Simplify The Expression:$\[ -2 \sqrt{54} - 3 \sqrt{24} + 2 \sqrt{5} \\]

Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students and professionals alike. In this article, we will focus on simplifying the given expression: $-2 \sqrt{54} - 3 \sqrt{24} + 2 \sqrt{5}$. We will break down the process into manageable steps, using a combination of mathematical techniques and strategies to simplify the expression.

Understanding the Basics of Radical Expressions

Before we dive into the simplification process, it's essential to understand the basics of radical expressions. A radical expression is a mathematical expression that contains a square root or a higher-order root. 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: Breaking Down the Problem

To simplify the given expression, we need to break it down into smaller, more manageable parts. We will start by simplifying each radical expression individually, and then combine the simplified expressions to get the final result.

Simplifying the First Radical Expression: $-2 \sqrt{54}$

The first radical expression is $-2 \sqrt{54}$. To simplify this expression, we need to find the prime factorization of 54. The prime factorization of 54 is $2 \times 3^3$. We can rewrite the radical expression as:

$$-2 \sqrt{2 \times 3^3}$$

Using the property of radicals, we can rewrite the expression as:

$$-2 \times 3 \sqrt{2 \times 3}$$

Simplifying further, we get:

$$-6 \sqrt{6}$$

Simplifying the Second Radical Expression: $-3 \sqrt{24}$

The second radical expression is $-3 \sqrt{24}$. To simplify this expression, we need to find the prime factorization of 24. The prime factorization of 24 is $2^3 \times 3$. We can rewrite the radical expression as:

$$-3 \sqrt{2^3 \times 3}$$

Using the property of radicals, we can rewrite the expression as:

$$-3 \times 2 \sqrt{2 \times 3}$$

Simplifying further, we get:

$$-6 \sqrt{6}$$

Simplifying the Third Radical Expression: $2 \sqrt{5}$

The third radical expression is $2 \sqrt{5}$. This expression is already simplified, so we can move on to the next step.

Combining the Simplified Expressions

Now that we have simplified each radical expression individually, we can combine the simplified expressions to get the final result. We have:

$$-6 \sqrt{6} - 6 \sqrt{6} + 2 \sqrt{5}$$

Combining like terms, we get:

$$-12 \sqrt{6} + 2 \sqrt{5}$$

Conclusion

Simplifying radical expressions is a crucial skill for students and professionals alike. In this article, we have broken down the process of simplifying the given expression into manageable steps, using a combination of mathematical techniques and strategies. We have simplified each radical expression individually, and then combined the simplified expressions to get the final result. By following these steps, you can simplify any radical expression and get the final result.

Frequently Asked Questions

• Q: What is the prime factorization of 54? A: The prime factorization of 54 is $2 \times 3^3$.
• Q: What is the prime factorization of 24? A: The prime factorization of 24 is $2^3 \times 3$.
• Q: How do I simplify a radical expression? A: To simplify a radical expression, you need to find the prime factorization of the number inside the radical, and then use the property of radicals to rewrite the expression.

Final Answer

The final answer is: $\boxed{-12 \sqrt{6} + 2 \sqrt{5}}$

Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students and professionals alike. In our previous article, we broke down the process of simplifying the given expression into manageable steps, using a combination of mathematical techniques and strategies. In this article, we will provide a Q&A guide to help you understand and simplify radical expressions.

Q&A Guide

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

A: A radical expression is a mathematical expression that contains a square root or a higher-order root, while a rational expression is a mathematical expression that contains a fraction.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to find the prime factorization of the number inside the radical, and then use the property of radicals to rewrite the expression.

Q: What is the property of radicals?

A: The property of radicals states that the product of two or more radicals is equal to the product of the numbers inside the radicals. For example, $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$.

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

A: To simplify a radical expression with a coefficient, you need to factor out the coefficient from the radical expression. For example, $2 \sqrt{3} = \sqrt{4} \times \sqrt{3} = 2 \sqrt{3}$.

Q: What is the difference between a perfect square and a non-perfect square?

A: A perfect square is a number that can be expressed as the square of an integer, while a non-perfect square is a number that cannot be expressed as the square of an integer.

Q: How do I simplify a radical expression with a perfect square?

A: To simplify a radical expression with a perfect square, you can rewrite the expression as the product of the perfect square and the remaining radical. For example, $\sqrt{16} = \sqrt{4 \times 4} = 4 \sqrt{1} = 4$.

Q: What is the difference between a rationalizing denominator and a non-rationalizing denominator?

A: A rationalizing denominator is a denominator that can be expressed as a rational number, while a non-rationalizing denominator is a denominator that cannot be expressed as a rational number.

Q: How do I rationalize a denominator?

A: To rationalize a denominator, you need to multiply the numerator and denominator by the conjugate of the denominator. For example, $\frac{\sqrt{2}}{1 + \sqrt{2}} = \frac{\sqrt{2}}{1 + \sqrt{2}} \times \frac{1 - \sqrt{2}}{1 - \sqrt{2}} = \frac{\sqrt{2} - 2}{1 - 2} = -\sqrt{2} + 2$.

Conclusion

Simplifying radical expressions is a crucial skill for students and professionals alike. In this article, we have provided a Q&A guide to help you understand and simplify radical expressions. By following these steps and using the property of radicals, you can simplify any radical expression and get the final result.

Frequently Asked Questions

• Q: What is the difference between a radical expression and a rational expression? A: A radical expression is a mathematical expression that contains a square root or a higher-order root, while a rational expression is a mathematical expression that contains a fraction.
• Q: How do I simplify a radical expression? A: To simplify a radical expression, you need to find the prime factorization of the number inside the radical, and then use the property of radicals to rewrite the expression.
• Q: What is the property of radicals? A: The property of radicals states that the product of two or more radicals is equal to the product of the numbers inside the radicals.

Final Answer

The final answer is: $\boxed{-12 \sqrt{6} + 2 \sqrt{5}}$