Simplify And Rationalize The Denominator:$\frac{10}{2-\sqrt{3}}$

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Introduction

Rationalizing the denominator is a crucial step in simplifying complex fractions, especially when dealing with square roots. In this article, we will delve into the process of simplifying and rationalizing the denominator of the given fraction $\frac{10}{2-\sqrt{3}}$. We will explore the necessary steps, provide examples, and offer a comprehensive guide to help you master this essential math concept.

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 cleverly chosen value 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 fraction $\frac{10}{2-\sqrt{3}}$, the denominator contains a radical expression: $2-\sqrt{3}$. This is the expression we need to rationalize.

Step 2: Choose a Value to Multiply the Fraction

To rationalize the denominator, we need to multiply both the numerator and the denominator by a value that will eliminate the radical. In this case, we can choose the conjugate of the denominator, which is $2+\sqrt{3}$.

Step 3: Multiply the Fraction by the Chosen Value

Now, let's multiply the fraction by the conjugate of the denominator:

$$\frac{10}{2-\sqrt{3}} \cdot \frac{2+\sqrt{3}}{2+\sqrt{3}}$$

Step 4: Simplify the Expression

When we multiply the fraction, we get:

$$\frac{10(2+\sqrt{3})}{(2-\sqrt{3})(2+\sqrt{3})}$$

Step 5: Apply the Difference of Squares Formula

The denominator can be simplified using the difference of squares formula:

$$(a-b)(a+b) = a^2 - b^2$$

In this case, we have:

$$(2-\sqrt{3})(2+\sqrt{3}) = 2^2 - (\sqrt{3})^2$$

Step 6: Simplify the Denominator

Applying the difference of squares formula, we get:

$$2^2 - (\sqrt{3})^2 = 4 - 3 = 1$$

Step 7: Simplify the Numerator

Now, let's simplify the numerator:

$$10(2+\sqrt{3}) = 20 + 10\sqrt{3}$$

Step 8: Write the Final Simplified Fraction

Combining the simplified numerator and denominator, we get:

$$\frac{20 + 10\sqrt{3}}{1}$$

Conclusion

In this article, we have simplified and rationalized the denominator of the given fraction $\frac{10}{2-\sqrt{3}}$. We have followed the necessary steps, provided examples, and offered a comprehensive guide to help you master this essential math concept. By understanding the concept of rationalizing the denominator and applying the necessary steps, you can simplify complex fractions and make them easier to work with.

Frequently Asked Questions

• What is rationalizing the denominator? Rationalizing the denominator involves removing any radical expressions from the denominator of a fraction.
• Why is rationalizing the denominator important? Rationalizing the denominator is important because it simplifies complex fractions and makes them easier to work with.
• How do I rationalize the denominator of a fraction? To rationalize the denominator, multiply both the numerator and the denominator by the conjugate of the denominator.

Examples and Practice Problems

• Simplify and rationalize the denominator of the fraction $\frac{5}{3-\sqrt{2}}$.
• Simplify and rationalize the denominator of the fraction $\frac{7}{2+\sqrt{5}}$.

Additional Resources

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

Final Thoughts

Rationalizing the denominator is a crucial step in simplifying complex fractions. By understanding the concept and applying the necessary steps, you can simplify fractions and make them easier to work with. Remember to choose the conjugate of the denominator, multiply the fraction, simplify the expression, and apply the difference of squares formula. With practice and patience, you will become proficient in rationalizing the denominator and simplifying complex fractions.

Introduction

Rationalizing the denominator is a crucial step in simplifying complex fractions. In our previous article, we explored the process of simplifying and rationalizing the denominator of the fraction $\frac{10}{2-\sqrt{3}}$. In this article, we will address some of the most frequently asked questions about rationalizing the denominator.

Q&A

Q: What is rationalizing the denominator?

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

Q: Why is rationalizing the denominator important?

A: Rationalizing the denominator is important because it simplifies complex fractions and makes them easier to work with.

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

A: The conjugate of a denominator is the expression obtained by changing the sign of the radical in the denominator.

Q: How do I find the conjugate of a denominator?

A: To find the conjugate of a denominator, simply change the sign of the radical in the denominator.

Q: What is the difference of squares formula?

A: The difference of squares formula is $(a-b)(a+b) = a^2 - b^2$.

Q: How do I apply the difference of squares formula?

A: To apply the difference of squares formula, multiply the two binomials together and simplify the expression.

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

A: Yes, you can rationalize the denominator of a fraction with a negative exponent by following the same steps as before.

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

A: Yes, you can rationalize the denominator of a fraction with a complex number in the denominator by following the same steps as before.

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

A: To rationalize the denominator of a fraction with a variable in the denominator, follow the same steps as before and use the conjugate of the variable.

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

A: Yes, you can rationalize the denominator of a fraction with a radical in the numerator by following the same steps as before.

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 by following the same steps as before.

Q: How do I check if a fraction is rationalized?

A: To check if a fraction is rationalized, look for a radical expression in the denominator. If there is no radical expression, then the fraction is rationalized.

Examples and Practice Problems

• Rationalize the denominator of the fraction $\frac{5}{3-\sqrt{2}}$.
• Rationalize the denominator of the fraction $\frac{7}{2+\sqrt{5}}$.
• Rationalize the denominator of the fraction $\frac{10}{2-\sqrt{3}}$.
• Rationalize the denominator of the fraction $\frac{15}{3+\sqrt{2}}$.
• Rationalize the denominator of the fraction $\frac{20}{2-\sqrt{5}}$.

Additional Resources

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

Final Thoughts

Rationalizing the denominator is a crucial step in simplifying complex fractions. By understanding the concept and applying the necessary steps, you can simplify fractions and make them easier to work with. Remember to choose the conjugate of the denominator, multiply the fraction, simplify the expression, and apply the difference of squares formula. With practice and patience, you will become proficient in rationalizing the denominator and simplifying complex fractions.