Which Choice Is Equivalent To The Quotient Below? 45 9 \frac{\sqrt{45}}{\sqrt{9}} 9 ​ 45 ​ ​ A. 5 3 \frac{5}{3} 3 5 ​ B. 5 3 \frac{\sqrt{5}}{3} 3 5 ​ ​ C. 5 \sqrt{5} 5 ​ D. 5

Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will explore the process of simplifying radical expressions, with a focus on the quotient $\frac{\sqrt{45}}{\sqrt{9}}$. We will examine each choice and determine which one is equivalent to the given quotient.

Understanding Radical Expressions

A radical expression is a mathematical expression that contains a root or a radical sign. The most common radical sign is the square root sign, denoted by $\sqrt{}$. Radical expressions can be simplified by factoring the radicand (the number inside the radical sign) into its prime factors.

Simplifying the Quotient

To simplify the quotient $\frac{\sqrt{45}}{\sqrt{9}}$, we need to simplify each radical expression separately.

Simplifying the Radicand of the Numerator

The radicand of the numerator is 45. We can factor 45 into its prime factors as follows:

45 = 3 × 3 × 5

Since there are two factors of 3, we can rewrite the radicand as:

45 = 3² × 5

Now, we can simplify the radical expression by taking the square root of the perfect square factor:

√(3² × 5) = 3√5

Simplifying the Radicand of the Denominator

The radicand of the denominator is 9. We can factor 9 into its prime factors as follows:

9 = 3 × 3

Since there are two factors of 3, we can rewrite the radicand as:

9 = 3²

Now, we can simplify the radical expression by taking the square root of the perfect square factor:

√(3²) = 3

Simplifying the Quotient

Now that we have simplified the radicand of the numerator and the denominator, we can simplify the quotient by dividing the two simplified radical expressions:

$\frac{3\sqrt{5}}{3} = \sqrt{5}$

Evaluating the Choices

Now that we have simplified the quotient, we can evaluate each choice to determine which one is equivalent to the simplified quotient.

Choice A: $\frac{5}{3}$

This choice is not equivalent to the simplified quotient. The simplified quotient is $\sqrt{5}$, which is not equal to $\frac{5}{3}$.

Choice B: $\frac{\sqrt{5}}{3}$

This choice is not equivalent to the simplified quotient. The simplified quotient is $\sqrt{5}$, which is not equal to $\frac{\sqrt{5}}{3}$.

Choice C: $\sqrt{5}$

This choice is equivalent to the simplified quotient. The simplified quotient is $\sqrt{5}$, which is equal to $\sqrt{5}$.

Choice D: 5

This choice is not equivalent to the simplified quotient. The simplified quotient is $\sqrt{5}$, which is not equal to 5.

Conclusion

In conclusion, the correct choice is C. $\sqrt{5}$. This choice is equivalent to the simplified quotient $\frac{\sqrt{45}}{\sqrt{9}}$. We simplified the quotient by factoring the radicand of the numerator and the denominator, and then dividing the two simplified radical expressions. The simplified quotient is $\sqrt{5}$, which is equal to choice C.

Final Answer

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

Additional Resources

For more information on simplifying radical expressions, please refer to the following resources:

• Mathway: Simplifying Radical Expressions
• Khan Academy: Simplifying Radical Expressions
• Purplemath: Simplifying Radical Expressions
  Simplifying Radical Expressions: A Q&A Guide =====================================================

Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will explore the process of simplifying radical expressions, with a focus on the quotient $\frac{\sqrt{45}}{\sqrt{9}}$. We will also answer some frequently asked questions about simplifying radical expressions.

Q&A

Q: What is a radical expression?

A: A radical expression is a mathematical expression that contains a root or a radical sign. The most common radical sign is the square root sign, denoted by $\sqrt{}$.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to factor the radicand (the number inside the radical sign) into its prime factors. Then, you can take the square root of the perfect square factors.

Q: What is a perfect square factor?

A: A perfect square factor is a factor that can be expressed as the square of an integer. For example, 9 is a perfect square factor because it can be expressed as 3².

Q: How do I simplify the quotient $\frac{\sqrt{45}}{\sqrt{9}}$?

A: To simplify the quotient, you need to simplify each radical expression separately. First, simplify the radicand of the numerator: $\sqrt{45} = 3\sqrt{5}$. Then, simplify the radicand of the denominator: $\sqrt{9} = 3$. Finally, divide the two simplified radical expressions: $\frac{3\sqrt{5}}{3} = \sqrt{5}$.

Q: What is the final answer to the quotient $\frac{\sqrt{45}}{\sqrt{9}}$?

A: The final answer is $\boxed{\sqrt{5}}$.

Q: How do I know if a radical expression can be simplified?

A: A radical expression can be simplified if the radicand can be factored into its prime factors. If the radicand can be factored into perfect square factors, then the radical expression can be simplified.

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

A: A radical expression is a mathematical expression that contains a root or a radical sign, while an exponential expression is a mathematical expression that contains an exponent. For example, $\sqrt{5}$ is a radical expression, while $5^2$ is an exponential expression.

Q: How do I simplify a radical expression with a variable?

A: To simplify a radical expression with a variable, you need to factor the radicand into its prime factors. Then, you can take the square root of the perfect square factors. For example, $\sqrt{16x}$ can be simplified as $\sqrt{16} \cdot \sqrt{x} = 4\sqrt{x}$.

Conclusion

In conclusion, simplifying radical expressions is a crucial skill to master in mathematics. By understanding the process of simplifying radical expressions, you can solve a wide range of mathematical problems. We hope that this Q&A guide has been helpful in answering your questions about simplifying radical expressions.

Final Answer

The final answer is $\boxed{\sqrt{5}}$.

Additional Resources

For more information on simplifying radical expressions, please refer to the following resources:

• Mathway: Simplifying Radical Expressions
• Khan Academy: Simplifying Radical Expressions
• Purplemath: Simplifying Radical Expressions