Combine These Radicals: 8 5 + 2 45 8 \sqrt{5} + 2 \sqrt{45} 8 5 ​ + 2 45 ​ Options:A. 4 5 4 \sqrt{5} 4 5 ​ B. 10 5 10 \sqrt{5} 10 5 ​ C. 14 5 14 \sqrt{5} 14 5 ​ D. 74 45 74 \sqrt{45} 74 45 ​

Introduction

Radicals, also known as square roots, are an essential part of mathematics, particularly in algebra and geometry. When dealing with radical expressions, it's crucial to simplify them to make calculations easier and more manageable. In this article, we'll explore how to combine radicals, focusing on the given problem: $8 \sqrt{5} + 2 \sqrt{45}$. We'll break down the steps involved in simplifying radical expressions and provide a clear understanding of the process.

Understanding Radicals

A radical is a mathematical expression that represents the square root of a number. It's denoted by the symbol $\sqrt{}$. For example, $\sqrt{16}$ represents the square root of 16, which is equal to 4. Radicals can be simplified by finding the square root of the number inside the radical sign.

Simplifying Radical Expressions

To simplify a radical expression, we need to find the largest perfect square that divides the number inside the radical sign. This is done by factoring the number into its prime factors and then grouping the factors in pairs. The square root of each pair is then taken, and the remaining factors are left inside the radical sign.

Combining Radicals

When combining radicals, we need to find a common base or a common factor that can be used to simplify the expression. In the given problem, we have two radical expressions: $8 \sqrt{5}$ and $2 \sqrt{45}$. To combine these radicals, we need to find a common base or a common factor that can be used to simplify the expression.

Step 1: Simplify the Second Radical Expression

The second radical expression is $2 \sqrt{45}$. To simplify this expression, we need to find the largest perfect square that divides 45. The prime factorization of 45 is $3^2 \cdot 5$. Therefore, we can simplify the expression as follows:

$$2 \sqrt{45} = 2 \sqrt{3^2 \cdot 5} = 2 \cdot 3 \sqrt{5} = 6 \sqrt{5}$$

Step 2: Combine the Two Radical Expressions

Now that we have simplified the second radical expression, we can combine the two expressions as follows:

$$8 \sqrt{5} + 6 \sqrt{5} = (8 + 6) \sqrt{5} = 14 \sqrt{5}$$

Conclusion

In this article, we explored how to combine radicals by simplifying radical expressions. We broke down the steps involved in simplifying radical expressions and provided a clear understanding of the process. By following these steps, we can simplify complex radical expressions and make calculations easier and more manageable. The given problem, $8 \sqrt{5} + 2 \sqrt{45}$, was simplified to $14 \sqrt{5}$, which is the correct answer.

Final Answer

The final answer is: C. $14 \sqrt{5}$

Additional Resources

For more information on simplifying radical expressions and combining radicals, check out the following resources:

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

FAQs

Q: What is the difference between a radical and a square root? A: A radical is a mathematical expression that represents the square root of a number, while a square root is a specific type of radical.

Q: How do I simplify a radical expression? A: To simplify a radical expression, find the largest perfect square that divides the number inside the radical sign and then take the square root of each pair.

Introduction

Simplifying radical expressions is an essential skill in mathematics, particularly in algebra and geometry. In our previous article, we explored how to combine radicals by simplifying radical expressions. However, we know that there are many more questions and concerns that students and educators may have. In this article, we'll address some of the most frequently asked questions about simplifying radical expressions.

Q&A

Q: What is the difference between a radical and a square root?

A: A radical is a mathematical expression that represents the square root of a number, while a square root is a specific type of radical. For example, $\sqrt{16}$ is a square root, while $4 \sqrt{2}$ is a radical.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, find the largest perfect square that divides the number inside the radical sign and then take the square root of each pair. For example, to simplify $\sqrt{48}$, we can break it down as follows:

$$\sqrt{48} = \sqrt{16 \cdot 3} = \sqrt{16} \cdot \sqrt{3} = 4 \sqrt{3}$$

Q: Can I combine radicals with different bases?

A: Yes, you can combine radicals with different bases by finding a common factor or a common base that can be used to simplify the expression. For example, to combine $3 \sqrt{2}$ and $4 \sqrt{3}$, we can find a common base as follows:

$$3 \sqrt{2} + 4 \sqrt{3} = 3 \sqrt{2} + 4 \sqrt{3} = 3 \sqrt{2} + 4 \sqrt{3} = 3 \sqrt{2} + 4 \sqrt{3}$$

However, we can simplify this expression further by finding a common base:

$$3 \sqrt{2} + 4 \sqrt{3} = 3 \sqrt{2} + 4 \sqrt{3} = 3 \sqrt{2} + 4 \sqrt{3} = 3 \sqrt{2} + 4 \sqrt{3}$$

Q: How do I know if a radical expression can be simplified?

A: A radical expression can be simplified if the number inside the radical sign can be broken down into a product of perfect squares and other factors. For example, $\sqrt{48}$ can be simplified because 48 can be broken down into a product of perfect squares and other factors:

$$\sqrt{48} = \sqrt{16 \cdot 3} = \sqrt{16} \cdot \sqrt{3} = 4 \sqrt{3}$$

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

A: Yes, you can simplify a radical expression with a negative number inside the radical sign. However, you need to follow the rules of exponents and radicals. For example, to simplify $\sqrt{-16}$, we can break it down as follows:

$$\sqrt{-16} = \sqrt{-1 \cdot 16} = \sqrt{-1} \cdot \sqrt{16} = i \cdot 4 = 4i$$

Q: How do I simplify a radical expression with a variable inside the radical sign?

A: To simplify a radical expression with a variable inside the radical sign, you need to follow the same rules as before. For example, to simplify $\sqrt{2x}$, we can break it down as follows:

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

Q: Can I simplify a radical expression with a fraction inside the radical sign?

A: Yes, you can simplify a radical expression with a fraction inside the radical sign. However, you need to follow the rules of exponents and radicals. For example, to simplify $\sqrt{\frac{16}{9}}$, we can break it down as follows:

$$\sqrt{\frac{16}{9}} = \sqrt{\frac{4^2}{3^2}} = \frac{4}{3}$$

Conclusion

Simplifying radical expressions is an essential skill in mathematics, particularly in algebra and geometry. By following the rules and guidelines outlined in this article, you can simplify complex radical expressions and make calculations easier and more manageable. Remember to always follow the order of operations and to simplify radical expressions by finding the largest perfect square that divides the number inside the radical sign.

Final Answer

The final answer is: There is no final answer, as this is a Q&A article.

Additional Resources

For more information on simplifying radical expressions, check out the following resources:

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

FAQs

Q: What is the difference between a radical and a square root? A: A radical is a mathematical expression that represents the square root of a number, while a square root is a specific type of radical.

Q: How do I simplify a radical expression? A: To simplify a radical expression, find the largest perfect square that divides the number inside the radical sign and then take the square root of each pair.

Q: Can I combine radicals with different bases? A: Yes, you can combine radicals with different bases by finding a common factor or a common base that can be used to simplify the expression.

Q: How do I know if a radical expression can be simplified? A: A radical expression can be simplified if the number inside the radical sign can be broken down into a product of perfect squares and other factors.

Q: Can I simplify a radical expression with a negative number inside the radical sign? A: Yes, you can simplify a radical expression with a negative number inside the radical sign. However, you need to follow the rules of exponents and radicals.

Q: How do I simplify a radical expression with a variable inside the radical sign? A: To simplify a radical expression with a variable inside the radical sign, you need to follow the same rules as before.

Q: Can I simplify a radical expression with a fraction inside the radical sign? A: Yes, you can simplify a radical expression with a fraction inside the radical sign. However, you need to follow the rules of exponents and radicals.