Perform The Indicated Operations.${ 2 \sqrt{54} + 3 \sqrt{6} = }$

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 explore the process of simplifying radical expressions, with a focus on the given problem: $2 \sqrt{54} + 3 \sqrt{6}$. We will break down the solution into manageable steps, using a combination of mathematical techniques and logical reasoning.

Understanding Radical Expressions

A radical expression is a mathematical expression that contains a square root or other 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. Radical expressions can be simplified by factoring out perfect squares, which are numbers that can be expressed as the product of a whole number and itself.

Breaking Down the Problem

The given problem is $2 \sqrt{54} + 3 \sqrt{6}$. To simplify this expression, we need to break it down into smaller parts. We can start by factoring the square root of 54. The square root of 54 can be expressed as the square root of (9 x 6), because 9 is a perfect square and 6 is a perfect square.

Factoring the Square Root of 54

The square root of 54 can be factored as follows:

$$\sqrt{54} = \sqrt{9 \times 6}$$

We can simplify this expression by taking the square root of 9, which is 3, and multiplying it by the square root of 6.

$$\sqrt{54} = 3 \sqrt{6}$$

Substituting the Simplified Expression

Now that we have simplified the square root of 54, we can substitute this expression back into the original problem.

$$2 \sqrt{54} + 3 \sqrt{6} = 2 (3 \sqrt{6}) + 3 \sqrt{6}$$

Distributing the 2

We can distribute the 2 to the terms inside the parentheses.

$$2 (3 \sqrt{6}) + 3 \sqrt{6} = 6 \sqrt{6} + 3 \sqrt{6}$$

Combining Like Terms

We can combine the like terms, which are the terms that have the same variable and exponent.

$$6 \sqrt{6} + 3 \sqrt{6} = 9 \sqrt{6}$$

Conclusion

In this article, we have simplified the radical expression $2 \sqrt{54} + 3 \sqrt{6}$ by factoring the square root of 54 and combining like terms. We have used a combination of mathematical techniques and logical reasoning to arrive at the simplified expression. This problem demonstrates the importance of simplifying radical expressions, which is a crucial skill for students and professionals alike.

Tips and Tricks

• When simplifying radical expressions, look for perfect squares that can be factored out.
• Use the distributive property to distribute coefficients to terms inside parentheses.
• Combine like terms to simplify the expression.

Common Mistakes

• Failing to factor out perfect squares.
• Not distributing coefficients to terms inside parentheses.
• Not combining like terms.

Real-World Applications

Simplifying radical expressions has many real-world applications, including:

• Calculating the area and perimeter of geometric shapes.
• Solving problems in physics and engineering.
• Working with financial data and statistics.

Conclusion

Introduction

In our previous article, we explored the process of simplifying radical expressions, with a focus on the given problem: $2 \sqrt{54} + 3 \sqrt{6}$. We broke down the solution into manageable steps, using a combination of mathematical techniques and logical reasoning. In this article, we will answer some of the most frequently asked questions about simplifying radical expressions.

Q: What is a radical expression?

A: A radical expression is a mathematical expression that contains a square root or other root. The square root of a number is a value that, when multiplied by itself, gives the original number.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, look for perfect squares that can be factored out. Use the distributive property to distribute coefficients to terms inside parentheses. Finally, combine like terms to simplify the expression.

Q: What is a perfect square?

A: A perfect square is a number that can be expressed as the product of a whole number and itself. For example, 9 is a perfect square because it can be expressed as 3 x 3.

Q: How do I factor out perfect squares?

A: To factor out perfect squares, look for numbers that can be expressed as the product of a whole number and itself. For example, the square root of 54 can be factored as the square root of (9 x 6), because 9 is a perfect square and 6 is a perfect square.

Q: What is the distributive property?

A: The distributive property is a mathematical property that allows us to distribute coefficients to terms inside parentheses. For example, 2(3x) = 6x.

Q: How do I combine like terms?

A: To combine like terms, look for terms that have the same variable and exponent. For example, 6x + 3x can be combined to give 9x.

Q: What are some common mistakes to avoid when simplifying radical expressions?

A: Some common mistakes to avoid when simplifying radical expressions include:

• Failing to factor out perfect squares.
• Not distributing coefficients to terms inside parentheses.
• Not combining like terms.

Q: What are some real-world applications of simplifying radical expressions?

A: Simplifying radical expressions has many real-world applications, including:

• Calculating the area and perimeter of geometric shapes.
• Solving problems in physics and engineering.
• Working with financial data and statistics.

Q: How can I practice simplifying radical expressions?

A: You can practice simplifying radical expressions by working through problems in a textbook or online resource. You can also try simplifying radical expressions on your own, using a combination of mathematical techniques and logical reasoning.

Conclusion

In conclusion, simplifying radical expressions is a crucial skill for students and professionals alike. By following the steps outlined in this article, you can simplify complex radical expressions and arrive at the correct solution. Remember to look for perfect squares, distribute coefficients, and combine like terms to simplify radical expressions. With practice and patience, you can become proficient in simplifying radical expressions and tackle complex mathematical problems with confidence.

Additional Resources

• Textbooks and online resources for simplifying radical expressions.
• Practice problems and exercises for simplifying radical expressions.
• Online communities and forums for discussing simplifying radical expressions.

Frequently Asked Questions

• What is a radical expression?
• How do I simplify a radical expression?
• What is a perfect square?
• How do I factor out perfect squares?
• What is the distributive property?
• How do I combine like terms?
• What are some common mistakes to avoid when simplifying radical expressions?
• What are some real-world applications of simplifying radical expressions?
• How can I practice simplifying radical expressions?