Which Choice Is Equivalent To The Quotient Below? 36 4 \frac{\sqrt{36}}{\sqrt{4}} 4 36 A. 3 B. 9 C. 12 \sqrt{12} 12 D. 2 2 \frac{\sqrt{2}}{2} 2 2

Mar 12, 2025 by ADMIN 160 views

**Simplifying Radical Expressions: A Step-by-Step Guide**

Introduction

Radical expressions are an essential part of mathematics, and simplifying them is a crucial skill to master. In this article, we will explore the concept of simplifying radical expressions, focusing on the quotient $\frac{\sqrt{36}}{\sqrt{4}}$ . We will break down the problem step by step, using various techniques to simplify the expression and arrive at the correct answer.

Understanding Radical Expressions

A radical expression is a mathematical expression that contains a square root or a higher root of a number. The 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.

Simplifying the Quotient

The given quotient is $\frac{\sqrt{36}}{\sqrt{4}}$ . To simplify this expression, we need to find the square roots of 36 and 4.

Finding the Square Root of 36

The square root of 36 is 6, because 6 multiplied by 6 equals 36.

Finding the Square Root of 4

The square root of 4 is 2, because 2 multiplied by 2 equals 4.

Substituting the Square Roots

Now that we have found the square roots of 36 and 4, we can substitute them into the original quotient:

$\frac{\sqrt{36}}{\sqrt{4}} = \frac{6}{2}$

Simplifying the Fraction

To simplify the fraction, we can divide the numerator by the denominator:

$\frac{6}{2} = 3$

Conclusion

In conclusion, the quotient $\frac{\sqrt{36}}{\sqrt{4}}$ is equivalent to 3. This is because the square root of 36 is 6, and the square root of 4 is 2, making the quotient $\frac{6}{2}$ , which simplifies to 3.

Alternative Solutions

Let's examine the alternative solutions provided:

A. 3 B. 9 C. $\sqrt{12}$ D. $\frac{\sqrt{2}}{2}$

Analyzing Solution A

Solution A is 3, which is the correct answer we arrived at by simplifying the quotient.

Analyzing Solution B

Solution B is 9, which is not the correct answer. To verify this, let's try to simplify the quotient $\frac{\sqrt{36}}{\sqrt{4}}$ using a different approach.

Using a Different Approach

Let's try to simplify the quotient by first simplifying the square roots:

$\sqrt{36} = \sqrt{6^2} = 6$

$\sqrt{4} = \sqrt{2^2} = 2$

Now, we can substitute these simplified square roots into the original quotient:

$\frac{\sqrt{36}}{\sqrt{4}} = \frac{6}{2} = 3$

As we can see, solution B is not the correct answer.

Analyzing Solution C

Solution C is $\sqrt{12}$ , which is not the correct answer. To verify this, let's try to simplify the quotient $\frac{\sqrt{36}}{\sqrt{4}}$ using a different approach.

Using a Different Approach

Let's try to simplify the quotient by first simplifying the square roots:

$\sqrt{36} = \sqrt{6^2} = 6$

$\sqrt{4} = \sqrt{2^2} = 2$

Now, we can substitute these simplified square roots into the original quotient:

$\frac{\sqrt{36}}{\sqrt{4}} = \frac{6}{2} = 3$

As we can see, solution C is not the correct answer.

Analyzing Solution D

Solution D is $\frac{\sqrt{2}}{2}$ , which is not the correct answer. To verify this, let's try to simplify the quotient $\frac{\sqrt{36}}{\sqrt{4}}$ using a different approach.

Using a Different Approach

Let's try to simplify the quotient by first simplifying the square roots:

$\sqrt{36} = \sqrt{6^2} = 6$

$\sqrt{4} = \sqrt{2^2} = 2$

Now, we can substitute these simplified square roots into the original quotient:

$\frac{\sqrt{36}}{\sqrt{4}} = \frac{6}{2} = 3$

As we can see, solution D is not the correct answer.

Conclusion

In conclusion, the correct answer is solution A, which is 3. This is because the square root of 36 is 6, and the square root of 4 is 2, making the quotient $\frac{6}{2}$ , which simplifies to 3.

Final Thoughts

Introduction

Radical expressions are an essential part of mathematics, and simplifying them is a crucial skill to master. In our previous article, we explored the concept of simplifying radical expressions, focusing on the quotient $\frac{\sqrt{36}}{\sqrt{4}}$ . In this article, we will provide a Q&A guide to help you better understand the concept of simplifying radical expressions.

Q: What is a radical expression?

A: A radical expression is a mathematical expression that contains a square root or a higher root of a number.

Q: What is the square root of a number?

A: The square root of a number is a value that, when multiplied by itself, gives the original number.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to find the square roots of the numbers inside the radical sign. Then, you can simplify the expression by dividing the numerator by the denominator.

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

A: A radical expression is a mathematical expression that contains a square root or a higher root of a number, while a rational expression is a mathematical expression that contains a fraction.

Q: Can I simplify a radical expression by multiplying the numerator and denominator by the same value?

A: Yes, you can simplify a radical expression by multiplying the numerator and denominator by the same value. This is known as rationalizing the denominator.

Q: How do I rationalize the denominator of a radical expression?

A: To rationalize the denominator of a radical expression, you need to multiply the numerator and denominator by the conjugate of the denominator. The conjugate of a denominator is the same value, but with the opposite sign.

Q: What is the conjugate of a denominator?

A: The conjugate of a denominator is the same value, but with the opposite sign.

Q: Can I simplify a radical expression by using a calculator?

A: Yes, you can simplify a radical expression by using a calculator. However, it's always a good idea to simplify the expression by hand to ensure that you understand the underlying concepts.

Q: What are some common mistakes to avoid when simplifying radical expressions?

A: Some common mistakes to avoid when simplifying radical expressions include:

Not simplifying the expression by hand
Not rationalizing the denominator
Not using the correct order of operations
Not checking the final answer for accuracy

Q: How can I practice simplifying radical expressions?

A: You can practice simplifying radical expressions by working through problems in a textbook or online resource. You can also try simplifying radical expressions on your own, using a calculator or by hand.

Conclusion

Simplifying radical expressions is an essential skill in mathematics, and it requires a deep understanding of the underlying concepts. By following the steps outlined in this Q&A guide, you can better understand how to simplify radical expressions and arrive at the correct answer. Remember to always simplify the expression by hand, rationalize the denominator, and check the final answer for accuracy.