What Is 25 X 2 Y 2 X Y \frac{\sqrt{25 X^2 Y^2}}{\sqrt{x Y}} X Y ​ 25 X 2 Y 2 ​ ​ In Simplest Form? Assume X ≥ 0 X \geq 0 X ≥ 0 And Y ≥ 0 Y \geq 0 Y ≥ 0 .A. 5 X Y 5 \sqrt{x Y} 5 X Y ​ B. 25 X Y 25 \sqrt{x Y} 25 X Y ​ C. 5 X Y \sqrt{5 X Y} 5 X Y ​ D. 5 X Y X Y 5 X Y \sqrt{x Y} 5 X Y X Y ​

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Introduction

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When dealing with algebraic expressions, simplifying them is crucial to understand their behavior and make calculations easier. In this case, we are given the expression $\frac{\sqrt{25 x^2 y^2}}{\sqrt{x y}}$ and we need to simplify it. The given expression involves square roots, which can be simplified using the properties of radicals.

Simplifying the Expression

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To simplify the given expression, we can start by using the property of radicals that states $\sqrt{a^2} = |a|$. Since $x \geq 0$ and $y \geq 0$, we can rewrite the expression as:

$$\frac{\sqrt{25 x^2 y^2}}{\sqrt{x y}} = \frac{\sqrt{(5x)^2 (y)^2}}{\sqrt{x y}}$$

Applying the Property of Radicals

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Now, we can apply the property of radicals that states $\sqrt{a^2} = |a|$. In this case, we have:

$$\frac{\sqrt{(5x)^2 (y)^2}}{\sqrt{x y}} = \frac{|5x| |y|}{\sqrt{x y}}$$

Simplifying the Absolute Values

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Since $x \geq 0$ and $y \geq 0$, we can remove the absolute values:

$$\frac{|5x| |y|}{\sqrt{x y}} = \frac{5x y}{\sqrt{x y}}$$

Canceling Out the Common Factors

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Now, we can cancel out the common factors between the numerator and the denominator:

$$\frac{5x y}{\sqrt{x y}} = \frac{5x y}{\sqrt{x y}} \cdot \frac{\sqrt{x y}}{\sqrt{x y}}$$

Simplifying the Expression

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After canceling out the common factors, we are left with:

$$\frac{5x y}{\sqrt{x y}} \cdot \frac{\sqrt{x y}}{\sqrt{x y}} = 5 \sqrt{x y}$$

Conclusion

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In conclusion, the expression $\frac{\sqrt{25 x^2 y^2}}{\sqrt{x y}}$ simplifies to $5 \sqrt{x y}$.

Answer

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The correct answer is:

• A. $5 \sqrt{x y}$

Discussion

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This problem involves simplifying an algebraic expression using the properties of radicals. The key concept used in this problem is the property of radicals that states $\sqrt{a^2} = |a|$. This property allows us to simplify the expression by removing the absolute values and canceling out the common factors.

Related Problems

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• Simplify the expression $\frac{\sqrt{16 x^2 y^2}}{\sqrt{x y}}$.
• Simplify the expression $\frac{\sqrt{9 x^2 y^2}}{\sqrt{x y}}$.
• Simplify the expression $\frac{\sqrt{36 x^2 y^2}}{\sqrt{x y}}$.

Practice Problems

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• Simplify the expression $\frac{\sqrt{25 x^2 y^2}}{\sqrt{x y}}$.
• Simplify the expression $\frac{\sqrt{16 x^2 y^2}}{\sqrt{x y}}$.
• Simplify the expression $\frac{\sqrt{9 x^2 y^2}}{\sqrt{x y}}$.

Solutions

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• The solution to the problem $\frac{\sqrt{25 x^2 y^2}}{\sqrt{x y}}$ is $5 \sqrt{x y}$.
• The solution to the problem $\frac{\sqrt{16 x^2 y^2}}{\sqrt{x y}}$ is $4 \sqrt{x y}$.
• The solution to the problem $\frac{\sqrt{9 x^2 y^2}}{\sqrt{x y}}$ is $3 \sqrt{x y}$.

Conclusion

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In conclusion, simplifying algebraic expressions using the properties of radicals is a crucial skill in mathematics. By understanding and applying these properties, we can simplify complex expressions and make calculations easier.

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Introduction

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In our previous article, we discussed how to simplify the expression $\frac{\sqrt{25 x^2 y^2}}{\sqrt{x y}}$ using the properties of radicals. In this article, we will answer some frequently asked questions related to simplifying algebraic expressions with radicals.

Q&A

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Q: What is the property of radicals that states $\sqrt{a^2} = |a|$?

A: The property of radicals that states $\sqrt{a^2} = |a|$ is a fundamental concept in simplifying algebraic expressions with radicals. This property allows us to remove the absolute values and simplify the expression.

Q: How do I simplify an expression with a square root in the numerator and a square root in the denominator?

A: To simplify an expression with a square root in the numerator and a square root in the denominator, you can cancel out the common factors between the numerator and the denominator. This is done by multiplying the numerator and the denominator by the square root of the denominator.

Q: What is the difference between $\sqrt{a^2}$ and $|a|$?

A: $\sqrt{a^2}$ and $|a|$ are equivalent expressions. The square root of a squared number is equal to the absolute value of that number.

Q: Can I simplify an expression with a square root in the numerator and a non-square root in the denominator?

A: No, you cannot simplify an expression with a square root in the numerator and a non-square root in the denominator. In this case, you would need to rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator.

Q: How do I simplify an expression with multiple square roots in the numerator and denominator?

A: To simplify an expression with multiple square roots in the numerator and denominator, you can cancel out the common factors between the numerator and the denominator. This is done by multiplying the numerator and the denominator by the square root of the denominator.

Q: Can I simplify an expression with a negative number under the square root?

A: Yes, you can simplify an expression with a negative number under the square root. In this case, you would need to use the property of radicals that states $\sqrt{-a} = i \sqrt{a}$, where $i$ is the imaginary unit.

Examples

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Example 1: Simplifying an Expression with a Square Root in the Numerator and a Square Root in the Denominator

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

Solution:

$$\frac{\sqrt{16 x^2 y^2}}{\sqrt{x y}} = \frac{\sqrt{(4x)^2 (y)^2}}{\sqrt{x y}} = \frac{|4x| |y|}{\sqrt{x y}} = \frac{4x y}{\sqrt{x y}} = 4 \sqrt{x y}$$

Example 2: Simplifying an Expression with Multiple Square Roots in the Numerator and Denominator

Simplify the expression $\frac{\sqrt{36 x^2 y^2}}{\sqrt{x y}}$.

Solution:

$$\frac{\sqrt{36 x^2 y^2}}{\sqrt{x y}} = \frac{\sqrt{(6x)^2 (y)^2}}{\sqrt{x y}} = \frac{|6x| |y|}{\sqrt{x y}} = \frac{6x y}{\sqrt{x y}} = 6 \sqrt{x y}$$

Conclusion

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In conclusion, simplifying algebraic expressions with radicals is a crucial skill in mathematics. By understanding and applying the properties of radicals, we can simplify complex expressions and make calculations easier. We hope this Q&A article has been helpful in answering your questions and providing you with a better understanding of simplifying algebraic expressions with radicals.

Related Articles

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• Simplifying Algebraic Expressions with Radicals
• Properties of Radicals
• Simplifying Expressions with Multiple Square Roots

Practice Problems

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• Simplify the expression $\frac{\sqrt{25 x^2 y^2}}{\sqrt{x y}}$.
• Simplify the expression $\frac{\sqrt{16 x^2 y^2}}{\sqrt{x y}}$.
• Simplify the expression $\frac{\sqrt{9 x^2 y^2}}{\sqrt{x y}}$.

Solutions

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• The solution to the problem $\frac{\sqrt{25 x^2 y^2}}{\sqrt{x y}}$ is $5 \sqrt{x y}$.
• The solution to the problem $\frac{\sqrt{16 x^2 y^2}}{\sqrt{x y}}$ is $4 \sqrt{x y}$.
• The solution to the problem $\frac{\sqrt{9 x^2 y^2}}{\sqrt{x y}}$ is $3 \sqrt{x y}$.