$\frac{6}{-5 \sqrt{3}+1}$, Expressed With A Rational Denominator, Is:A. $\frac{-15 \sqrt{3}+3}{38}$ B. $\frac{-15 \sqrt{3}+3}{37}$ C. $\frac{-15 \sqrt{3}-3}{38}$ D. $\frac{-15 \sqrt{3}-3}{37}$

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Introduction

Rationalizing the denominator is a crucial step in simplifying complex fractions, especially when dealing with square roots or other irrational numbers. In this article, we will delve into the process of rationalizing the denominator and apply it to the given expression: 6−53+1\frac{6}{-5 \sqrt{3}+1}. We will explore the different options and determine the correct answer.

Understanding Rationalizing the Denominator

Rationalizing the denominator involves multiplying both the numerator and the denominator by a specific value to eliminate any radical expressions in the denominator. This process is essential in simplifying complex fractions and making them easier to work with.

Why Rationalize the Denominator?

Rationalizing the denominator is necessary for several reasons:

  • It eliminates any radical expressions in the denominator, making the fraction easier to work with.
  • It allows for the simplification of complex fractions.
  • It enables the comparison of fractions with different denominators.

Applying Rationalizing the Denominator to the Given Expression

To rationalize the denominator of the given expression, we need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of a binomial expression a+ba + b is a−ba - b. In this case, the conjugate of −53+1-5 \sqrt{3} + 1 is −53−1-5 \sqrt{3} - 1.

Step 1: Multiply the Numerator and Denominator by the Conjugate

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

6−53+1⋅−53−1−53−1\frac{6}{-5 \sqrt{3}+1} \cdot \frac{-5 \sqrt{3} - 1}{-5 \sqrt{3} - 1}

Step 2: Simplify the Expression

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

6(−53−1)(−53+1)(−53−1)\frac{6(-5 \sqrt{3} - 1)}{(-5 \sqrt{3} + 1)(-5 \sqrt{3} - 1)}

−303−625⋅3−12\frac{-30 \sqrt{3} - 6}{25 \cdot 3 - 1^2}

−303−674\frac{-30 \sqrt{3} - 6}{74}

Step 3: Simplify the Fraction

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor (GCD). In this case, the GCD of −303−6-30 \sqrt{3} - 6 and 7474 is 22. Dividing both the numerator and the denominator by 22 gives us:

−153−337\frac{-15 \sqrt{3} - 3}{37}

Conclusion

In conclusion, the correct answer is −153−337\frac{-15 \sqrt{3} - 3}{37}. This is the result of rationalizing the denominator of the given expression and simplifying the fraction.

Final Answer

The final answer is −153−337\boxed{\frac{-15 \sqrt{3} - 3}{37}}.

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Introduction

Rationalizing the denominator is a crucial step in simplifying complex fractions, especially when dealing with square roots or other irrational numbers. In this article, we will delve into the process of rationalizing the denominator and provide answers to frequently asked questions.

Q&A: Rationalizing the Denominator

Q: What is rationalizing the denominator?

A: Rationalizing the denominator involves multiplying both the numerator and the denominator by a specific value to eliminate any radical expressions in the denominator.

Q: Why is rationalizing the denominator necessary?

A: Rationalizing the denominator is necessary to eliminate any radical expressions in the denominator, making the fraction easier to work with. It also allows for the simplification of complex fractions and enables the comparison of fractions with different denominators.

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

A: To rationalize the denominator, you need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of a binomial expression a+ba + b is a−ba - b.

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 −53+1-5 \sqrt{3} + 1 is −53−1-5 \sqrt{3} - 1.

Q: How do I simplify a fraction after rationalizing the denominator?

A: To simplify a fraction after rationalizing the denominator, you can divide both the numerator and the denominator by their greatest common divisor (GCD).

Q: What is the greatest common divisor (GCD)?

A: The greatest common divisor (GCD) of two numbers is the largest number that divides both numbers without leaving a remainder.

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. However, you need to follow the same steps as rationalizing the denominator of a fraction with a positive exponent.

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. However, you need to follow the same steps as rationalizing the denominator of a fraction with a radical in the denominator.

Examples of Rationalizing the Denominator

Example 1: Rationalizing the Denominator of a Fraction with a Radical in the Denominator

Rationalize the denominator of the fraction: 6−53+1\frac{6}{-5 \sqrt{3}+1}

Solution:

6−53+1⋅−53−1−53−1\frac{6}{-5 \sqrt{3}+1} \cdot \frac{-5 \sqrt{3} - 1}{-5 \sqrt{3} - 1}

6(−53−1)(−53+1)(−53−1)\frac{6(-5 \sqrt{3} - 1)}{(-5 \sqrt{3} + 1)(-5 \sqrt{3} - 1)}

−303−625⋅3−12\frac{-30 \sqrt{3} - 6}{25 \cdot 3 - 1^2}

−303−674\frac{-30 \sqrt{3} - 6}{74}

Example 2: Rationalizing the Denominator of a Fraction with a Negative Exponent

Rationalize the denominator of the fraction: 6−53−1+1\frac{6}{-5 \sqrt{3}^{-1}+1}

Solution:

6−53−1+1⋅−53−1−53−1\frac{6}{-5 \sqrt{3}^{-1}+1} \cdot \frac{-5 \sqrt{3} - 1}{-5 \sqrt{3} - 1}

6(−53−1)(−53−1+1)(−53−1)\frac{6(-5 \sqrt{3} - 1)}{(-5 \sqrt{3}^{-1} + 1)(-5 \sqrt{3} - 1)}

−303−625⋅3−1−12\frac{-30 \sqrt{3} - 6}{25 \cdot 3^{-1} - 1^2}

−303−6253−1\frac{-30 \sqrt{3} - 6}{\frac{25}{3} - 1}

−303−6223\frac{-30 \sqrt{3} - 6}{\frac{22}{3}}

Conclusion

In conclusion, rationalizing the denominator is a crucial step in simplifying complex fractions, especially when dealing with square roots or other irrational numbers. By following the steps outlined in this article, you can rationalize the denominator of any fraction and simplify it to its simplest form.

Final Answer

The final answer is −153−337\boxed{\frac{-15 \sqrt{3} - 3}{37}}.