To Rationalize The Denominator Of 2 10 3 11 \frac{2 \sqrt{10}}{3 \sqrt{11}} 3 11 ​ 2 10 ​ ​ , You Should Multiply The Expression By Which Fraction?A. 10 10 \frac{\sqrt{10}}{\sqrt{10}} 10 ​ 10 ​ ​ B. 11 11 \frac{\sqrt{11}}{\sqrt{11}} 11 ​ 11 ​ ​ C.

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Introduction

Rationalizing the denominator is a crucial step in simplifying complex fractions, especially those involving square roots. It's a technique used to eliminate the radical from the denominator, making the expression easier to work with. In this article, we'll explore how to rationalize the denominator of a fraction, using the example of 210311\frac{2 \sqrt{10}}{3 \sqrt{11}}. We'll also discuss the importance of this technique and provide a step-by-step guide on how to do it.

What is Rationalizing the Denominator?

Rationalizing the denominator involves multiplying the fraction by a cleverly chosen expression that eliminates the radical from the denominator. This is done by multiplying both the numerator and the denominator by the same expression, which is usually a radical. The goal is to get rid of the radical in the denominator, making the fraction easier to simplify.

Why is Rationalizing the Denominator Important?

Rationalizing the denominator is an essential technique in mathematics, particularly in algebra and calculus. It helps to simplify complex fractions, making them easier to work with. By eliminating the radical from the denominator, we can perform operations such as addition, subtraction, multiplication, and division more easily. Additionally, rationalizing the denominator is a crucial step in solving equations and inequalities involving fractions.

How to Rationalize the Denominator: A Step-by-Step Guide

Now that we've discussed the importance of rationalizing the denominator, let's move on to the step-by-step guide. We'll use the example of 210311\frac{2 \sqrt{10}}{3 \sqrt{11}} to illustrate the process.

Step 1: Identify the Radical in the Denominator

The first step is to identify the radical in the denominator. In this case, the radical is 11\sqrt{11}.

Step 2: Choose the Correct Expression to Multiply

To rationalize the denominator, we need to multiply the fraction by an expression that will eliminate the radical from the denominator. In this case, we need to multiply by 1111\frac{\sqrt{11}}{\sqrt{11}}.

Step 3: Multiply the Numerator and Denominator

Now that we've chosen the correct expression to multiply, we can multiply the numerator and denominator by 1111\frac{\sqrt{11}}{\sqrt{11}}.

210311⋅1111=210⋅11311⋅11\frac{2 \sqrt{10}}{3 \sqrt{11}} \cdot \frac{\sqrt{11}}{\sqrt{11}} = \frac{2 \sqrt{10} \cdot \sqrt{11}}{3 \sqrt{11} \cdot \sqrt{11}}

Step 4: Simplify the Expression

Now that we've multiplied the numerator and denominator, we can simplify the expression.

210⋅11311⋅11=21103⋅11\frac{2 \sqrt{10} \cdot \sqrt{11}}{3 \sqrt{11} \cdot \sqrt{11}} = \frac{2 \sqrt{110}}{3 \cdot 11}

Step 5: Simplify the Fraction

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

21103⋅11=211033\frac{2 \sqrt{110}}{3 \cdot 11} = \frac{2 \sqrt{110}}{33}

Conclusion

Rationalizing the denominator is a crucial step in simplifying complex fractions, especially those involving square roots. By following the step-by-step guide outlined in this article, you can learn how to rationalize the denominator of a fraction. Remember to identify the radical in the denominator, choose the correct expression to multiply, multiply the numerator and denominator, simplify the expression, and finally simplify the fraction. With practice, you'll become proficient in rationalizing the denominator and be able to simplify complex fractions with ease.

Common Mistakes to Avoid

When rationalizing the denominator, there are several common mistakes to avoid. Here are a few:

  • Not identifying the radical in the denominator: Make sure to identify the radical in the denominator before attempting to rationalize it.
  • Choosing the wrong expression to multiply: Choose the correct expression to multiply, which is usually a radical.
  • Not multiplying the numerator and denominator: Make sure to multiply both the numerator and denominator by the chosen expression.
  • Not simplifying the expression: Simplify the expression after multiplying the numerator and denominator.

Practice Problems

To practice rationalizing the denominator, try the following problems:

  1. Rationalize the denominator of 3243\frac{3 \sqrt{2}}{4 \sqrt{3}}.
  2. Rationalize the denominator of 5527\frac{5 \sqrt{5}}{2 \sqrt{7}}.
  3. Rationalize the denominator of 211313\frac{2 \sqrt{11}}{3 \sqrt{13}}.

Answer Key

  1. 32⋅343⋅3=3612=64\frac{3 \sqrt{2} \cdot \sqrt{3}}{4 \sqrt{3} \cdot \sqrt{3}} = \frac{3 \sqrt{6}}{12} = \frac{\sqrt{6}}{4}
  2. 55⋅727⋅7=53514\frac{5 \sqrt{5} \cdot \sqrt{7}}{2 \sqrt{7} \cdot \sqrt{7}} = \frac{5 \sqrt{35}}{14}
  3. 211⋅13313⋅13=214339\frac{2 \sqrt{11} \cdot \sqrt{13}}{3 \sqrt{13} \cdot \sqrt{13}} = \frac{2 \sqrt{143}}{39}
    Rationalizing the Denominator: A Q&A Guide =====================================================

Introduction

Rationalizing the denominator is a crucial step in simplifying complex fractions, especially those involving square roots. In our previous article, we explored the step-by-step guide on how to rationalize the denominator. In this article, we'll answer some frequently asked questions about rationalizing the denominator, providing you with a deeper understanding of this important technique.

Q&A

Q: What is the purpose of rationalizing the denominator?

A: The purpose of rationalizing the denominator is to eliminate the radical from the denominator, making the fraction easier to simplify and work with.

Q: Why do I need to rationalize the denominator?

A: You need to rationalize the denominator to simplify complex fractions, especially those involving square roots. This is essential in algebra and calculus, where you'll encounter many fractions with radicals in the denominator.

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

A: You should rationalize the denominator whenever you encounter a fraction with a radical in the denominator. This is a crucial step in simplifying complex fractions.

Q: What is the correct expression to multiply to rationalize the denominator?

A: The correct expression to multiply is usually a radical, which is the same as the radical in the denominator. For example, if the denominator is 11\sqrt{11}, you should multiply by 1111\frac{\sqrt{11}}{\sqrt{11}}.

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

A: Yes, you can rationalize the denominator of a fraction with a cube root. The process is similar to rationalizing the denominator of a fraction with a square root.

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

A: After rationalizing the denominator, you should simplify the expression by multiplying the numerator and denominator by the same expression, and then simplifying the resulting fraction.

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

A: Yes, you can rationalize the denominator of a fraction with a negative sign. The process is the same as rationalizing the denominator of a fraction with a positive sign.

Q: What are some common mistakes to avoid when rationalizing the denominator?

A: Some common mistakes to avoid when rationalizing the denominator include not identifying the radical in the denominator, choosing the wrong expression to multiply, not multiplying the numerator and denominator, and not simplifying the expression.

Real-World Applications

Rationalizing the denominator has many real-world applications, including:

  • Engineering: Rationalizing the denominator is essential in engineering, where you'll encounter many complex fractions involving square roots.
  • Physics: Rationalizing the denominator is crucial in physics, where you'll encounter many complex fractions involving square roots and other radicals.
  • Computer Science: Rationalizing the denominator is important in computer science, where you'll encounter many complex fractions involving square roots and other radicals.

Conclusion

Rationalizing the denominator is a crucial step in simplifying complex fractions, especially those involving square roots. By following the step-by-step guide and answering the frequently asked questions, you'll become proficient in rationalizing the denominator and be able to simplify complex fractions with ease. Remember to identify the radical in the denominator, choose the correct expression to multiply, multiply the numerator and denominator, simplify the expression, and finally simplify the fraction.

Practice Problems

To practice rationalizing the denominator, try the following problems:

  1. Rationalize the denominator of 3243\frac{3 \sqrt{2}}{4 \sqrt{3}}.
  2. Rationalize the denominator of 5527\frac{5 \sqrt{5}}{2 \sqrt{7}}.
  3. Rationalize the denominator of 211313\frac{2 \sqrt{11}}{3 \sqrt{13}}.

Answer Key

  1. 32⋅343⋅3=3612=64\frac{3 \sqrt{2} \cdot \sqrt{3}}{4 \sqrt{3} \cdot \sqrt{3}} = \frac{3 \sqrt{6}}{12} = \frac{\sqrt{6}}{4}
  2. 55⋅727⋅7=53514\frac{5 \sqrt{5} \cdot \sqrt{7}}{2 \sqrt{7} \cdot \sqrt{7}} = \frac{5 \sqrt{35}}{14}
  3. 211⋅13313⋅13=214339\frac{2 \sqrt{11} \cdot \sqrt{13}}{3 \sqrt{13} \cdot \sqrt{13}} = \frac{2 \sqrt{143}}{39}