Applying the Property to the Given Quotient

Oct 13, 2025 by miodeya.com 77 views

Understanding the Property of Radicals

In mathematics, radicals are a fundamental concept that deals with the square root of a number. The property ${$ \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$}$ is a crucial one to understand when simplifying radical expressions. This property states that the quotient of two square roots is equal to the square root of the quotient of the numbers inside the square roots.

Applying the Property to the Given Quotient

Now, let's apply this property to the given quotient ${$ \frac{\sqrt{30}}{\sqrt{5}}$}$. Using the property, we can rewrite this expression as ${$ \sqrt{\frac{30}{5}}$}$. Simplifying the fraction inside the square root, we get ${$ \sqrt{6}$}$.

Evaluating the Answer Choices

Now that we have simplified the given quotient, let's evaluate the answer choices to determine which one is equivalent to ${$ \sqrt{6}$}$.

A. 36: This is not equivalent to ${$ \sqrt{6}$}$ because 36 is a perfect square, whereas ${$ \sqrt{6}$}$ is an irrational number.
B. -6: This is also not equivalent to ${$ \sqrt{6}$}$ because -6 is a negative number, whereas ${$ \sqrt{6}$}$ is a positive number.
**C. ${$ -\sqrt6}$}$** This is equivalent to ${$ \sqrt{6$}$ because the negative sign is outside the square root, making it a negative number.
D. 6: This is not equivalent to ${$ \sqrt{6}$}$ because 6 is a perfect square, whereas ${$ \sqrt{6}$}$ is an irrational number.
**E. ${$ \sqrt6}$}$** This is equivalent to ${$ \sqrt{6$}$ because it is the same expression.

Conclusion

In conclusion, the correct answer is C. ${$ -\sqrt{6}$}$ because it is equivalent to the simplified expression ${$ \sqrt{6}$}$. This problem demonstrates the importance of understanding the property of radicals and applying it to simplify radical expressions.

Additional Tips and Examples

Here are some additional tips and examples to help you understand the property of radicals and simplify radical expressions:

Tip 1: When simplifying radical expressions, always look for perfect squares inside the square root.
Tip 2: When simplifying radical expressions, always simplify the fraction inside the square root.
Example 1: Simplify the expression ${$ \sqrt{\frac{24}{9}}$}$. Using the property, we can rewrite this expression as ${$ \sqrt{\frac{4}{3}}$}$. Simplifying the fraction inside the square root, we get ${$ \frac{2}{\sqrt{3}}$}$.
Example 2: Simplify the expression ${$ \sqrt{\frac{81}{16}}$}$. Using the property, we can rewrite this expression as ${$ \sqrt{\frac{9}{4}}$}$. Simplifying the fraction inside the square root, we get ${$ \frac{3}{2}$}$.

Q: What is the property of radicals?

A: The property of radicals states that the quotient of two square roots is equal to the square root of the quotient of the numbers inside the square roots. This can be expressed as ${$ \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$}$.

Q: How do I simplify radical expressions?

A: To simplify radical expressions, follow these steps:

Look for perfect squares inside the square root: If there are any perfect squares inside the square root, simplify them by taking the square root of the perfect square.
Simplify the fraction inside the square root: If there is a fraction inside the square root, simplify it by dividing the numerator and denominator by their greatest common divisor.
Apply the property of radicals: Use the property of radicals to rewrite the expression as the square root of the quotient of the numbers inside the square roots.

Q: What is the difference between a rational number and an irrational number?

A: A rational number is a number that can be expressed as the quotient of two integers, where the denominator is non-zero. Examples of rational numbers include 3, 4, and 12/5. An irrational number, on the other hand, is a number that cannot be expressed as the quotient of two integers. Examples of irrational numbers include ${$ \sqrt{2}$}$ and ${$ \pi$}$.

Q: Can I simplify radical expressions with negative numbers?

A: Yes, you can simplify radical expressions with negative numbers. When simplifying radical expressions with negative numbers, remember to take the negative sign outside the square root. For example, ${$ \sqrt{-16}$}$ can be simplified as ${$ -4$}$.

Q: What is the difference between a radical expression and an exponential expression?

A: A radical expression is an expression that contains a square root or other root. Examples of radical expressions include ${$ \sqrt{16}$}$ and ${$ \sqrt[3]{27}$}$. An exponential expression, on the other hand, is an expression that contains a power or exponent. Examples of exponential expressions include ${ $2^3\$}$ and ${ $5^2\$}$ .

Q: Can I simplify radical expressions with variables?

A: Yes, you can simplify radical expressions with variables. When simplifying radical expressions with variables, remember to follow the same steps as when simplifying radical expressions with numbers. For example, ${$ \sqrt{16x^2}$}$ can be simplified as ${ $4x\$}$ .

Q: What is the importance of simplifying radical expressions?

A: Simplifying radical expressions is important because it helps to:

Simplify complex expressions: Simplifying radical expressions can help to simplify complex expressions and make them easier to work with.
Reduce errors: Simplifying radical expressions can help to reduce errors and make calculations more accurate.
Improve understanding: Simplifying radical expressions can help to improve understanding of mathematical concepts and relationships.

By following these tips and examples, you can become more confident in simplifying radical expressions and applying the property of radicals.