Express In Simplest Radical Form:${ 4x \sqrt{192} - \sqrt{3x^2} }$

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 given expression: $4x \sqrt{192} - \sqrt{3x^2}$. We will break down the process into manageable steps, using the properties of radicals and square roots to simplify the expression.

Understanding Radicals and Square Roots

Before we dive into simplifying the expression, let's review the basics of radicals and square roots.

• A radical is a symbol that represents the square root of a number. It is denoted by the symbol $\sqrt{}$.
• A square root is a number that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because $4 \times 4 = 16$.
• The property of radicals states that $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$.

Step 1: Simplify the Radical Expression Inside the Square Root

Let's start by simplifying the radical expression inside the square root: $\sqrt{192}$.

• We can break down 192 into its prime factors: $192 = 2^6 \times 3$.
• Using the property of radicals, we can rewrite $\sqrt{192}$ as $\sqrt{2^6 \times 3}$.
• Simplifying further, we get $\sqrt{2^6 \times 3} = 2^3 \times \sqrt{3} = 8\sqrt{3}$.

Step 2: Simplify the Second Radical Expression

Now, let's simplify the second radical expression: $\sqrt{3x^2}$.

• We can rewrite $\sqrt{3x^2}$ as $\sqrt{3} \times \sqrt{x^2}$.
• Using the property of radicals, we can simplify further: $\sqrt{3} \times \sqrt{x^2} = \sqrt{3} \times x = x\sqrt{3}$.

Step 3: Substitute the Simplified Radical Expressions

Now that we have simplified the radical expressions, let's substitute them back into the original expression: $4x \sqrt{192} - \sqrt{3x^2}$.

• Substituting the simplified radical expressions, we get $4x \times 8\sqrt{3} - x\sqrt{3}$.
• Simplifying further, we get $32x\sqrt{3} - x\sqrt{3}$.

Step 4: Combine Like Terms

Now that we have simplified the expression, let's combine like terms: $32x\sqrt{3} - x\sqrt{3}$.

• Combining like terms, we get $(32x - x)\sqrt{3}$.
• Simplifying further, we get $31x\sqrt{3}$.

Conclusion

In this article, we simplified the given expression: $4x \sqrt{192} - \sqrt{3x^2}$. We broke down the process into manageable steps, using the properties of radicals and square roots to simplify the expression. We simplified the radical expressions inside the square root, substituted them back into the original expression, and combined like terms to get the final simplified expression: $31x\sqrt{3}$.

Final Answer

Introduction

In our previous article, we simplified the given expression: $4x \sqrt{192} - \sqrt{3x^2}$. We broke down the process into manageable steps, using the properties of radicals and square roots to simplify the expression. In this article, we will answer some frequently asked questions about simplifying radical expressions.

Q&A

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

A: A radical is a symbol that represents the square root of a number. It is denoted by the symbol $\sqrt{}$. A square root is a number 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 break down the number inside the square root into its prime factors. Then, use the property of radicals to rewrite the expression. Finally, simplify the expression by combining like terms.

Q: What is the property of radicals?

A: The property of radicals states that $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$. This means that you can break down a radical expression into smaller parts and simplify each part separately.

Q: How do I simplify a radical expression with a variable?

A: To simplify a radical expression with a variable, you need to follow the same steps as simplifying a radical expression with a number. However, you need to be careful when simplifying the variable part of the expression.

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 be careful when simplifying the negative part of the expression.

Q: How do I know when to simplify a radical expression?

A: You should simplify a radical expression when it is necessary to do so. For example, if you have a radical expression in a numerator or denominator, you may need to simplify it to cancel out common factors.

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

A: Yes, you can simplify a radical expression with a fraction. However, you need to be careful when simplifying the fraction part of the expression.

Q: How do I simplify a radical expression with a decimal?

A: You cannot simplify a radical expression with a decimal. However, you can approximate the decimal part of the expression and simplify the radical part separately.

Common Mistakes to Avoid

When simplifying radical expressions, there are several common mistakes to avoid:

• Not breaking down the number inside the square root into its prime factors: This can make it difficult to simplify the expression.
• Not using the property of radicals: This can make it difficult to simplify the expression.
• Not combining like terms: This can make the expression more complicated than it needs to be.
• Not being careful when simplifying the variable part of the expression: This can lead to errors in the final answer.

Conclusion

In this article, we answered some frequently asked questions about simplifying radical expressions. We covered topics such as the difference between a radical and a square root, how to simplify a radical expression, and common mistakes to avoid. By following the steps outlined in this article, you should be able to simplify radical expressions with ease.

Final Tips

• Practice, practice, practice: The more you practice simplifying radical expressions, the more comfortable you will become with the process.
• Use the property of radicals: This is a powerful tool for simplifying radical expressions.
• Combine like terms: This will help you simplify the expression and make it easier to read.
• Be careful when simplifying the variable part of the expression: This can lead to errors in the final answer.

Final Answer

The final answer is: Simplifying radical expressions is a crucial skill for students and professionals alike. By following the steps outlined in this article, you should be able to simplify radical expressions with ease.