How Would The Fraction $\frac{5}{1-\sqrt{3}}$ Be Rewritten If Its Denominator Is Rationalized Using The Difference Of Squares?A. $\frac{5-5 \sqrt{5}}{-2}$ B. $\frac{5+5 \sqrt{3}}{2}$ C. $\frac{5-5 \sqrt{3}}{2}$ D.

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**Rationalizing the Denominator of a Fraction: A Step-by-Step Guide** ===========================================================

Introduction

Rationalizing the denominator of a fraction is an essential skill in algebra and mathematics. It involves removing any radicals from the denominator of a fraction, making it easier to work with and simplifying complex expressions. In this article, we will explore how to rationalize the denominator of a fraction using the difference of squares, and apply this technique to the given fraction 51−3\frac{5}{1-\sqrt{3}}.

What is Rationalizing the Denominator?

Rationalizing the denominator of a fraction involves multiplying both the numerator and the denominator by a cleverly chosen expression, such that the denominator becomes a rational number (i.e., a number without any radicals). This process is essential in algebra, as it allows us to simplify complex expressions and perform operations like addition, subtraction, multiplication, and division.

The Difference of Squares Formula

The difference of squares formula is a fundamental concept in algebra, which states that:

a2−b2=(a+b)(a−b)a^2 - b^2 = (a + b)(a - b)

This formula can be used to rationalize the denominator of a fraction by multiplying both the numerator and the denominator by the conjugate of the denominator.

Rationalizing the Denominator of 51−3\frac{5}{1-\sqrt{3}}

To rationalize the denominator of 51−3\frac{5}{1-\sqrt{3}}, we will use the difference of squares formula. The conjugate of the denominator 1−31-\sqrt{3} is 1+31+\sqrt{3}. We will multiply both the numerator and the denominator by this conjugate:

51−3⋅1+31+3\frac{5}{1-\sqrt{3}} \cdot \frac{1+\sqrt{3}}{1+\sqrt{3}}

Using the difference of squares formula, we can simplify the denominator:

(1−3)(1+3)=12−(3)2=1−3=−2(1-\sqrt{3})(1+\sqrt{3}) = 1^2 - (\sqrt{3})^2 = 1 - 3 = -2

So, the fraction becomes:

5(1+3)−2\frac{5(1+\sqrt{3})}{-2}

Now, we can simplify the numerator:

5(1+3)=5+535(1+\sqrt{3}) = 5 + 5\sqrt{3}

Therefore, the rationalized fraction is:

5+53−2\frac{5+5\sqrt{3}}{-2}

Conclusion

In this article, we have explored how to rationalize the denominator of a fraction using the difference of squares formula. We have applied this technique to the given fraction 51−3\frac{5}{1-\sqrt{3}} and obtained the rationalized fraction 5+53−2\frac{5+5\sqrt{3}}{-2}. This process is essential in algebra, as it allows us to simplify complex expressions and perform operations like addition, subtraction, multiplication, and division.

Answer

The correct answer is:

A. 5−55−2\frac{5-5 \sqrt{5}}{-2}

However, this is not the correct answer. The correct answer is:

B. 5+532\frac{5+5 \sqrt{3}}{2}