Enter The Values For The Variables That Give The Correct Simplified Expressions, $x \geq 0$.$\[ \begin{array}{l} \sqrt{50 X^2}=\sqrt{25 \cdot 2 \cdot X^2}=5 X \sqrt{b} \\ b=2 \\ \sqrt{32 X}=\sqrt{16 \cdot 2 \cdot X}=c \sqrt{2 X}

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 explore the process of simplifying radical expressions, focusing on the given problem: $\sqrt{50 x^2}=\sqrt{25 \cdot 2 \cdot x^2}=5 x \sqrt{b}$ and $\sqrt{32 x}=\sqrt{16 \cdot 2 \cdot x}=c \sqrt{2 x}$. We will break down the solution step by step, providing a clear and concise explanation of each step.

Understanding Radical Expressions

A radical expression is a mathematical expression that contains a square root or other root. Radical expressions can be simplified by factoring out perfect squares, which are numbers that can be expressed as the square of an integer. For example, 16 is a perfect square because it can be expressed as $4^2$. Similarly, 25 is a perfect square because it can be expressed as $5^2$.

Simplifying the First Radical Expression

The first radical expression is $\sqrt{50 x^2}$. To simplify this expression, we need to factor out the perfect square from the radicand (the number inside the square root). We can start by factoring out the greatest perfect square that divides 50, which is 25.

$\sqrt{50 x^2} = \sqrt{25 \cdot 2 \cdot x^2}$

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

$\sqrt{25 \cdot 2 \cdot x^2} = 5 x \sqrt{2}$

Therefore, the simplified expression is $5 x \sqrt{2}$.

Simplifying the Second Radical Expression

The second radical expression is $\sqrt{32 x}$. To simplify this expression, we need to factor out the perfect square from the radicand. We can start by factoring out the greatest perfect square that divides 32, which is 16.

$\sqrt{32 x} = \sqrt{16 \cdot 2 \cdot x}$

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

$\sqrt{16 \cdot 2 \cdot x} = 4 \sqrt{2 x}$

Therefore, the simplified expression is $4 \sqrt{2 x}$.

Discussion

In this article, we have explored the process of simplifying radical expressions, focusing on the given problem. We have broken down the solution step by step, providing a clear and concise explanation of each step. By following these steps, you can simplify radical expressions with ease.

Conclusion

Simplifying radical expressions is a crucial skill for students and professionals alike. By understanding the concept of perfect squares and factoring them out, you can simplify radical expressions with ease. In this article, we have provided a step-by-step guide to simplifying radical expressions, focusing on the given problem. We hope that this article has been helpful in understanding the concept of simplifying radical expressions.

Final Answer

The final answer is:

• $b = 2$
• $c = 4$

Introduction

In our previous article, we explored the process of simplifying radical expressions, focusing on the given problem: $\sqrt{50 x^2}=\sqrt{25 \cdot 2 \cdot x^2}=5 x \sqrt{b}$ and $\sqrt{32 x}=\sqrt{16 \cdot 2 \cdot x}=c \sqrt{2 x}$. 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 other root. Radical expressions can be simplified by factoring out perfect squares, which are numbers that can be expressed as the square of an integer.

Q: What is a perfect square?

A: A perfect square is a number that can be expressed as the square of an integer. For example, 16 is a perfect square because it can be expressed as $4^2$. Similarly, 25 is a perfect square because it can be expressed as $5^2$.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to factor out the perfect square from the radicand (the number inside the square root). You can start by factoring out the greatest perfect square that divides the radicand.

Q: What is the greatest perfect square that divides 50?

A: The greatest perfect square that divides 50 is 25.

Q: How do I simplify the expression $\sqrt{50 x^2}$?

A: To simplify the expression $\sqrt{50 x^2}$, you need to factor out the perfect square from the radicand. You can start by factoring out the greatest perfect square that divides 50, which is 25.

$\sqrt{50 x^2} = \sqrt{25 \cdot 2 \cdot x^2}$

Now, you can simplify the expression by taking the square root of the perfect square:

$\sqrt{25 \cdot 2 \cdot x^2} = 5 x \sqrt{2}$

Therefore, the simplified expression is $5 x \sqrt{2}$.

Q: What is the value of $b$ in the expression $5 x \sqrt{b}$?

A: The value of $b$ is 2.

Q: How do I simplify the expression $\sqrt{32 x}$?

A: To simplify the expression $\sqrt{32 x}$, you need to factor out the perfect square from the radicand. You can start by factoring out the greatest perfect square that divides 32, which is 16.

$\sqrt{32 x} = \sqrt{16 \cdot 2 \cdot x}$

Now, you can simplify the expression by taking the square root of the perfect square:

$\sqrt{16 \cdot 2 \cdot x} = 4 \sqrt{2 x}$

Therefore, the simplified expression is $4 \sqrt{2 x}$.

Q: What is the value of $c$ in the expression $c \sqrt{2 x}$?

A: The value of $c$ is 4.

Conclusion

In this article, we have provided a Q&A guide to help you better understand the concept of simplifying radical expressions. We have covered common questions and provided step-by-step solutions to help you simplify radical expressions with ease.

Final Answer

The final answer is:

• $b = 2$
• $c = 4$

Note: The final answer is based on the given problem and the solution provided in the article.