What Is The Following Quotient?$\[ \frac{\sqrt{120}}{\sqrt{30}} \\]A. 2 B. 4 C. $2 \sqrt{10}$ D. $3 \sqrt{10}$

Introduction

When dealing with square roots in mathematics, it's essential to understand the properties and rules that govern their behavior. One of the most critical concepts is simplifying square roots, which involves expressing them in their most basic form. In this article, we will delve into the world of square roots and explore how to simplify them, with a focus on the given quotient: $\frac{\sqrt{120}}{\sqrt{30}}$. We will examine the properties of square roots, learn how to simplify them, and finally, determine the value of the given quotient.

Understanding Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 multiplied by 4 equals 16. In mathematical notation, this is represented as $\sqrt{16} = 4$. Square roots can be expressed as both positive and negative values, but in this article, we will focus on the positive square root.

Simplifying Square Roots

Simplifying square roots involves expressing them in their most basic form. This is achieved by finding the largest perfect square that divides the number inside the square root. For instance, the square root of 120 can be simplified as follows:

$\sqrt{120} = \sqrt{4 \times 30} = \sqrt{4} \times \sqrt{30} = 2\sqrt{30}$

In this example, we identified the largest perfect square that divides 120, which is 4. We then expressed the square root of 120 as the product of the square root of 4 and the square root of 30.

Simplifying the Quotient

Now that we have a better understanding of simplifying square roots, let's apply this knowledge to the given quotient: $\frac{\sqrt{120}}{\sqrt{30}}$. We can simplify the numerator and denominator separately:

$\frac{\sqrt{120}}{\sqrt{30}} = \frac{2\sqrt{30}}{\sqrt{30}}$

Canceling Out the Common Factor

Notice that the numerator and denominator share a common factor, which is the square root of 30. We can cancel out this common factor to simplify the quotient:

$\frac{2\sqrt{30}}{\sqrt{30}} = 2$

Conclusion

In conclusion, the given quotient $\frac{\sqrt{120}}{\sqrt{30}}$ simplifies to 2. This is achieved by simplifying the square roots in the numerator and denominator, and then canceling out the common factor. By understanding the properties and rules of square roots, we can simplify complex expressions and arrive at the correct solution.

Final Answer

The final answer is 2.

Additional Tips and Tricks

• When simplifying square roots, always look for the largest perfect square that divides the number inside the square root.
• Use the property of square roots that $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ to simplify expressions.
• Cancel out common factors between the numerator and denominator to simplify quotients.

Common Mistakes to Avoid

• Failing to identify the largest perfect square that divides the number inside the square root.
• Not using the property of square roots to simplify expressions.
• Not canceling out common factors between the numerator and denominator.

Real-World Applications

Simplifying square roots has numerous real-world applications, including:

• Calculating distances and heights in geometry and trigonometry.
• Determining the area and perimeter of shapes in geometry.
• Solving equations and inequalities in algebra.

Conclusion

In conclusion, simplifying square roots is a critical concept in mathematics that has numerous real-world applications. By understanding the properties and rules of square roots, we can simplify complex expressions and arrive at the correct solution. In this article, we explored how to simplify the given quotient $\frac{\sqrt{120}}{\sqrt{30}}$ and arrived at the final answer of 2.

Introduction

In our previous article, we explored the concept of simplifying square roots and applied it to the given quotient $\frac{\sqrt{120}}{\sqrt{30}}$. We arrived at the final answer of 2, but we know that there are many more questions and scenarios that require our attention. In this article, we will address some of the most frequently asked questions related to quotient simplification and provide clear and concise answers.

Q&A: Quotient Simplification

Q: What is the largest perfect square that divides 120?

A: The largest perfect square that divides 120 is 4.

Q: How do I simplify the square root of 120?

A: To simplify the square root of 120, we can express it as the product of the square root of 4 and the square root of 30: $\sqrt{120} = \sqrt{4 \times 30} = \sqrt{4} \times \sqrt{30} = 2\sqrt{30}$.

Q: What is the value of the given quotient $\frac{\sqrt{120}}{\sqrt{30}}$?

A: The value of the given quotient $\frac{\sqrt{120}}{\sqrt{30}}$ is 2.

Q: How do I simplify the quotient $\frac{\sqrt{180}}{\sqrt{45}}$?

A: To simplify the quotient $\frac{\sqrt{180}}{\sqrt{45}}$, we can first simplify the square roots in the numerator and denominator:

$\frac{\sqrt{180}}{\sqrt{45}} = \frac{\sqrt{36 \times 5}}{\sqrt{9 \times 5}} = \frac{\sqrt{36} \times \sqrt{5}}{\sqrt{9} \times \sqrt{5}} = \frac{6\sqrt{5}}{3\sqrt{5}} = 2$

Q: What is the largest perfect square that divides 180?

A: The largest perfect square that divides 180 is 36.

Q: How do I simplify the square root of 180?

A: To simplify the square root of 180, we can express it as the product of the square root of 36 and the square root of 5: $\sqrt{180} = \sqrt{36 \times 5} = \sqrt{36} \times \sqrt{5} = 6\sqrt{5}$.

Q: What is the value of the given quotient $\frac{\sqrt{180}}{\sqrt{45}}$?

A: The value of the given quotient $\frac{\sqrt{180}}{\sqrt{45}}$ is 2.

Conclusion

In conclusion, quotient simplification is a critical concept in mathematics that requires a deep understanding of square roots and their properties. By addressing some of the most frequently asked questions related to quotient simplification, we have provided clear and concise answers that will help you navigate complex expressions and arrive at the correct solution. Remember to always look for the largest perfect square that divides the number inside the square root, use the property of square roots to simplify expressions, and cancel out common factors between the numerator and denominator.

Additional Tips and Tricks

• When simplifying square roots, always look for the largest perfect square that divides the number inside the square root.
• Use the property of square roots that $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ to simplify expressions.
• Cancel out common factors between the numerator and denominator to simplify quotients.

Common Mistakes to Avoid

• Failing to identify the largest perfect square that divides the number inside the square root.
• Not using the property of square roots to simplify expressions.
• Not canceling out common factors between the numerator and denominator.

Real-World Applications

Simplifying square roots has numerous real-world applications, including:

• Calculating distances and heights in geometry and trigonometry.
• Determining the area and perimeter of shapes in geometry.
• Solving equations and inequalities in algebra.

Conclusion

In conclusion, simplifying square roots is a critical concept in mathematics that has numerous real-world applications. By understanding the properties and rules of square roots, we can simplify complex expressions and arrive at the correct solution. In this article, we addressed some of the most frequently asked questions related to quotient simplification and provided clear and concise answers.