Select The Correct Answer.What Is This Expression In Simplified Form? 32 2 \frac{\sqrt{32}}{\sqrt{2}} 2 32 A. 2 B. 30 \sqrt{30} 30 C. 4 D. 16

Mar 12, 2025 by ADMIN 152 views

Simplifying Radical Expressions: A Step-by-Step Guide

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Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students and professionals alike. In this article, we will focus on simplifying the expression $\frac{\sqrt{32}}{\sqrt{2}}$ and explore the different options available.

Understanding the Expression

The given expression is $\frac{\sqrt{32}}{\sqrt{2}}$ . To simplify this expression, we need to understand the properties of square roots. 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 Expression

To simplify the expression $\frac{\sqrt{32}}{\sqrt{2}}$ , we can start by simplifying the square roots individually. The square root of 32 can be simplified as follows:

$\sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2}$

Similarly, the square root of 2 remains the same.

Canceling Out Common Factors

Now that we have simplified the square roots, we can cancel out common factors in the numerator and denominator. In this case, we can cancel out the $\sqrt{2}$ in the numerator and denominator:

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

Conclusion

In conclusion, the simplified form of the expression $\frac{\sqrt{32}}{\sqrt{2}}$ is 4. This is the correct answer among the options provided.

Options Analysis

Let's analyze the options provided:

A. 2: This is incorrect, as the simplified form of the expression is not 2.

B. $\sqrt{30}$ : This is incorrect, as the simplified form of the expression is not $\sqrt{30}$ .

C. 4: This is the correct answer, as we have simplified the expression to 4.

D. 16: This is incorrect, as the simplified form of the expression is not 16.

Final Answer

The final answer is C. 4.

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Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students and professionals alike. In our previous article, we explored the process of simplifying the expression $\frac{\sqrt{32}}{\sqrt{2}}$ . In this article, we will provide a Q&A guide to help you better understand the concept of simplifying radical expressions.

Q&A

Q: What is a radical expression?

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

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to identify the square root of the number inside the radical sign and simplify it. You can also cancel out common factors in the numerator and denominator.

Q: What is the difference between a square root and a higher root?

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

Q: How do I simplify a square root of a product?

A: To simplify a square root of a product, you need to break down the product into its prime factors and simplify the square root of each factor.

Q: Can I simplify a radical expression with a variable?

A: Yes, you can simplify a radical expression with a variable. However, you need to follow the same rules as simplifying a radical expression with a number.

Q: What is the property of square roots that allows us to simplify radical expressions?

A: The property of square roots that allows us to simplify radical expressions is the property that states that the square root of a product is equal to the product of the square roots.

Q: Can I simplify a radical expression with a negative number?

A: Yes, you can simplify a radical expression with a negative number. However, you need to follow the same rules as simplifying a radical expression with a positive number.

Examples

Example 1: Simplifying a Square Root of a Product

Simplify the expression $\sqrt{18}$ .

$\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}$

Example 2: Simplifying a Radical Expression with a Variable

Simplify the expression $\sqrt{16x^2}$ .

$\sqrt{16x^2} = \sqrt{16} \times \sqrt{x^2} = 4x$

Example 3: Simplifying a Radical Expression with a Negative Number

Simplify the expression $\sqrt{-16}$ .

$\sqrt{-16} = \sqrt{-1} \times \sqrt{16} = 4i$

Conclusion

In conclusion, simplifying radical expressions is a crucial skill for students and professionals alike. By understanding the properties of square roots and following the rules for simplifying radical expressions, you can simplify even the most complex expressions.

Final Tips

Always simplify the square root of a number before simplifying the radical expression.
Use the property of square roots that states that the square root of a product is equal to the product of the square roots.
Follow the same rules for simplifying radical expressions with variables and negative numbers.

Final Answer

The final answer is that simplifying radical expressions is a crucial skill that requires understanding the properties of square roots and following the rules for simplifying radical expressions.