Complete The Following Sentence:The Division $\frac{4+5i}{7-7i}$ Is Performed By Multiplying The Numerator And Denominator By $\square$.

Introduction

When dealing with complex fractions, rationalizing the denominator is a crucial step in simplifying the expression. This process involves multiplying both the numerator and denominator by a specific value to eliminate any imaginary components from the denominator. In this article, we will explore the concept of rationalizing the denominator and provide a step-by-step guide on how to simplify complex fractions.

What is Rationalizing the Denominator?

Rationalizing the denominator is a mathematical technique used to simplify complex fractions by eliminating any imaginary components from the denominator. This is achieved by multiplying both the numerator and denominator by a specific value, known as the conjugate of the denominator. The conjugate of a complex number is obtained by changing the sign of the imaginary part.

Why is Rationalizing the Denominator Important?

Rationalizing the denominator is an essential step in simplifying complex fractions because it allows us to express the fraction in a more manageable form. By eliminating the imaginary components from the denominator, we can perform arithmetic operations more easily and accurately. Additionally, rationalizing the denominator is a crucial step in solving equations and inequalities involving complex numbers.

How to Rationalize the Denominator

To rationalize the denominator, we need to multiply both 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, if the denominator is $7-7i$, the conjugate is $7+7i$.

Step 1: Identify the Conjugate of the Denominator

The first step in rationalizing the denominator is to identify the conjugate of the denominator. In this case, the denominator is $7-7i$, so the conjugate is $7+7i$.

Step 2: Multiply the Numerator and Denominator by the Conjugate

Once we have identified the conjugate of the denominator, we need to multiply both the numerator and denominator by this value. In this case, we multiply the numerator and denominator by $7+7i$.

Step 3: Simplify the Expression

After multiplying the numerator and denominator by the conjugate, we need to simplify the expression. This involves combining like terms and eliminating any imaginary components from the denominator.

Example: Rationalizing the Denominator in the Expression $\frac{4+5i}{7-7i}$

Let's apply the steps outlined above to rationalize the denominator in the expression $\frac{4+5i}{7-7i}$.

Step 1: Identify the Conjugate of the Denominator

The conjugate of the denominator $7-7i$ is $7+7i$.

Step 2: Multiply the Numerator and Denominator by the Conjugate

We multiply the numerator and denominator by $7+7i$:

$$\frac{4+5i}{7-7i} \cdot \frac{7+7i}{7+7i}$$

Step 3: Simplify the Expression

We simplify the expression by combining like terms and eliminating any imaginary components from the denominator:

$$\frac{(4+5i)(7+7i)}{(7-7i)(7+7i)}$$

$$=\frac{28+28i+35i+35i^2}{49-49i^2}$$

$$=\frac{28+63i-35}{49+49}$$

$$=\frac{-7+63i}{98}$$

$$=\frac{-7}{98}+\frac{63i}{98}$$

$$=-\frac{1}{14}+\frac{9i}{14}$$

Conclusion

Rationalizing the denominator is a crucial step in simplifying complex fractions. By multiplying both the numerator and denominator by the conjugate of the denominator, we can eliminate any imaginary components from the denominator and express the fraction in a more manageable form. In this article, we have provided a step-by-step guide on how to rationalize the denominator and have applied this technique to the expression $\frac{4+5i}{7-7i}$. We hope that this article has provided a clear understanding of the concept of rationalizing the denominator and has equipped readers with the skills and knowledge necessary to simplify complex fractions.

Introduction

In our previous article, we explored the concept of rationalizing the denominator and provided a step-by-step guide on how to simplify complex fractions. In this article, we will answer some of the most frequently asked questions about rationalizing the denominator.

Q: What is the purpose of rationalizing the denominator?

A: The purpose of rationalizing the denominator is to eliminate any imaginary components from the denominator, making it easier to perform arithmetic operations and simplify complex fractions.

Q: How do I know when to rationalize the denominator?

A: You should rationalize the denominator whenever you encounter a complex fraction, especially when the denominator contains imaginary components.

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 $7-7i$ is $7+7i$.

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

A: To multiply the numerator and denominator by the conjugate, you simply multiply both the numerator and denominator by the conjugate value. For example, if the denominator is $7-7i$, you would multiply the numerator and denominator by $7+7i$.

Q: What if the denominator is a binomial?

A: If the denominator is a binomial, you can use the formula $(a+b)(a-b) = a^2 - b^2$ to simplify the expression.

Q: Can I rationalize the denominator of a fraction with a negative denominator?

A: Yes, you can rationalize the denominator of a fraction with a negative denominator by multiplying both the numerator and denominator by the conjugate of the denominator.

Q: How do I simplify the expression after rationalizing the denominator?

A: After rationalizing the denominator, you should simplify the expression by combining like terms and eliminating any imaginary components from the denominator.

Q: What if I have a fraction with a complex numerator and denominator?

A: If you have a fraction with a complex numerator and denominator, you can rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator.

Q: Can I use rationalizing the denominator to simplify expressions with complex coefficients?

A: Yes, you can use rationalizing the denominator to simplify expressions with complex coefficients by multiplying both the numerator and denominator by the conjugate of the denominator.

Q: Are there any shortcuts for rationalizing the denominator?

A: Yes, there are shortcuts for rationalizing the denominator, such as using the formula $(a+b)(a-b) = a^2 - b^2$ or multiplying both the numerator and denominator by the conjugate of the denominator.

Conclusion

Rationalizing the denominator is a crucial step in simplifying complex fractions. By understanding the concept of rationalizing the denominator and applying the techniques outlined in this article, you can simplify complex fractions and express them in a more manageable form. We hope that this Q&A guide has provided a clear understanding of the concept of rationalizing the denominator and has equipped readers with the skills and knowledge necessary to simplify complex fractions.

Additional Resources

For further practice and review, we recommend the following resources:

• Khan Academy: Rationalizing the Denominator
• Mathway: Rationalizing the Denominator
• Wolfram Alpha: Rationalizing the Denominator

Practice Problems

Try the following practice problems to test your understanding of rationalizing the denominator:

1. Rationalize the denominator of the expression $\frac{3+4i}{5-2i}$.
2. Simplify the expression $\frac{2-3i}{7+4i}$ by rationalizing the denominator.
3. Rationalize the denominator of the expression $\frac{1+2i}{3-4i}$.

We hope that this Q&A guide has provided a clear understanding of the concept of rationalizing the denominator and has equipped readers with the skills and knowledge necessary to simplify complex fractions.