Simplify And Rationalize The Denominator: − 9 6 − 2 \frac{-9}{\sqrt{6}-\sqrt{2}} 6 ​ − 2 ​ − 9 ​

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Introduction

Rationalizing the denominator is a crucial step in simplifying complex fractions, especially those involving square roots. In this article, we will focus on simplifying and rationalizing the denominator of the given expression: −96−2\frac{-9}{\sqrt{6}-\sqrt{2}}. We will break down the process into manageable steps, making it easier to understand and apply.

Understanding the Concept of Rationalizing the Denominator

Rationalizing the denominator involves removing any radical expressions from the denominator of a fraction. This is achieved by multiplying both the numerator and the denominator by a suitable expression that eliminates the radical. The goal is to simplify the fraction and make it easier to work with.

Step 1: Identify the Radical Expression in the Denominator

In the given expression, −96−2\frac{-9}{\sqrt{6}-\sqrt{2}}, the denominator contains two square roots: 6\sqrt{6} and 2\sqrt{2}. To rationalize the denominator, we need to eliminate these radicals.

Step 2: Multiply the Numerator and Denominator by the Conjugate

The conjugate of a binomial expression a−ba-b is a+ba+b. In this case, the conjugate of 6−2\sqrt{6}-\sqrt{2} is 6+2\sqrt{6}+\sqrt{2}. To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate:

−96−2⋅6+26+2\frac{-9}{\sqrt{6}-\sqrt{2}} \cdot \frac{\sqrt{6}+\sqrt{2}}{\sqrt{6}+\sqrt{2}}

Step 3: Simplify the Expression

Now, we simplify the expression by multiplying the numerators and denominators:

−9(6+2)(6−2)(6+2)\frac{-9(\sqrt{6}+\sqrt{2})}{(\sqrt{6}-\sqrt{2})(\sqrt{6}+\sqrt{2})}

Step 4: Apply the Difference of Squares Formula

The denominator can be simplified using the difference of squares formula: (a−b)(a+b)=a2−b2(a-b)(a+b) = a^2 - b^2. In this case, we have:

(6−2)(6+2)=62−22=6−2=4(\sqrt{6}-\sqrt{2})(\sqrt{6}+\sqrt{2}) = \sqrt{6}^2 - \sqrt{2}^2 = 6 - 2 = 4

Step 5: Simplify the Expression Further

Now, we substitute the simplified denominator back into the expression:

−9(6+2)4\frac{-9(\sqrt{6}+\sqrt{2})}{4}

Step 6: Distribute the Negative Sign

Finally, we distribute the negative sign to the terms inside the parentheses:

−96−924\frac{-9\sqrt{6}-9\sqrt{2}}{4}

Conclusion

In this article, we simplified and rationalized the denominator of the given expression: −96−2\frac{-9}{\sqrt{6}-\sqrt{2}}. We broke down the process into manageable steps, making it easier to understand and apply. By following these steps, you can simplify and rationalize the denominator of any complex fraction involving square roots.

Frequently Asked Questions

Q: What is rationalizing the denominator?

A: Rationalizing the denominator involves removing any radical expressions from the denominator of a fraction.

Q: How do I rationalize the denominator of a complex fraction?

A: To rationalize the denominator, multiply both the numerator and the denominator by the conjugate of the denominator.

Q: What is the conjugate of a binomial expression?

A: The conjugate of a binomial expression a−ba-b is a+ba+b.

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

A: Simplify the expression by multiplying the numerators and denominators, and then apply any relevant formulas, such as the difference of squares formula.

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

Final Thoughts

Rationalizing the denominator is an essential skill in mathematics, especially when working with complex fractions involving square roots. By following the steps outlined in this article, you can simplify and rationalize the denominator of any complex fraction. Remember to identify the radical expression in the denominator, multiply the numerator and denominator by the conjugate, simplify the expression, and apply any relevant formulas. With practice, you will become proficient in rationalizing the denominator and simplifying complex fractions.

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Introduction

Rationalizing the denominator is a crucial step in simplifying complex fractions, especially those involving square roots. In our previous article, we provided a step-by-step guide on how to simplify and rationalize the denominator of a given expression. However, we understand that some readers may still have questions or need further clarification on this topic. In this article, we will address some of the most frequently asked questions about rationalizing the denominator.

Q&A: Rationalizing the Denominator

Q: What is the purpose of rationalizing the denominator?

A: The purpose of rationalizing the denominator is to remove any radical expressions from the denominator of a fraction, making it easier to work with and simplifying the expression.

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

A: You need to rationalize the denominator if the denominator contains any radical expressions, such as square roots.

Q: What is the conjugate of a binomial expression?

A: The conjugate of a binomial expression a−ba-b is a+ba+b. For example, the conjugate of 6−2\sqrt{6}-\sqrt{2} is 6+2\sqrt{6}+\sqrt{2}.

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

A: To multiply the numerator and denominator by the conjugate, simply multiply the two expressions together. For example:

−96−2⋅6+26+2\frac{-9}{\sqrt{6}-\sqrt{2}} \cdot \frac{\sqrt{6}+\sqrt{2}}{\sqrt{6}+\sqrt{2}}

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 after multiplying by the conjugate.

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

A: To simplify the expression, multiply the numerators and denominators together, and then apply any relevant formulas, such as the difference of squares formula.

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

A: Yes, you can rationalize the denominator of a fraction with a negative sign in the denominator. Simply multiply the numerator and denominator by the conjugate, and then simplify the expression.

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

A: Some common mistakes to avoid when rationalizing the denominator include:

  • Not multiplying the numerator and denominator by the conjugate
  • Not simplifying the expression after rationalizing the denominator
  • Not applying the difference of squares formula when necessary

Tips and Tricks

Tip 1: Make sure to identify the radical expression in the denominator before rationalizing.

Tip 2: Multiply the numerator and denominator by the conjugate to eliminate the radical expression.

Tip 3: Simplify the expression after rationalizing the denominator by multiplying the numerators and denominators together.

Tip 4: Apply the difference of squares formula when necessary to simplify the denominator.

Conclusion

Rationalizing the denominator is an essential skill in mathematics, especially when working with complex fractions involving square roots. By following the steps outlined in this article and avoiding common mistakes, you can simplify and rationalize the denominator of any complex fraction. Remember to identify the radical expression in the denominator, multiply the numerator and denominator by the conjugate, simplify the expression, and apply any relevant formulas. With practice, you will become proficient in rationalizing the denominator and simplifying complex fractions.

Frequently Asked Questions

Q: What is the difference between rationalizing the denominator and simplifying the expression?

A: Rationalizing the denominator involves removing any radical expressions from the denominator of a fraction, while simplifying the expression involves making the fraction easier to work with by combining like terms or canceling out common factors.

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 radical expressions in the denominator.

Q: How do I rationalize the denominator of a fraction with a negative sign in the numerator?

A: To rationalize the denominator of a fraction with a negative sign in the numerator, simply multiply the numerator and denominator by the conjugate, and then simplify the expression.

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

Final Thoughts

Rationalizing the denominator is an essential skill in mathematics, especially when working with complex fractions involving square roots. By following the steps outlined in this article and avoiding common mistakes, you can simplify and rationalize the denominator of any complex fraction. Remember to identify the radical expression in the denominator, multiply the numerator and denominator by the conjugate, simplify the expression, and apply any relevant formulas. With practice, you will become proficient in rationalizing the denominator and simplifying complex fractions.