Simplify The Radical Expression By Rationalizing The Denominator.$\[ \frac{4}{\sqrt{21}} \\]A. \[$\frac{\sqrt{441}}{21}\$\]B. \[$4 \sqrt{21}\$\]C. \[$21 \sqrt{4}\$\]D. \[$\frac{4 \sqrt{21}}{21}\$\]

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Introduction

Rationalizing the denominator is a process used to simplify radical expressions by removing any radicals from the denominator. This is a crucial concept in algebra and is often used to simplify complex fractions. In this article, we will explore how to simplify the radical expression by rationalizing the denominator.

What is Rationalizing the Denominator?

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

Step-by-Step Guide to Rationalizing the Denominator

To rationalize the denominator, follow these steps:

  1. Identify the radical in the denominator: The first step is to identify the radical in the denominator. In this case, the radical is √21.
  2. Multiply the numerator and denominator by the radical: To eliminate the radical in the denominator, multiply both the numerator and the denominator by the radical. In this case, we will multiply both the numerator and the denominator by √21.
  3. Simplify the expression: After multiplying the numerator and denominator by the radical, simplify the expression. This will involve multiplying the numbers together and combining like terms.

Example: Simplify the Radical Expression

Let's use the example given in the problem statement: 421{ \frac{4}{\sqrt{21}} }

To simplify this expression, we will follow the steps outlined above.

  1. Identify the radical in the denominator: The radical in the denominator is √21.
  2. Multiply the numerator and denominator by the radical: Multiply both the numerator and the denominator by √21.

421×2121=42121{ \frac{4}{\sqrt{21}} \times \frac{\sqrt{21}}{\sqrt{21}} = \frac{4\sqrt{21}}{21} }

  1. Simplify the expression: The expression is already simplified, so we can stop here.

Answer

The simplified radical expression is 42121{ \frac{4\sqrt{21}}{21} }

Conclusion

Rationalizing the denominator is a process used to simplify radical expressions by removing any radicals from the denominator. By following the steps outlined above, we can simplify complex fractions and make them easier to work with. In this article, we used the example 421{ \frac{4}{\sqrt{21}} } to demonstrate how to simplify the radical expression by rationalizing the denominator.

Comparison of Options

Let's compare the options given in the problem statement:

A. 44121{ \frac{\sqrt{441}}{21} }

B. 421{ 4 \sqrt{21} }

C. 214{ 21 \sqrt{4} }

D. 42121{ \frac{4 \sqrt{21}}{21} }

The correct answer is D. 42121{ \frac{4 \sqrt{21}}{21} }

Why is Option D Correct?

Option D is correct because it is the simplified radical expression. The process of rationalizing the denominator involved multiplying both the numerator and the denominator by √21, which resulted in the expression 42121{ \frac{4\sqrt{21}}{21} }

Why are Options A, B, and C Incorrect?

Options A, B, and C are incorrect because they do not represent the simplified radical expression. Option A is incorrect because it does not have the correct numerator. Option B is incorrect because it does not have the correct denominator. Option C is incorrect because it does not have the correct numerator and denominator.

Final Thoughts

Q: What is rationalizing the denominator?

A: Rationalizing the denominator is a process used to simplify radical expressions by removing any radicals from the denominator. This is done by multiplying both the numerator and the denominator by a radical that will eliminate the radical in the denominator.

Q: Why is rationalizing the denominator important?

A: Rationalizing the denominator is important because it allows us to simplify complex fractions and make them easier to work with. This is especially useful in algebra and other mathematical applications where complex fractions are common.

Q: How do I rationalize the denominator?

A: To rationalize the denominator, follow these steps:

  1. Identify the radical in the denominator: The first step is to identify the radical in the denominator.
  2. Multiply the numerator and denominator by the radical: To eliminate the radical in the denominator, multiply both the numerator and the denominator by the radical.
  3. Simplify the expression: After multiplying the numerator and denominator by the radical, simplify the expression.

Q: What if the denominator has multiple radicals?

A: If the denominator has multiple radicals, you will need to multiply the numerator and denominator by each radical individually. For example, if the denominator is √a√b, you will need to multiply the numerator and denominator by √ab.

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

A: Yes, you can rationalize the denominator of a fraction with a variable in the denominator. However, you will need to use the same process as before, multiplying the numerator and denominator by the radical.

Q: What if the numerator and denominator have the same radical?

A: If the numerator and denominator have the same radical, you can simplify the fraction by canceling out the radical. For example, if the fraction is aa{ \frac{\sqrt{a}}{\sqrt{a}} }, you can simplify it to 1.

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 will need to use the same process as before, multiplying the numerator and denominator by the radical.

Q: What if the denominator is a perfect square?

A: If the denominator is a perfect square, you can simplify the fraction by taking the square root of the denominator. For example, if the fraction is a4{ \frac{a}{4} } and 4 is a perfect square, you can simplify it to a2{ \frac{a}{2} }.

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

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

Q: What if I get a negative number in the denominator after rationalizing?

A: If you get a negative number in the denominator after rationalizing, you can simplify the fraction by multiplying the numerator and denominator by the negative of the radical. For example, if the fraction is a−b{ \frac{a}{-\sqrt{b}} }, you can simplify it to −ab{ \frac{-a}{\sqrt{b}} }.

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

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

Conclusion

Rationalizing the denominator is a process used to simplify radical expressions by removing any radicals from the denominator. By following the steps outlined above, we can simplify complex fractions and make them easier to work with. In this article, we answered some frequently asked questions about rationalizing the denominator and provided examples to illustrate the process.