Rationalize The Denominator Of The Fraction Below. What Is The New Denominator?$\[ \frac{6}{4+\sqrt{5}} \\]A. -9 B. 21 C. -1 D. 11

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Introduction

Rationalizing the denominator of a fraction is a process used to eliminate any radical expressions in the denominator. This is particularly important when working with fractions that have square roots or other radicals in the denominator. In this article, we will explore the process of rationalizing the denominator of a given fraction and determine the new denominator.

What is Rationalizing the Denominator?

Rationalizing the denominator is a technique used to remove any radical expressions from the denominator of a fraction. This is done by multiplying both the numerator and the denominator by a specific value that will eliminate the radical in the denominator. The goal of rationalizing the denominator is to simplify the fraction and make it easier to work with.

The Process of Rationalizing the Denominator

To rationalize the denominator of a fraction, we need to follow these steps:

  1. Identify the radical in the denominator: The first step is to identify the radical expression in the denominator. In this case, the denominator is 4+54+\sqrt{5}.
  2. Multiply by the conjugate: To eliminate the radical in the denominator, we need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of 4+54+\sqrt{5} is 4−54-\sqrt{5}.
  3. Simplify the expression: After multiplying, we need to simplify the expression by combining like terms.

Rationalizing the Denominator of the Given Fraction

Now that we have understood the process of rationalizing the denominator, let's apply it to the given fraction:

64+5\frac{6}{4+\sqrt{5}}

Step 1: Identify the radical in the denominator

The radical in the denominator is 5\sqrt{5}.

Step 2: Multiply by the conjugate

To eliminate the radical in the denominator, we need to multiply both the numerator and the denominator by the conjugate of the denominator, which is 4−54-\sqrt{5}.

64+5⋅4−54−5\frac{6}{4+\sqrt{5}} \cdot \frac{4-\sqrt{5}}{4-\sqrt{5}}

Step 3: Simplify the expression

After multiplying, we need to simplify the expression by combining like terms.

6(4−5)(4+5)(4−5)\frac{6(4-\sqrt{5})}{(4+\sqrt{5})(4-\sqrt{5})}

Using the difference of squares formula, we can simplify the denominator:

(4+5)(4−5)=42−(5)2=16−5=11(4+\sqrt{5})(4-\sqrt{5}) = 4^2 - (\sqrt{5})^2 = 16 - 5 = 11

So, the simplified expression is:

6(4−5)11\frac{6(4-\sqrt{5})}{11}

The New Denominator

The new denominator is 1111.

Conclusion

Rationalizing the denominator of a fraction is an important process used to eliminate any radical expressions in the denominator. By following the steps outlined in this article, we can simplify fractions and make them easier to work with. In this case, we rationalized the denominator of the given fraction and determined the new denominator to be 1111.

Answer

Introduction

Rationalizing the denominator of a fraction is a process used to eliminate any radical expressions in the denominator. In our previous article, we explored the process of rationalizing the denominator of a given fraction and determined the new denominator. In this article, we will answer some frequently asked questions about rationalizing the denominator.

Q&A

Q: What is the purpose of rationalizing the denominator?

A: The purpose of rationalizing the denominator is to eliminate any radical expressions in the denominator of a fraction. This is done to simplify the fraction and make it easier to work with.

Q: How do I know if I need to rationalize the denominator?

A: You need to rationalize the denominator if the denominator contains a radical expression, such as a square root.

Q: What is the conjugate of a denominator?

A: The conjugate of a denominator is a value that, when multiplied by the denominator, will eliminate the radical expression. For example, the conjugate of 4+54+\sqrt{5} is 4−54-\sqrt{5}.

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 follow these steps:

  1. Multiply the numerator by the conjugate.
  2. Multiply the denominator by the conjugate.
  3. Simplify the expression by combining like terms.

Q: What is the difference of squares formula?

A: The difference of squares formula is (a+b)(a−b)=a2−b2(a+b)(a-b) = a^2 - b^2. This formula can be used to simplify the denominator when multiplying by the conjugate.

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

A: Yes, you can rationalize the denominator of a fraction with a negative number in the denominator. However, you need to follow the same steps as before, including multiplying by the conjugate.

Q: Can I rationalize the denominator of a fraction with a decimal number in the denominator?

A: No, you cannot rationalize the denominator of a fraction with a decimal number in the denominator. Rationalizing the denominator only works for fractions with radical expressions in the denominator.

Q: How do I know if I have rationalized the denominator correctly?

A: To check if you have rationalized the denominator correctly, you need to simplify the expression and make sure that there are no radical expressions in the denominator.

Common Mistakes to Avoid

When rationalizing the denominator, there are several common mistakes to avoid:

  • Not multiplying by the conjugate: Make sure to multiply both the numerator and denominator by the conjugate.
  • Not simplifying the expression: Make sure to simplify the expression by combining like terms.
  • Not checking for radical expressions: Make sure to check if there are any radical expressions in the denominator after rationalizing.

Conclusion

Rationalizing the denominator of a fraction is an important process used to eliminate any radical expressions in the denominator. By following the steps outlined in this article and avoiding common mistakes, you can simplify fractions and make them easier to work with. If you have any further questions or need help with rationalizing the denominator, feel free to ask.

Additional Resources

For more information on rationalizing the denominator, check out 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 fraction 32\frac{3}{\sqrt{2}}.
  2. Rationalize the denominator of the fraction 41+3\frac{4}{1+\sqrt{3}}.
  3. Rationalize the denominator of the fraction 25−2\frac{2}{\sqrt{5}-2}.

Answer Key

  1. 322\frac{3\sqrt{2}}{2}
  2. 4(3−1)2\frac{4(\sqrt{3}-1)}{2}
  3. 2(5+2)3\frac{2(\sqrt{5}+2)}{3}