How Would The Fraction 7 1 − 5 \frac{7}{1-\sqrt{5}} 1 − 5 ​ 7 ​ Be Rewritten If Its Denominator Is Rationalized Using The Difference Of Squares?A. − 7 + 7 5 4 -\frac{7+7\sqrt{5}}{4} − 4 7 + 7 5 ​ ​ B. − 7 + 7 5 6 -\frac{7+7\sqrt{5}}{6} − 6 7 + 7 5 ​ ​ C. 7 − 7 5 6 \frac{7-7\sqrt{5}}{6} 6 7 − 7 5 ​ ​ D.

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Rationalizing the Denominator Using the Difference of Squares

Rationalizing the denominator of a fraction is a process that eliminates any radical expressions in the denominator. This is particularly useful when dealing with fractions that have irrational numbers in the denominator. In this case, we are given the fraction $\frac{7}{1-\sqrt{5}}$ and we are asked to rewrite it with a rationalized denominator using the difference of squares.

Understanding the Difference of Squares

The difference of squares is a fundamental algebraic identity that states: $a^2 - b^2 = (a - b)(a + b)$. This identity 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

To rationalize the denominator of the given fraction, we need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of $1-\sqrt{5}$ is $1+\sqrt{5}$. Therefore, we multiply both the numerator and the denominator by $1+\sqrt{5}$.

$\frac{7}{1-\sqrt{5}} \cdot \frac{1+\sqrt{5}}{1+\sqrt{5}}$

Applying the Difference of Squares

Now, we can apply the difference of squares identity to simplify the expression.

$\frac{7(1+\sqrt{5})}{(1-\sqrt{5})(1+\sqrt{5})}$

Using the difference of squares identity, we can simplify the denominator as follows:

$(1-\sqrt{5})(1+\sqrt{5}) = 1^2 - (\sqrt{5})^2 = 1 - 5 = -4$

Therefore, the expression becomes:

$\frac{7(1+\sqrt{5})}{-4}$

Simplifying the Expression

Now, we can simplify the expression by distributing the 7 to the terms inside the parentheses.

$\frac{7 + 7\sqrt{5}}{-4}$

Final Answer

The final answer is $\frac{7 + 7\sqrt{5}}{-4}$. However, we need to express this in the form of the given options. To do this, we can multiply both the numerator and the denominator by -1 to get:

$-\frac{7 + 7\sqrt{5}}{4}$

This is the correct answer.

Conclusion

In this article, we have shown how to rationalize the denominator of the fraction $\frac{7}{1-\sqrt{5}}$ using the difference of squares. We have applied the difference of squares identity to simplify the expression and have obtained the final answer in the form of one of the given options.

Final Answer

The final answer is $-\frac{7+7\sqrt{5}}{4}$.

Frequently Asked Questions

In this article, we will answer some frequently asked questions related to rationalizing the denominator using the difference of squares.

Q: What is the difference of squares?

A: The difference of squares is a fundamental algebraic identity that states: $a^2 - b^2 = (a - b)(a + b)$. This identity can be used to rationalize the denominator of a fraction by multiplying both the numerator and the denominator by the conjugate of the denominator.

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

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

Q: What is the conjugate of a binomial expression?

A: The conjugate of a binomial expression $a-b$ is $a+b$. For example, the conjugate of $1-\sqrt{5}$ is $1+\sqrt{5}$.

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

A: To apply the difference of squares identity, you need to multiply both the numerator and the denominator by the conjugate of the denominator. Then, you can simplify the expression using the difference of squares identity.

Q: What is the difference of squares identity?

A: The difference of squares identity is: $a^2 - b^2 = (a - b)(a + b)$. This identity can be used to simplify expressions that involve the difference of two squares.

Q: Can I use the difference of squares identity to rationalize the denominator of a fraction with a denominator that is not a binomial expression?

A: No, the difference of squares identity can only be used to rationalize the denominator of a fraction with a denominator that is a binomial expression.

Q: How do I know if a fraction has a rationalized denominator?

A: A fraction has a rationalized denominator if the denominator is a rational number (i.e., a number that can be expressed as the ratio of two integers).

Q: Can I rationalize the denominator of a fraction with a denominator that is a rational number?

A: No, a fraction with a denominator that is a rational number already has a rationalized denominator.

Examples and Practice Problems

In this section, we will provide some examples and practice problems to help you understand how to rationalize the denominator using the difference of squares.

Example 1

Rationalize the denominator of the fraction $\frac{3}{1+\sqrt{2}}$.

Solution:

$\frac{3}{1+\sqrt{2}} \cdot \frac{1-\sqrt{2}}{1-\sqrt{2}}$

Applying the difference of squares identity, we get:

$\frac{3(1-\sqrt{2})}{(1+\sqrt{2})(1-\sqrt{2})}$

Simplifying the expression, we get:

$\frac{3-3\sqrt{2}}{1-2} = \frac{3-3\sqrt{2}}{-1} = -3+3\sqrt{2}$

Example 2

Rationalize the denominator of the fraction $\frac{4}{\sqrt{3}-1}$.

Solution:

$\frac{4}{\sqrt{3}-1} \cdot \frac{\sqrt{3}+1}{\sqrt{3}+1}$

Applying the difference of squares identity, we get:

$\frac{4(\sqrt{3}+1)}{(\sqrt{3}-1)(\sqrt{3}+1)}$

Simplifying the expression, we get:

$\frac{4\sqrt{3}+4}{3-1} = \frac{4\sqrt{3}+4}{2} = 2\sqrt{3}+2$

Conclusion

In this article, we have answered some frequently asked questions related to rationalizing the denominator using the difference of squares. We have also provided some examples and practice problems to help you understand how to rationalize the denominator using the difference of squares.

Final Answer

The final answer is $-\frac{7+7\sqrt{5}}{4}$.