractions.

Q3: Why do we multiply by the conjugate?

A3: We multiply by the conjugate to eliminate the imaginary part from the denominator. This is done by multiplying both the numerator and denominator by the conjugate of the denominator.

Q4: How do we simplify a complex fraction?

A4: To simplify a complex fraction, we can use the technique of multiplying both the numerator and denominator by the conjugate of the denominator. This will eliminate the imaginary part from the denominator.

Q5: What is the final value of the complex fraction $\frac{15i}{2+i}$?

A5: The final value of the complex fraction $\frac{15i}{2+i}$ is $6i + 3$.

Q6: Can we use other techniques to simplify complex fractions?

A6: Yes, we can use other techniques to simplify complex fractions, such as multiplying by the reciprocal of the denominator or using the distributive property.

Q7: What are some common applications of complex fractions?

A7: Complex fractions have various applications in mathematics, including algebra, geometry, and analysis. They are also used in physics and engineering to model real-world problems.

Q8: How do we handle complex fractions with multiple complex numbers?

A8: To handle complex fractions with multiple complex numbers, we can use the technique of multiplying by the conjugate of each complex number.

Q9: Can we simplify complex fractions with negative numbers?

A9: Yes, we can simplify complex fractions with negative numbers by using the same techniques as for positive numbers.

Q10: What are some common mistakes to avoid when simplifying complex fractions?

A10: Some common mistakes to avoid when simplifying complex fractions include forgetting to multiply by the conjugate, not simplifying the numerator, and not checking the final answer.

Conclusion

In this article, we have answered some frequently asked questions related to complex fractions and conjugates. We have discussed the technique of multiplying by the conjugate and its applications in mathematics. We have also provided some common mistakes to avoid when simplifying complex fractions.

Related Topics

• Complex numbers
• Conjugate of a complex number
• Multiplying complex fractions
• Simplifying complex fractions
• Algebra
• Geometry
• Analysis
• Physics
• Engineering

References

• [1] "Complex Numbers" by Math Open Reference
• [2] "Conjugate of a Complex Number" by Math Is Fun
• [3] "Multiplying Complex Fractions" by Purplemath

Keywords

• Complex fractions
• Conjugate of a complex number
• Multiplying complex fractions
• Simplifying complex fractions
• Complex numbers
• Imaginary unit
• Algebra
• Geometry
• Analysis
• Physics
• Engineering