Which Of The Following Is Equivalent To The Fraction Below After Rationalizing The Denominator And Simplifying? 10 2 \frac{10}{\sqrt{2}} 2 10 A. 5 2 5 \sqrt{2} 5 2 B. 5 C. 2 \sqrt{2} 2 D. 5 \sqrt{5} 5

Mar 11, 2025 by ADMIN 212 views

**Rationalizing the Denominator and Simplifying Fractions**

Understanding Rationalizing the Denominator

Rationalizing the denominator is a process used to eliminate any radicals from the denominator of a fraction. This is particularly important when dealing with fractions that have square roots or other radicals in the denominator. The goal of rationalizing the denominator is to simplify the fraction and make it easier to work with.

The Process of Rationalizing the Denominator

To rationalize the denominator, we need to multiply both the numerator and the denominator by a value that will eliminate the radical from the denominator. This value is typically the conjugate of the denominator. The conjugate of a binomial expression is found by changing the sign of the second term. For example, the conjugate of $a + b$ is $a - b$ .

Rationalizing the Denominator of a Fraction

Let's consider the fraction $\frac{10}{\sqrt{2}}$ . To rationalize the denominator, we need to multiply both the numerator and the denominator by the conjugate of the denominator, which is $\sqrt{2}$ . This will eliminate the radical from the denominator.

$\frac{10}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{10\sqrt{2}}{2}$

Simplifying the Fraction

Now that we have rationalized the denominator, we can simplify the fraction by dividing the numerator and the denominator by their greatest common divisor (GCD). In this case, the GCD of $10\sqrt{2}$ and $2$ is $2$ . Dividing both the numerator and the denominator by $2$ gives us:

$\frac{10\sqrt{2}}{2} = 5\sqrt{2}$

Comparing the Simplified Fraction to the Answer Choices

Now that we have simplified the fraction, we can compare it to the answer choices to see which one is equivalent.

A. $5 \sqrt{2}$ B. 5 C. $\sqrt{2}$ D. $\sqrt{5}$

The simplified fraction $5\sqrt{2}$ is equivalent to answer choice A.

Conclusion

In conclusion, rationalizing the denominator and simplifying a fraction involves multiplying both the numerator and the denominator by the conjugate of the denominator to eliminate any radicals from the denominator. This process can be used to simplify fractions and make them easier to work with. In this case, the simplified fraction $5\sqrt{2}$ is equivalent to answer choice A.

Key Takeaways

Rationalizing the denominator involves multiplying both the numerator and the denominator by the conjugate of the denominator.
The conjugate of a binomial expression is found by changing the sign of the second term.
Simplifying a fraction involves dividing the numerator and the denominator by their greatest common divisor (GCD).
Rationalizing the denominator and simplifying a fraction can be used to make fractions easier to work with.

Additional Examples

Rationalize the denominator of the fraction $\frac{3}{\sqrt{3}}$ and simplify.
Rationalize the denominator of the fraction $\frac{4}{\sqrt{5}}$ and simplify.

Answer Key

A. $5 \sqrt{2}$ B. 5 C. $\sqrt{2}$ D. $\sqrt{5}$

Final Thoughts

Q: What is rationalizing the denominator?

A: Rationalizing the denominator is a process used to eliminate any radicals from the denominator of a fraction. This is particularly important when dealing with fractions that have square roots or other radicals in the denominator.

Q: Why is rationalizing the denominator important?

A: Rationalizing the denominator is important because it allows us to simplify fractions and make them easier to work with. When a fraction has a radical in the denominator, it can be difficult to perform operations such as addition and subtraction. By rationalizing the denominator, we can eliminate the radical and make the fraction easier to work with.

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 is found by changing the sign of the second term.

Q: What is the conjugate of a binomial expression?

A: The conjugate of a binomial expression is found by changing the sign of the second term. For example, the conjugate of $a + b$ is $a - b$ .

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

A: To simplify a fraction after rationalizing the denominator, you need to divide 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) is the largest number that divides both the numerator and the denominator of a fraction.

Q: Can you give an example of rationalizing the denominator and simplifying a fraction?

A: Let's consider the fraction $\frac{10}{\sqrt{2}}$ . To rationalize the denominator, we need to multiply both the numerator and the denominator by the conjugate of the denominator, which is $\sqrt{2}$ . This will eliminate the radical from the denominator.

$\frac{10}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{10\sqrt{2}}{2}$

Now that we have rationalized the denominator, we can simplify the fraction by dividing the numerator and the denominator by their GCD, which is 2.

$\frac{10\sqrt{2}}{2} = 5\sqrt{2}$

Q: What are some common mistakes to avoid when rationalizing the denominator?

A: Some common mistakes to avoid when rationalizing the denominator include:

Not multiplying both the numerator and the denominator by the conjugate of the denominator.
Not simplifying the fraction after rationalizing the denominator.
Not using the correct conjugate of the denominator.

Q: Can you give some examples of rationalizing the denominator and simplifying fractions?

A: Here are some examples of rationalizing the denominator and simplifying fractions:

Rationalize the denominator of the fraction $\frac{3}{\sqrt{3}}$ and simplify.
Rationalize the denominator of the fraction $\frac{4}{\sqrt{5}}$ and simplify.
Rationalize the denominator of the fraction $\frac{6}{\sqrt{6}}$ and simplify.

Q: What are some real-world applications of rationalizing the denominator and simplifying fractions?

A: Rationalizing the denominator and simplifying fractions has many real-world applications, including:

Calculating probabilities and statistics.
Working with finance and economics.
Solving problems in physics and engineering.

Q: Can you give some tips for mastering rationalizing the denominator and simplifying fractions?

A: Here are some tips for mastering rationalizing the denominator and simplifying fractions:

Practice, practice, practice! The more you practice, the more comfortable you will become with rationalizing the denominator and simplifying fractions.
Start with simple examples and gradually move on to more complex ones.
Use online resources and study guides to help you understand the concept.
Ask your teacher or tutor for help if you are struggling.

Conclusion

Rationalizing the denominator and simplifying fractions is an important concept in mathematics that can be used to make fractions easier to work with. By following the steps outlined in this article, you can simplify fractions and make them more manageable. Remember to practice regularly and seek help if you are struggling.