Simplify 3 9 + 5 I \frac{3}{9+5i} 9 + 5 I 3 ​ .

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Introduction

Complex fractions, also known as complex numbers, are a fundamental concept in mathematics. They are used to represent numbers that have both real and imaginary parts. In this article, we will focus on simplifying complex fractions, specifically the expression 39+5i\frac{3}{9+5i}.

What are Complex Fractions?

Complex fractions are numbers that have both real and imaginary parts. They are typically represented in the form a+bia+bi, where aa is the real part and bb is the imaginary part. The imaginary part is denoted by the letter ii, which is defined as the square root of −1-1. Complex fractions can be added, subtracted, multiplied, and divided just like real numbers.

Simplifying Complex Fractions

Simplifying complex fractions involves expressing them in their simplest form. This is done by multiplying the numerator and denominator by the conjugate of the denominator. The conjugate of a complex number is obtained by changing the sign of the imaginary part. For example, the conjugate of 9+5i9+5i is 9−5i9-5i.

Step 1: Multiply the Numerator and Denominator by the Conjugate

To simplify the expression 39+5i\frac{3}{9+5i}, we need to multiply the numerator and denominator by the conjugate of the denominator, which is 9−5i9-5i.

39+5i⋅9−5i9−5i\frac{3}{9+5i} \cdot \frac{9-5i}{9-5i}

Step 2: Expand the Expression

Next, we need to expand the expression by multiplying the numerator and denominator.

=3(9−5i)(9+5i)(9−5i)= \frac{3(9-5i)}{(9+5i)(9-5i)}

Step 3: Simplify the Expression

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

=27−15i81−25i2= \frac{27-15i}{81-25i^2}

Step 4: Simplify the Denominator

Since i2=−1i^2 = -1, we can simplify the denominator.

=27−15i81+25= \frac{27-15i}{81+25}

Step 5: Simplify the Expression

Finally, we can simplify the expression by dividing the numerator and denominator by their greatest common divisor.

=27−15i106= \frac{27-15i}{106}

Conclusion

Simplifying complex fractions involves expressing them in their simplest form. This is done by multiplying the numerator and denominator by the conjugate of the denominator. In this article, we simplified the expression 39+5i\frac{3}{9+5i} by multiplying the numerator and denominator by the conjugate of the denominator and then simplifying the expression. The final simplified expression is 27−15i106\frac{27-15i}{106}.

Example Use Cases

Complex fractions have many practical applications in mathematics and science. Here are a few example use cases:

  • Electrical Engineering: Complex fractions are used to represent impedance in electrical circuits.
  • Signal Processing: Complex fractions are used to represent filters in signal processing.
  • Control Systems: Complex fractions are used to represent transfer functions in control systems.

Tips and Tricks

Here are a few tips and tricks for simplifying complex fractions:

  • Use the Conjugate: The conjugate of a complex number is obtained by changing the sign of the imaginary part. Multiplying the numerator and denominator by the conjugate of the denominator is a common technique for simplifying complex fractions.
  • Simplify the Denominator: Simplifying the denominator is an important step in simplifying complex fractions. This involves multiplying the denominator by the conjugate of the denominator and then simplifying the expression.
  • Use the Greatest Common Divisor: The greatest common divisor (GCD) of two numbers is the largest number that divides both numbers without leaving a remainder. Dividing the numerator and denominator by their GCD is a common technique for simplifying complex fractions.

Conclusion

Simplifying complex fractions involves expressing them in their simplest form. This is done by multiplying the numerator and denominator by the conjugate of the denominator and then simplifying the expression. In this article, we simplified the expression 39+5i\frac{3}{9+5i} by multiplying the numerator and denominator by the conjugate of the denominator and then simplifying the expression. The final simplified expression is 27−15i106\frac{27-15i}{106}. Complex fractions have many practical applications in mathematics and science, and simplifying them is an important step in solving problems involving complex numbers.

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Q: What is a complex fraction?

A: A complex fraction is a number that has both real and imaginary parts. It is typically represented in the form a+bia+bi, where aa is the real part and bb is the imaginary part.

Q: How do I simplify a complex fraction?

A: To simplify a complex fraction, you need to multiply the numerator and denominator by the conjugate of the denominator. The conjugate of a complex number is obtained by changing the sign of the imaginary part.

Q: What is the conjugate of a complex number?

A: The conjugate of a complex number is obtained by changing the sign of the imaginary part. For example, the conjugate of 9+5i9+5i is 9−5i9-5i.

Q: How do I multiply the numerator and denominator by the conjugate?

A: To multiply the numerator and denominator by the conjugate, you need to multiply the numerator and denominator by the conjugate of the denominator. For example, to simplify the expression 39+5i\frac{3}{9+5i}, you would multiply the numerator and denominator by the conjugate of the denominator, which is 9−5i9-5i.

Q: What is the final simplified expression for 39+5i\frac{3}{9+5i}?

A: The final simplified expression for 39+5i\frac{3}{9+5i} is 27−15i106\frac{27-15i}{106}.

Q: What are some common applications of complex fractions?

A: Complex fractions have many practical applications in mathematics and science. Some common applications include:

  • Electrical Engineering: Complex fractions are used to represent impedance in electrical circuits.
  • Signal Processing: Complex fractions are used to represent filters in signal processing.
  • Control Systems: Complex fractions are used to represent transfer functions in control systems.

Q: How do I simplify a complex fraction with a complex numerator?

A: To simplify a complex fraction with a complex numerator, you need to multiply the numerator and denominator by the conjugate of the denominator. This will eliminate the complex part of the numerator.

Q: What is the greatest common divisor (GCD) of two numbers?

A: The greatest common divisor (GCD) of two numbers is the largest number that divides both numbers without leaving a remainder. Dividing the numerator and denominator by their GCD is a common technique for simplifying complex fractions.

Q: How do I divide the numerator and denominator by their GCD?

A: To divide the numerator and denominator by their GCD, you need to find the GCD of the numerator and denominator and then divide both numbers by the GCD.

Q: What are some tips and tricks for simplifying complex fractions?

A: Here are a few tips and tricks for simplifying complex fractions:

  • Use the Conjugate: The conjugate of a complex number is obtained by changing the sign of the imaginary part. Multiplying the numerator and denominator by the conjugate of the denominator is a common technique for simplifying complex fractions.
  • Simplify the Denominator: Simplifying the denominator is an important step in simplifying complex fractions. This involves multiplying the denominator by the conjugate of the denominator and then simplifying the expression.
  • Use the Greatest Common Divisor: The greatest common divisor (GCD) of two numbers is the largest number that divides both numbers without leaving a remainder. Dividing the numerator and denominator by their GCD is a common technique for simplifying complex fractions.

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

A: Here are a few common mistakes to avoid when simplifying complex fractions:

  • Not using the conjugate: Failing to use the conjugate of the denominator can make it difficult to simplify the expression.
  • Not simplifying the denominator: Failing to simplify the denominator can make it difficult to simplify the expression.
  • Not using the greatest common divisor: Failing to use the greatest common divisor (GCD) of the numerator and denominator can make it difficult to simplify the expression.

Q: How do I check my work when simplifying complex fractions?

A: To check your work when simplifying complex fractions, you need to multiply the numerator and denominator by the conjugate of the denominator and then simplify the expression. You can also use a calculator to check your work.

Q: What are some resources for learning more about simplifying complex fractions?

A: Here are a few resources for learning more about simplifying complex fractions:

  • Textbooks: There are many textbooks available that cover the topic of simplifying complex fractions.
  • Online tutorials: There are many online tutorials available that cover the topic of simplifying complex fractions.
  • Practice problems: There are many practice problems available that can help you practice simplifying complex fractions.