Evaluate The Expression: $\[ -7 \sqrt{63} \\]

Feb 26, 2025 by ADMIN 46 views

**Evaluating Expressions with Radicals: A Step-by-Step Guide**

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Introduction

When dealing with mathematical expressions that involve radicals, it's essential to understand the rules and properties of radicals to simplify and evaluate them correctly. In this article, we will focus on evaluating the expression ${-7 \sqrt{63}}$ and explore the concepts and techniques involved in simplifying radical expressions.

Understanding Radicals

A radical is a mathematical expression that represents a number that can be expressed as the product of a perfect square and another number. The radical symbol, denoted by $\sqrt{}$ , is used to indicate that the expression inside the symbol is a radical. For example, $\sqrt{16}$ is a radical because it can be expressed as $4 \times 4$ , where $4$ is a perfect square.

Properties of Radicals

There are several properties of radicals that are essential to understand when simplifying and evaluating radical expressions. These properties include:

Product Property: $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$
Quotient Property: $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$
Power Property: $\sqrt{a^2} = a$ (if $a$ is a positive number)
Zero Property: $\sqrt{0} = 0$

Simplifying Radical Expressions

To simplify a radical expression, we need to find the largest perfect square that divides the number inside the radical symbol. We can then rewrite the expression as the product of the perfect square and another number.

For example, let's simplify the expression $\sqrt{63}$ . We can start by finding the largest perfect square that divides $63$ . Since $63 = 9 \times 7$ , we can rewrite the expression as $\sqrt{9 \times 7}$ . Using the product property, we can simplify this expression as $\sqrt{9} \times \sqrt{7} = 3 \sqrt{7}$ .

Evaluating the Expression

Now that we have a good understanding of radicals and their properties, let's evaluate the expression ${-7 \sqrt{63}}$. To do this, we need to simplify the radical expression inside the square root.

As we discussed earlier, we can simplify $\sqrt{63}$ as $3 \sqrt{7}$ . Therefore, the expression ${-7 \sqrt{63}}$ can be rewritten as ${-7 \times 3 \sqrt{7}}$.

Using the product property, we can simplify this expression as ${-21 \sqrt{7}}$.

Conclusion

In this article, we evaluated the expression ${-7 \sqrt{63}}$ and explored the concepts and techniques involved in simplifying radical expressions. We discussed the properties of radicals, including the product property, quotient property, power property, and zero property. We also learned how to simplify radical expressions by finding the largest perfect square that divides the number inside the radical symbol.

By understanding and applying these concepts, we can simplify and evaluate radical expressions with confidence. Whether you're a student or a professional, mastering the art of simplifying radical expressions is essential for success in mathematics and beyond.

Frequently Asked Questions

Q: What is a radical?

A: A radical is a mathematical expression that represents a number that can be expressed as the product of a perfect square and another number.

Q: What are the properties of radicals?

A: The properties of radicals include the product property, quotient property, power property, and zero property.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to find the largest perfect square that divides the number inside the radical symbol and rewrite the expression as the product of the perfect square and another number.

Q: What is the product property of radicals?

A: The product property of radicals states that $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ .

Q: What is the quotient property of radicals?

A: The quotient property of radicals states that $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$ .

Q: What is the power property of radicals?

A: The power property of radicals states that $\sqrt{a^2} = a$ (if $a$ is a positive number).

Q: What is the zero property of radicals?

A: The zero property of radicals states that $\sqrt{0} = 0$ .

Additional Resources

If you're looking for additional resources to help you master the art of simplifying radical expressions, here are a few suggestions:

Math textbooks: There are many excellent math textbooks available that cover radical expressions in detail.
Online resources: Websites like Khan Academy, Mathway, and Wolfram Alpha offer a wealth of information and resources on radical expressions.
Practice problems: Practice problems are an excellent way to reinforce your understanding of radical expressions. You can find practice problems online or in math textbooks.

By following these tips and resources, you'll be well on your way to becoming a master of simplifying radical expressions. Happy learning!

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Introduction

Simplifying radical expressions can be a challenging task, especially for those who are new to the concept. In this article, we will address some of the most frequently asked questions about simplifying radical expressions, providing you with a better understanding of the subject.

Q: What is a radical?

A: A radical is a mathematical expression that represents a number that can be expressed as the product of a perfect square and another number. The radical symbol, denoted by $\sqrt{}$ , is used to indicate that the expression inside the symbol is a radical.

Q: What are the properties of radicals?

A: The properties of radicals include the product property, quotient property, power property, and zero property. These properties are essential to understand when simplifying and evaluating radical expressions.

Product Property

The product property of radicals states that $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ . This property allows us to simplify radical expressions by combining the radicals.

Quotient Property

The quotient property of radicals states that $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$ . This property allows us to simplify radical expressions by dividing the radicals.

Power Property

The power property of radicals states that $\sqrt{a^2} = a$ (if $a$ is a positive number). This property allows us to simplify radical expressions by raising the number inside the radical to a power.

Zero Property

The zero property of radicals states that $\sqrt{0} = 0$ . This property allows us to simplify radical expressions by setting the number inside the radical to zero.

Q: How do I simplify a radical expression?

Here's a step-by-step guide to simplifying a radical expression:

Find the largest perfect square: Find the largest perfect square that divides the number inside the radical symbol.
Rewrite the expression: Rewrite the expression as the product of the perfect square and another number.
Simplify the expression: Simplify the expression by combining the radicals.

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

A: A radical is a mathematical expression that represents a number that can be expressed as the product of a perfect square and another number. A rational number, on the other hand, is a number that can be expressed as the ratio of two integers.

Q: Can I simplify a radical expression with a negative number?

A: Yes, you can simplify a radical expression with a negative number. To do this, you need to follow the same steps as before, but with the negative 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, but with the variable.

Q: Can I simplify a radical expression with a decimal number?

A: Yes, you can simplify a radical expression with a decimal number. To do this, you need to follow the same steps as before, but with the decimal number.

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

A: To simplify a radical expression with a fraction, you need to follow the same steps as before, but with the fraction.

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

A: Yes, you can simplify a radical expression with a negative fraction. To do this, you need to follow the same steps as before, but with the negative fraction.

Q: How do I simplify a radical expression with a complex number?

A: To simplify a radical expression with a complex number, you need to follow the same steps as before, but with the complex number.