Select The Correct Answer.Simplify The Following Radical Expression: $\sqrt{20}$A. $2 \sqrt{5}$ B. $5 \sqrt{2}$ C. $4 \sqrt{5}$ D. $10 \sqrt{5}$

Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will focus on simplifying the radical expression $\sqrt{20}$ and explore the correct answer among the given options.

Understanding 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 Radical Expressions

To simplify a radical expression, we need to find the largest perfect square that divides the number inside the square root. A perfect square is a number that can be expressed as the product of an integer with itself. For example, 16 is a perfect square because it can be expressed as 4 multiplied by 4.

Simplifying $\sqrt{20}$

To simplify $\sqrt{20}$, we need to find the largest perfect square that divides 20. We can start by listing the factors of 20:

• 1
• 2
• 4
• 5
• 10
• 20

We can see that 4 is a perfect square because it can be expressed as 2 multiplied by 2. Therefore, we can rewrite $\sqrt{20}$ as $\sqrt{4 \times 5}$.

Using the Properties of Square Roots

We know that the square root of a product is equal to the product of the square roots. Therefore, we can rewrite $\sqrt{4 \times 5}$ as $\sqrt{4} \times \sqrt{5}$.

Simplifying the Square Root of 4

The square root of 4 is equal to 2, because 2 multiplied by 2 equals 4. Therefore, we can rewrite $\sqrt{4}$ as 2.

Combining the Results

We have simplified $\sqrt{20}$ to $2 \times \sqrt{5}$. Therefore, the correct answer is:

$2 \sqrt{5}$

Conclusion

Simplifying radical expressions is an essential skill in mathematics. By understanding the properties of square roots and perfect squares, we can simplify complex expressions and arrive at the correct answer. In this article, we have simplified the radical expression $\sqrt{20}$ and arrived at the correct answer, $2 \sqrt{5}$.

Common Mistakes to Avoid

When simplifying radical expressions, it's essential to avoid common mistakes. Here are a few:

• Not identifying perfect squares: Make sure to identify perfect squares and rewrite the expression accordingly.
• Not using the properties of square roots: Remember that the square root of a product is equal to the product of the square roots.
• Not simplifying the expression: Take the time to simplify the expression and arrive at the correct answer.

Practice Problems

To practice simplifying radical expressions, try the following problems:

• Simplify $\sqrt{48}$.
• Simplify $\sqrt{75}$.
• Simplify $\sqrt{100}$.

Answer Key

• $\sqrt{48} = 4 \sqrt{3}$
• $\sqrt{75} = 5 \sqrt{3}$
• $\sqrt{100} = 10$

Final Thoughts

Introduction

In our previous article, we explored the concept of simplifying radical expressions and applied it to the expression $\sqrt{20}$. In this article, we will continue to delve deeper into the world of radical expressions and answer some frequently asked questions.

Q&A Session

Q: What is a radical expression?

A: 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.

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 square root. A perfect square is a number that can be expressed as the product of an integer with itself.

Q: What is a perfect square?

A: A perfect square is a number that can be expressed as the product of an integer with itself. For example, 16 is a perfect square because it can be expressed as 4 multiplied by 4.

Q: How do I identify perfect squares?

A: To identify perfect squares, you can look for numbers that have an even exponent. For example, 16 is a perfect square because it has an exponent of 2, which is even.

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

A: A square root is a value that, when multiplied by itself, gives the original number. A perfect square is a number that can be expressed as the product of an integer with itself.

Q: Can I simplify a radical expression with a variable inside the square root?

A: Yes, you can simplify a radical expression with a variable inside the square root. To do this, you need to find the largest perfect square that divides the variable.

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

A: To simplify a radical expression with a variable inside the square root, you need to find the largest perfect square that divides the variable. For example, if you have $\sqrt{16x}$, you can simplify it to $4\sqrt{x}$.

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

A: Yes, you can simplify a radical expression with a fraction inside the square root. To do this, you need to find the largest perfect square that divides the numerator and denominator.

Q: How do I simplify a radical expression with a fraction inside the square root?

A: To simplify a radical expression with a fraction inside the square root, you need to find the largest perfect square that divides the numerator and denominator. For example, if you have $\sqrt{\frac{16}{9}}$, you can simplify it to $\frac{4}{3}$.

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

A: Yes, you can simplify a radical expression with a negative number inside the square root. To do this, you need to find the largest perfect square that divides the absolute value of the number.

Q: How do I simplify a radical expression with a negative number inside the square root?

A: To simplify a radical expression with a negative number inside the square root, you need to find the largest perfect square that divides the absolute value of the number. For example, if you have $\sqrt{-16}$, you can simplify it to $4i$, where $i$ is the imaginary unit.

Conclusion

Simplifying radical expressions is a crucial skill in mathematics. By understanding the properties of square roots and perfect squares, we can simplify complex expressions and arrive at the correct answer. In this article, we have answered some frequently asked questions and provided examples to help you understand the concept better.

Practice Problems

To practice simplifying radical expressions, try the following problems:

• Simplify $\sqrt{48}$.
• Simplify $\sqrt{75}$.
• Simplify $\sqrt{100}$.
• Simplify $\sqrt{16x}$.
• Simplify $\sqrt{\frac{16}{9}}$.
• Simplify $\sqrt{-16}$.

Answer Key

• $\sqrt{48} = 4 \sqrt{3}$
• $\sqrt{75} = 5 \sqrt{3}$
• $\sqrt{100} = 10$
• $\sqrt{16x} = 4 \sqrt{x}$
• $\sqrt{\frac{16}{9}} = \frac{4}{3}$
• $\sqrt{-16} = 4i$

Final Thoughts

Simplifying radical expressions is a crucial skill in mathematics. By understanding the properties of square roots and perfect squares, we can simplify complex expressions and arrive at the correct answer. Remember to practice regularly and seek help when needed. With time and practice, you'll become proficient in simplifying radical expressions and arrive at the correct answer every time.