Simplify The Expression: { \sqrt{-100 X^8 Y^3}$}$

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us to solve problems more efficiently. When dealing with expressions involving square roots, it's essential to understand the properties of radicals and how to manipulate them. In this article, we will simplify the given expression $\sqrt{-100 x^8 y^3}$ using various mathematical techniques.

Understanding the Properties of Radicals

Before we dive into simplifying the expression, let's review the properties of radicals. The square root of a number is a value that, when multiplied by itself, gives the original number. In mathematical notation, this is represented as $\sqrt{x} = y \iff x = y^2$. We can also extend this concept to higher-order roots, such as cube roots or fourth roots.

One of the key properties of radicals is that they can be simplified by factoring out perfect squares. For example, $\sqrt{16} = \sqrt{4 \cdot 4} = 4 \sqrt{1} = 4$. This property is essential in simplifying expressions involving square roots.

Simplifying the Expression

Now that we have a good understanding of the properties of radicals, let's simplify the given expression $\sqrt{-100 x^8 y^3}$. To simplify this expression, we need to factor out perfect squares and apply the properties of radicals.

First, let's factor out the perfect squares from the expression:

$$\sqrt{-100 x^8 y^3} = \sqrt{-100} \cdot \sqrt{x^8} \cdot \sqrt{y^3}$$

Next, let's simplify each of these terms separately:

$$\sqrt{-100} = \sqrt{-1} \cdot \sqrt{100} = 10i$$

$$\sqrt{x^8} = x^4$$

$$\sqrt{y^3} = y \sqrt{y}$$

Now, let's substitute these simplified terms back into the original expression:

$$\sqrt{-100 x^8 y^3} = 10i x^4 y \sqrt{y}$$

Conclusion

In this article, we simplified the expression $\sqrt{-100 x^8 y^3}$ using various mathematical techniques. We reviewed the properties of radicals and applied them to factor out perfect squares and simplify the expression. The final simplified expression is $10i x^4 y \sqrt{y}$.

Final Answer

The final answer is $\boxed{10i x^4 y \sqrt{y}}$.

Related Topics

• Simplifying expressions involving square roots
• Properties of radicals
• Factoring out perfect squares
• Simplifying expressions using mathematical techniques

Further Reading

If you want to learn more about simplifying expressions involving square roots, I recommend checking out the following resources:

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

FAQs

• Q: What is the property of radicals that allows us to simplify expressions? A: The property of radicals that allows us to simplify expressions is that they can be simplified by factoring out perfect squares.
• Q: How do we simplify expressions involving square roots? A: We simplify expressions involving square roots by factoring out perfect squares and applying the properties of radicals.
• Q: What is the final simplified expression for $\sqrt{-100 x^8 y^3}$? A: The final simplified expression for $\sqrt{-100 x^8 y^3}$ is $10i x^4 y \sqrt{y}$.
  

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Introduction

In our previous article, we simplified the expression $\sqrt{-100 x^8 y^3}$ using various mathematical techniques. In this article, we will answer some frequently asked questions (FAQs) related to simplifying expressions involving square roots.

Q&A

Q: What is the property of radicals that allows us to simplify expressions?

A: The property of radicals that allows us to simplify expressions is that they can be simplified by factoring out perfect squares. This property is essential in simplifying expressions involving square roots.

Q: How do we simplify expressions involving square roots?

A: We simplify expressions involving square roots by factoring out perfect squares and applying the properties of radicals. This involves breaking down the expression into smaller parts, simplifying each part separately, and then combining the simplified parts.

Q: What is the final simplified expression for $\sqrt{-100 x^8 y^3}$?

A: The final simplified expression for $\sqrt{-100 x^8 y^3}$ is $10i x^4 y \sqrt{y}$.

Q: Can we simplify expressions involving negative numbers under the square root?

A: Yes, we can simplify expressions involving negative numbers under the square root. In the case of $\sqrt{-100 x^8 y^3}$, we can factor out the perfect square $-100$ as $10i$, which is a negative number.

Q: How do we handle expressions with variables under the square root?

A: When handling expressions with variables under the square root, we need to consider the properties of radicals and the rules of exponents. We can simplify expressions involving variables under the square root by factoring out perfect squares and applying the properties of radicals.

Q: Can we simplify expressions involving fractions under the square root?

A: Yes, we can simplify expressions involving fractions under the square root. In the case of $\sqrt{\frac{1}{x^2}}$, we can simplify the expression by factoring out the perfect square $\frac{1}{x^2}$ as $\frac{1}{x}$.

Q: How do we simplify expressions involving multiple square roots?

A: When simplifying expressions involving multiple square roots, we need to consider the properties of radicals and the rules of exponents. We can simplify expressions involving multiple square roots by factoring out perfect squares and applying the properties of radicals.

Q: Can we simplify expressions involving complex numbers under the square root?

A: Yes, we can simplify expressions involving complex numbers under the square root. In the case of $\sqrt{-100 x^8 y^3}$, we can factor out the perfect square $-100$ as $10i$, which is a complex number.

Conclusion

In this article, we answered some frequently asked questions (FAQs) related to simplifying expressions involving square roots. We covered topics such as the property of radicals, simplifying expressions involving negative numbers, variables, fractions, and multiple square roots, and complex numbers.

Final Answer

The final answer is $\boxed{10i x^4 y \sqrt{y}}$.

Related Topics

• Simplifying expressions involving square roots
• Properties of radicals
• Factoring out perfect squares
• Simplifying expressions using mathematical techniques
• Complex numbers
• Variables under the square root
• Fractions under the square root
• Multiple square roots

Further Reading

If you want to learn more about simplifying expressions involving square roots, I recommend checking out the following resources:

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

FAQs

• Q: What is the property of radicals that allows us to simplify expressions? A: The property of radicals that allows us to simplify expressions is that they can be simplified by factoring out perfect squares.
• Q: How do we simplify expressions involving square roots? A: We simplify expressions involving square roots by factoring out perfect squares and applying the properties of radicals.
• Q: What is the final simplified expression for $\sqrt{-100 x^8 y^3}$? A: The final simplified expression for $\sqrt{-100 x^8 y^3}$ is $10i x^4 y \sqrt{y}$.