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 explore the quotient of two square roots and provide a step-by-step guide on how to simplify it.

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. The square root of a number can be represented using the symbol √. For instance, √16 can be read as "the square root of 16."

Simplifying Square Roots

To simplify a square root, we need to find the largest perfect square that divides the number inside the square root. A perfect square is a number that can be expressed as the product of an integer with itself. For example, 16 is a perfect square because it can be expressed as 4 × 4.

The Quotient of Two Square Roots

Now, let's focus on the given quotient: $\frac{\sqrt{120}}{\sqrt{30}}$. To simplify this expression, we need to find the largest perfect square that divides both 120 and 30.

Breaking Down the Numbers

Let's break down the numbers 120 and 30 into their prime factors:

• 120 = 2 × 2 × 2 × 3 × 5
• 30 = 2 × 3 × 5

Finding the Largest Perfect Square

From the prime factorization of 120 and 30, we can see that the largest perfect square that divides both numbers is 2 × 2 × 3 × 5, which equals 60.

Simplifying the Quotient

Now that we have found the largest perfect square that divides both 120 and 30, we can simplify the quotient:

$\frac{\sqrt{120}}{\sqrt{30}} = \frac{\sqrt{2 \times 2 \times 2 \times 3 \times 5}}{\sqrt{2 \times 3 \times 5}}$

Canceling Out Common Factors

We can cancel out the common factors in the numerator and denominator:

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

Simplifying the Expression

Now, we can simplify the expression by canceling out the common factors:

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

Evaluating the Expression

Finally, we can evaluate the expression:

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

Conclusion

In conclusion, the quotient of two square roots can be simplified by finding the largest perfect square that divides both numbers. By breaking down the numbers into their prime factors and canceling out common factors, we can simplify the expression and arrive at the final answer.

Final Answer

The final answer is $\boxed{2 \sqrt{10}}$

Introduction

In our previous article, we explored the concept of simplifying square roots and applied it to a specific quotient. In this article, we will delve deeper into the world of square roots and provide a comprehensive guide to help you understand and simplify quotients of square roots.

Q&A: Quotient of Square Roots

Q1: What is the quotient of two square roots?

A1: The quotient of two square roots is the result of dividing one square root by another. For example, $\frac{\sqrt{120}}{\sqrt{30}}$ is a quotient of two square roots.

Q2: How do I simplify a quotient of two square roots?

A2: To simplify a quotient of two square roots, you need to find the largest perfect square that divides both numbers inside the square roots. Then, cancel out the common factors in the numerator and denominator.

Q3: What is the largest perfect square that divides both 120 and 30?

A3: The largest perfect square that divides both 120 and 30 is 60. This is because 60 is the largest number that can be expressed as the product of an integer with itself, and it divides both 120 and 30.

Q4: How do I cancel out common factors in the numerator and denominator?

A4: To cancel out common factors in the numerator and denominator, you need to identify the common factors and divide both the numerator and denominator by them. In the case of the quotient $\frac{\sqrt{120}}{\sqrt{30}}$, we can cancel out the common factors as follows:

$\frac{\sqrt{120}}{\sqrt{30}} = \frac{\sqrt{2 \times 2 \times 2 \times 3 \times 5}}{\sqrt{2 \times 3 \times 5}}$

Q5: What is the simplified form of the quotient $\frac{\sqrt{120}}{\sqrt{30}}$?

A5: The simplified form of the quotient $\frac{\sqrt{120}}{\sqrt{30}}$ is $\sqrt{2}$.

Q6: Can I simplify a quotient of two square roots with different bases?

A6: Yes, you can simplify a quotient of two square roots with different bases. However, you need to find the largest perfect square that divides both numbers inside the square roots, and then cancel out the common factors in the numerator and denominator.

Q7: How do I handle a quotient of two square roots with a negative number inside the square root?

A7: When dealing with a quotient of two square roots with a negative number inside the square root, you need to remember that the square root of a negative number is an imaginary number. In this case, you can simplify the quotient by finding the largest perfect square that divides both numbers inside the square roots, and then cancel out the common factors in the numerator and denominator.

Q8: Can I simplify a quotient of two square roots with a fraction inside the square root?

A8: Yes, you can simplify a quotient of two square roots with a fraction inside the square root. However, you need to find the largest perfect square that divides both numbers inside the square roots, and then cancel out the common factors in the numerator and denominator.

Q9: How do I handle a quotient of two square roots with a variable inside the square root?

A9: When dealing with a quotient of two square roots with a variable inside the square root, you need to remember that the variable can be a constant or a variable expression. In this case, you can simplify the quotient by finding the largest perfect square that divides both numbers inside the square roots, and then cancel out the common factors in the numerator and denominator.

Q10: Can I simplify a quotient of two square roots with a complex number inside the square root?

A10: Yes, you can simplify a quotient of two square roots with a complex number inside the square root. However, you need to find the largest perfect square that divides both numbers inside the square roots, and then cancel out the common factors in the numerator and denominator.

Conclusion

In conclusion, simplifying quotients of square roots requires a deep understanding of the properties and rules that govern square roots. By following the steps outlined in this article, you can simplify even the most complex quotients of square roots and arrive at the final answer.

Final Answer

The final answer is $\boxed{2 \sqrt{10}}$