Select The Correct Answer.What Is This Expression In Simplified Form? 128 2 \frac{\sqrt{128}}{\sqrt{2}} 2 ​ 128 ​ ​ A. 64 B. 4 C. 8 D. 130 \sqrt{130} 130 ​

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Introduction

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Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will explore the process of simplifying radical expressions, with a focus on the given expression: $\frac{\sqrt{128}}{\sqrt{2}}$. We will break down the steps involved in simplifying this expression and provide a clear explanation of the reasoning behind each step.

Understanding the Expression

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The given expression is $\frac{\sqrt{128}}{\sqrt{2}}$. To simplify this expression, we need to understand the properties of square roots and how they interact with each other.

Properties of Square Roots

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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

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To simplify the expression $\frac{\sqrt{128}}{\sqrt{2}}$, we need to break down the square roots into their prime factors.

Prime Factorization

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The prime factorization of 128 is $2^7$, and the prime factorization of 2 is simply 2.

Simplifying the Expression

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Now that we have the prime factorization of the numbers involved, we can simplify the expression.

$\frac{\sqrt{128}}{\sqrt{2}} = \frac{\sqrt{2^7}}{\sqrt{2}}$

Using the property of square roots that $\sqrt{a^2} = a$, we can simplify the expression further.

$\frac{\sqrt{2^7}}{\sqrt{2}} = \frac{2^3\sqrt{2}}{\sqrt{2}}$

Now, we can cancel out the common factor of $\sqrt{2}$ in the numerator and denominator.

$\frac{2^3\sqrt{2}}{\sqrt{2}} = 2^3$

Final Answer

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The simplified form of the expression $\frac{\sqrt{128}}{\sqrt{2}}$ is $2^3$, which equals 8.

Conclusion

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Simplifying radical expressions requires a deep understanding of the properties of square roots and how they interact with each other. By breaking down the square roots into their prime factors and using the properties of square roots, we can simplify complex expressions and arrive at a final answer.

Final Answer Options

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Based on the simplified expression, we can now evaluate the answer options.

A. 64 B. 4 C. 8 D. $\sqrt{130}$

The correct answer is C. 8.

Frequently Asked Questions

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Q: What is the simplified form of the expression $\frac{\sqrt{128}}{\sqrt{2}}$?

A: The simplified form of the expression is $2^3$, which equals 8.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to break down the square roots into their prime factors and use the properties of square roots to simplify the expression.

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

A: The property of square roots that allows us to simplify the expression is $\sqrt{a^2} = a$.

Q: Can I simplify a radical expression by canceling out common factors?

A: Yes, you can simplify a radical expression by canceling out common factors. In this case, we canceled out the common factor of $\sqrt{2}$ in the numerator and denominator.

Additional Resources

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For more information on simplifying radical expressions, check out the following resources:

• Khan Academy: Simplifying Radical Expressions
• Mathway: Simplifying Radical Expressions
• Wolfram Alpha: Simplifying Radical Expressions

By following the steps outlined in this article, you can simplify complex radical expressions and arrive at a final answer. Remember to break down the square roots into their prime factors and use the properties of square roots to simplify the expression.

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Introduction

---

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In our previous article, we explored the process of simplifying radical expressions, with a focus on the given expression: $\frac{\sqrt{128}}{\sqrt{2}}$. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques involved in simplifying radical expressions.

Q&A Guide

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Q: What is the simplified form of the expression $\frac{\sqrt{128}}{\sqrt{2}}$?

A: The simplified form of the expression is $2^3$, which equals 8.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to break down the square roots into their prime factors and use the properties of square roots to simplify the expression.

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

A: The property of square roots that allows us to simplify the expression is $\sqrt{a^2} = a$.

Q: Can I simplify a radical expression by canceling out common factors?

A: Yes, you can simplify a radical expression by canceling out common factors. In this case, we canceled out the common factor of $\sqrt{2}$ in the numerator and denominator.

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

A: A radical expression is an expression that contains a square root, while a rational expression is an expression that contains a fraction.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to factor the numerator and denominator, and then cancel out any common factors.

Q: What is the order of operations for simplifying radical expressions?

A: The order of operations for simplifying radical expressions is:

1. Break down the square roots into their prime factors.
2. Use the properties of square roots to simplify the expression.
3. Cancel out any common factors.

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 remember that the square root of a negative number is an imaginary number.

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 before. However, you need to be careful when canceling out common factors, as the variable may not be a perfect square.

Common Mistakes

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Mistake 1: Not breaking down the square roots into their prime factors.

A: Failing to break down the square roots into their prime factors can lead to incorrect simplifications.

Mistake 2: Not using the properties of square roots to simplify the expression.

A: Failing to use the properties of square roots to simplify the expression can lead to incorrect simplifications.

Mistake 3: Not canceling out common factors.

A: Failing to cancel out common factors can lead to incorrect simplifications.

Tips and Tricks

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Tip 1: Use the properties of square roots to simplify the expression.

A: Using the properties of square roots can help you simplify the expression more efficiently.

Tip 2: Break down the square roots into their prime factors.

A: Breaking down the square roots into their prime factors can help you simplify the expression more efficiently.

Tip 3: Cancel out common factors.

A: Canceling out common factors can help you simplify the expression more efficiently.

Conclusion

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Simplifying radical expressions requires a deep understanding of the properties of square roots and how they interact with each other. By following the steps outlined in this article, you can simplify complex radical expressions and arrive at a final answer. Remember to break down the square roots into their prime factors, use the properties of square roots to simplify the expression, and cancel out any common factors.

Additional Resources

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For more information on simplifying radical expressions, check out the following resources:

• Khan Academy: Simplifying Radical Expressions
• Mathway: Simplifying Radical Expressions
• Wolfram Alpha: Simplifying Radical Expressions

By following the steps outlined in this article and practicing with different types of radical expressions, you can become proficient in simplifying radical expressions and tackle more complex math problems with confidence.