Assuming $x$ And $y$ Are Both Positive, Write The Following Expression In Simplest Radical Form.$\sqrt{4 X^2 Y^2}$

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Introduction

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Radical expressions are an essential part of mathematics, and simplifying them is a crucial skill to master. In this article, we will focus on simplifying the expression $\sqrt{4 x^2 y^2}$, assuming that $x$ and $y$ are both positive. We will break down the process into manageable steps and provide a clear explanation of each step.

Understanding the Expression

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The given expression is $\sqrt{4 x^2 y^2}$. To simplify this expression, we need to understand the properties of radicals and exponents. The square root of a number is a value that, when multiplied by itself, gives the original number. In this case, we have a square root of a product of two variables, $x$ and $y$, each raised to the power of 2.

Step 1: Simplify the Coefficient

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The first step in simplifying the expression is to simplify the coefficient, which is 4. We can rewrite 4 as $2^2$, since 4 is equal to 2 squared. This gives us:

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

Step 2: Apply the Product Rule

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The product rule of radicals states that the square root of a product is equal to the product of the square roots. In this case, we can rewrite the expression as:

$$\sqrt{(2^2) x^2 y^2} = \sqrt{2^2} \cdot \sqrt{x^2} \cdot \sqrt{y^2}$$

Step 3: Simplify the Square Roots

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Now, we can simplify each square root individually. The square root of $2^2$ is 2, since 2 squared is equal to 4. The square root of $x^2$ is $x$, since $x$ squared is equal to $x^2$. The square root of $y^2$ is $y$, since $y$ squared is equal to $y^2$. This gives us:

$$\sqrt{2^2} \cdot \sqrt{x^2} \cdot \sqrt{y^2} = 2 \cdot x \cdot y$$

Step 4: Simplify the Expression

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Now, we can simplify the expression by combining the terms. We have:

$$2 \cdot x \cdot y = 2xy$$

Therefore, the simplified form of the expression $\sqrt{4 x^2 y^2}$ is $2xy$.

Conclusion

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In this article, we simplified the expression $\sqrt{4 x^2 y^2}$, assuming that $x$ and $y$ are both positive. We broke down the process into manageable steps and provided a clear explanation of each step. By applying the product rule of radicals and simplifying the square roots, we arrived at the simplified form of the expression, which is $2xy$.

Frequently Asked Questions

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Q: What is the simplified form of the expression $\sqrt{4 x^2 y^2}$?

A: The simplified form of the expression $\sqrt{4 x^2 y^2}$ is $2xy$.

Q: What is the product rule of radicals?

A: The product rule of radicals states that the square root of a product is equal to the product of the square roots.

Q: How do I simplify a square root expression?

A: To simplify a square root expression, you can apply the product rule of radicals and simplify the square roots individually.

Final Thoughts

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Simplifying radical expressions is an essential skill in mathematics, and it requires a clear understanding of the properties of radicals and exponents. By following the steps outlined in this article, you can simplify complex radical expressions and arrive at the simplified form. Remember to apply the product rule of radicals and simplify the square roots individually to arrive at the final answer.

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Introduction

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In our previous article, we simplified the expression $\sqrt{4 x^2 y^2}$, assuming that $x$ and $y$ are both positive. We broke down the process into manageable steps and provided a clear explanation of each step. In this article, we will continue to provide a Q&A guide on simplifying radical expressions.

Q&A: Simplifying Radical Expressions

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Q: What is the simplified form of the expression $\sqrt{9 x^2 y^2}$?

A: The simplified form of the expression $\sqrt{9 x^2 y^2}$ is $3xy$.

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

A: To simplify a radical expression with a coefficient, you can rewrite the coefficient as a product of a perfect square and a remaining factor. Then, you can simplify the square root of the perfect square and leave the remaining factor inside the square root.

Q: What is the product rule of radicals?

A: The product rule of radicals states that the square root of a product is equal to the product of the square roots.

Q: How do I simplify a square root expression with multiple variables?

A: To simplify a square root expression with multiple variables, you can apply the product rule of radicals and simplify the square roots individually.

Q: What is the simplified form of the expression $\sqrt{16 x^2 y^2}$?

A: The simplified form of the expression $\sqrt{16 x^2 y^2}$ is $4xy$.

Q: How do I simplify a radical expression with a negative coefficient?

A: To simplify a radical expression with a negative coefficient, you can rewrite the coefficient as a product of a perfect square and a remaining factor. Then, you can simplify the square root of the perfect square and leave the remaining factor inside the square root.

Q: What is the simplified form of the expression $\sqrt{25 x^2 y^2}$?

A: The simplified form of the expression $\sqrt{25 x^2 y^2}$ is $5xy$.

Tips and Tricks

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Tip 1: Simplify the Coefficient First

When simplifying a radical expression, it's often helpful to simplify the coefficient first. This can make it easier to simplify the square roots and arrive at the final answer.

Tip 2: Apply the Product Rule of Radicals

The product rule of radicals states that the square root of a product is equal to the product of the square roots. This is a powerful tool for simplifying radical expressions.

Tip 3: Simplify the Square Roots Individually

When simplifying a square root expression, it's often helpful to simplify the square roots individually. This can make it easier to arrive at the final answer.

Conclusion

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In this article, we provided a Q&A guide on simplifying radical expressions. We covered a range of topics, from simplifying radical expressions with coefficients to simplifying square root expressions with multiple variables. By following the tips and tricks outlined in this article, you can simplify complex radical expressions and arrive at the final answer.

Frequently Asked Questions

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Q: What is the simplified form of the expression $\sqrt{36 x^2 y^2}$?

A: The simplified form of the expression $\sqrt{36 x^2 y^2}$ is $6xy$.

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

A: To simplify a radical expression with a variable in the denominator, you can rewrite the expression as a fraction and simplify the numerator and denominator separately.

Q: What is the simplified form of the expression $\sqrt{49 x^2 y^2}$?

A: The simplified form of the expression $\sqrt{49 x^2 y^2}$ is $7xy$.

Final Thoughts

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Simplifying radical expressions is an essential skill in mathematics, and it requires a clear understanding of the properties of radicals and exponents. By following the tips and tricks outlined in this article, you can simplify complex radical expressions and arrive at the final answer. Remember to simplify the coefficient first, apply the product rule of radicals, and simplify the square roots individually to arrive at the final answer.