Which Choice Is Equivalent To The Fraction Below When $x$ Is An Appropriate Value?Hint: Rationalize The Denominator And Simplify.$\frac{5}{5+\sqrt{10 X}}$A. $\frac{5-\sqrt{10 X}}{5-2 X}$ B. $\frac{5-\sqrt{10 X}}{25-10

Introduction

Rationalizing the denominator is a crucial step in simplifying fractions, especially when dealing with expressions involving square roots. In this article, we will explore the process of rationalizing the denominator and simplifying the given fraction $\frac{5}{5+\sqrt{10 x}}$. We will also examine the two options provided, A and B, and determine which one is equivalent to the given fraction.

Understanding Rationalizing the Denominator

Rationalizing the denominator involves eliminating any square roots from the denominator of a fraction. This is achieved by multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of a binomial expression $a + b$ is $a - b$. In the case of the given fraction, the denominator is $5 + \sqrt{10 x}$, so the conjugate is $5 - \sqrt{10 x}$.

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, which is $5 - \sqrt{10 x}$.

$$\frac{5}{5+\sqrt{10 x}} \cdot \frac{5-\sqrt{10 x}}{5-\sqrt{10 x}}$$

Step 2: Simplify the Expression

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

$$\frac{5(5-\sqrt{10 x})}{(5+\sqrt{10 x})(5-\sqrt{10 x})}$$

Using the difference of squares formula, $(a+b)(a-b) = a^2 - b^2$, we can simplify the denominator.

$$\frac{5(5-\sqrt{10 x})}{25 - 10 x}$$

Step 3: Simplify the Numerator

Now, we simplify the numerator by multiplying the terms.

$$\frac{25 - 5\sqrt{10 x}}{25 - 10 x}$$

Comparing with the Options

Now that we have simplified the fraction, we can compare it with the two options provided.

Option A: $\frac{5-\sqrt{10 x}}{5-2 x}$

To determine if this option is equivalent to the simplified fraction, we need to rationalize the denominator of this option as well.

$$\frac{5-\sqrt{10 x}}{5-2 x} \cdot \frac{5+2 x}{5+2 x}$$

Simplifying the expression, we get:

$$\frac{(5-\sqrt{10 x})(5+2 x)}{25 - 10 x}$$

Expanding the numerator, we get:

$$\frac{25 + 10 x - 5\sqrt{10 x} - 2 x\sqrt{10 x}}{25 - 10 x}$$

Rearranging the terms, we get:

$$\frac{25 - 5\sqrt{10 x} + (10 x - 2 x\sqrt{10 x})}{25 - 10 x}$$

Factoring out the common term, we get:

$$\frac{25 - 5\sqrt{10 x} + 2 x(5 - \sqrt{10 x})}{25 - 10 x}$$

Simplifying further, we get:

$$\frac{25 - 5\sqrt{10 x} + 10 x - 2 x\sqrt{10 x}}{25 - 10 x}$$

Rearranging the terms, we get:

$$\frac{(25 - 10 x) - 5\sqrt{10 x} + 2 x\sqrt{10 x}}{25 - 10 x}$$

Simplifying further, we get:

$$\frac{25 - 10 x - 3\sqrt{10 x}}{25 - 10 x}$$

Cancelling out the common term, we get:

$$\frac{25 - 10 x}{25 - 10 x} - \frac{3\sqrt{10 x}}{25 - 10 x}$$

Simplifying further, we get:

$$1 - \frac{3\sqrt{10 x}}{25 - 10 x}$$

This is not equivalent to the simplified fraction.

Option B: $\frac{5-\sqrt{10 x}}{25-10 x}$

This option is already in the simplified form, and it is equivalent to the simplified fraction.

Conclusion

In conclusion, the correct answer is Option B: $\frac{5-\sqrt{10 x}}{25-10 x}$. This option is equivalent to the simplified fraction, while Option A is not. By rationalizing the denominator and simplifying the expression, we can determine which option is correct.

Final Answer

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Introduction

Rationalizing the denominator is a crucial step in simplifying fractions, especially when dealing with expressions involving square roots. In this article, we will explore the process of rationalizing the denominator and simplifying the given fraction $\frac{5}{5+\sqrt{10 x}}$. We will also examine the two options provided, A and B, and determine which one is equivalent to the given fraction.

Q&A

Q: What is rationalizing the denominator?

A: Rationalizing the denominator involves eliminating any square roots from the denominator of a fraction. This is achieved by multiplying 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 $a + b$ is $a - b$. In the case of the given fraction, the denominator is $5 + \sqrt{10 x}$, so the conjugate is $5 - \sqrt{10 x}$.

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.

Q: What is the difference of squares formula?

A: The difference of squares formula is $(a+b)(a-b) = a^2 - b^2$. This formula can be used to simplify the denominator of a fraction.

Q: How do I simplify the numerator of a fraction?

A: To simplify the numerator, you need to multiply the terms together.

Q: How do I compare two fractions?

A: To compare two fractions, you need to simplify them and then compare the numerators and denominators.

Q: What is the final answer?

A: The final answer is $\boxed{B}$.

Common Mistakes

• Not rationalizing the denominator
• Not simplifying the numerator
• Not comparing the fractions correctly

Tips and Tricks

• Always rationalize the denominator before simplifying the fraction
• Use the difference of squares formula to simplify the denominator
• Simplify the numerator by multiplying the terms together
• Compare the fractions by simplifying them and then comparing the numerators and denominators

Conclusion

In conclusion, rationalizing the denominator is a crucial step in simplifying fractions, especially when dealing with expressions involving square roots. By following the steps outlined in this article, you can simplify fractions and determine which option is correct. Remember to always rationalize the denominator, simplify the numerator, and compare the fractions correctly.

Final Answer

The final answer is $\boxed{B}$.