Simplify The Expression:$\[ \sqrt{\frac{3}{10}} \\]Options:A. \[$\frac{\sqrt{3}}{10}\$\]B. \[$\frac{\sqrt{3}}{5}\$\]C. \[$\frac{\sqrt{30}}{10}\$\]D. \[$\frac{\sqrt{15}}{5}\$\]

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Understanding the Problem

When dealing with square roots of fractions, it's essential to simplify the expression by finding the square root of the numerator and the denominator separately. In this case, we have the expression $\sqrt{\frac{3}{10}}$, and we need to simplify it.

Simplifying the Expression

To simplify the expression, we can start by finding the square root of the numerator and the denominator separately. The square root of 3 is $\sqrt{3}$, and the square root of 10 is $\sqrt{10}$. However, we can simplify the square root of 10 further by finding its prime factors.

Prime Factorization of 10

The prime factorization of 10 is $2 \times 5$. Therefore, we can rewrite the square root of 10 as $\sqrt{2 \times 5}$.

Simplifying the Square Root of 10

Using the property of square roots that allows us to split the square root of a product into the product of the square roots, we can simplify the square root of 10 as follows:

$$\sqrt{2 \times 5} = \sqrt{2} \times \sqrt{5}$$

Simplifying the Expression

Now that we have simplified the square root of 10, we can rewrite the original expression as follows:

$$\sqrt{\frac{3}{10}} = \frac{\sqrt{3}}{\sqrt{2} \times \sqrt{5}}$$

Rationalizing the Denominator

To rationalize the denominator, we can multiply the numerator and the denominator by the conjugate of the denominator, which is $\sqrt{2} \times \sqrt{5}$. This will eliminate the square root in the denominator.

Rationalizing the Denominator

Multiplying the numerator and the denominator by $\sqrt{2} \times \sqrt{5}$, we get:

$$\frac{\sqrt{3}}{\sqrt{2} \times \sqrt{5}} \times \frac{\sqrt{2} \times \sqrt{5}}{\sqrt{2} \times \sqrt{5}} = \frac{\sqrt{3} \times \sqrt{2} \times \sqrt{5}}{(\sqrt{2} \times \sqrt{5})^2}$$

Simplifying the Expression

Using the property of exponents that states $(a^m)^n = a^{m \times n}$, we can simplify the denominator as follows:

$$(\sqrt{2} \times \sqrt{5})^2 = (\sqrt{2})^2 \times (\sqrt{5})^2 = 2 \times 5 = 10$$

Simplifying the Expression

Now that we have simplified the denominator, we can rewrite the expression as follows:

$$\frac{\sqrt{3} \times \sqrt{2} \times \sqrt{5}}{10} = \frac{\sqrt{30}}{10}$$

Conclusion

Therefore, the simplified expression is $\frac{\sqrt{30}}{10}$.

Final Answer

The final answer is:

$\boxed{\frac{\sqrt{30}}{10}}$

Discussion

The correct answer is C. $\frac{\sqrt{30}}{10}$. This is because we simplified the expression by finding the square root of the numerator and the denominator separately, and then rationalized the denominator to eliminate the square root in the denominator.

Common Mistakes

Some common mistakes that students make when simplifying expressions with square roots include:

• Not simplifying the square root of the numerator and the denominator separately
• Not rationalizing the denominator
• Not using the property of exponents to simplify the denominator

Tips for Simplifying Expressions with Square Roots

To simplify expressions with square roots, follow these tips:

• Simplify the square root of the numerator and the denominator separately
• Rationalize the denominator to eliminate the square root in the denominator
• Use the property of exponents to simplify the denominator

Practice Problems

Practice simplifying expressions with square roots by trying the following problems:

• $\sqrt{\frac{4}{9}}$
• $\sqrt{\frac{9}{16}}$
• $\sqrt{\frac{16}{25}}$

Solutions

The solutions to the practice problems are:

• $\frac{2}{3}$
• $\frac{3}{4}$
• $\frac{4}{5}$
  

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Understanding the Problem

When dealing with square roots of fractions, it's essential to simplify the expression by finding the square root of the numerator and the denominator separately. In this case, we have the expression $\sqrt{\frac{3}{10}}$, and we need to simplify it.

Q&A

Q: What is the first step in simplifying the expression $\sqrt{\frac{3}{10}}$?

A: The first step in simplifying the expression $\sqrt{\frac{3}{10}}$ is to find the square root of the numerator and the denominator separately.

Q: How do I find the square root of a fraction?

A: To find the square root of a fraction, you can find the square root of the numerator and the denominator separately. For example, the square root of $\frac{3}{10}$ is equal to $\frac{\sqrt{3}}{\sqrt{10}}$.

Q: Can I simplify the square root of 10 further?

A: Yes, you can simplify the square root of 10 further by finding its prime factors. The prime factorization of 10 is $2 \times 5$, so we can rewrite the square root of 10 as $\sqrt{2 \times 5}$.

Q: How do I simplify the square root of 10?

A: To simplify the square root of 10, you can use the property of square roots that allows you to split the square root of a product into the product of the square roots. Therefore, we can simplify the square root of 10 as follows:

$$\sqrt{2 \times 5} = \sqrt{2} \times \sqrt{5}$$

Q: How do I rationalize the denominator?

A: To rationalize the denominator, you can multiply the numerator and the denominator by the conjugate of the denominator. In this case, the conjugate of the denominator is $\sqrt{2} \times \sqrt{5}$.

Q: What is the final simplified expression?

A: The final simplified expression is $\frac{\sqrt{30}}{10}$.

Q: Why is the correct answer C. $\frac{\sqrt{30}}{10}$?

A: The correct answer is C. $\frac{\sqrt{30}}{10}$ because we simplified the expression by finding the square root of the numerator and the denominator separately, and then rationalized the denominator to eliminate the square root in the denominator.

Q: What are some common mistakes that students make when simplifying expressions with square roots?

A: Some common mistakes that students make when simplifying expressions with square roots include:

• Not simplifying the square root of the numerator and the denominator separately
• Not rationalizing the denominator
• Not using the property of exponents to simplify the denominator

Q: What are some tips for simplifying expressions with square roots?

A: To simplify expressions with square roots, follow these tips:

• Simplify the square root of the numerator and the denominator separately
• Rationalize the denominator to eliminate the square root in the denominator
• Use the property of exponents to simplify the denominator

Practice Problems

Practice simplifying expressions with square roots by trying the following problems:

• $\sqrt{\frac{4}{9}}$
• $\sqrt{\frac{9}{16}}$
• $\sqrt{\frac{16}{25}}$

Solutions

The solutions to the practice problems are:

• $\frac{2}{3}$
• $\frac{3}{4}$
• $\frac{4}{5}$

Conclusion

Simplifying expressions with square roots requires careful attention to detail and a thorough understanding of the properties of square roots. By following the steps outlined in this article, you can simplify expressions with square roots and arrive at the correct solution.

Final Answer

The final answer is:

$\boxed{\frac{\sqrt{30}}{10}}$