Which Expressions Are Listed In The Simplest Form? Check All That Apply.- $5 \sqrt{3b}$- $2 \sqrt{21}$- $x \sqrt{8}$- $2y \sqrt{36}$- $\sqrt{5}$- $c \sqrt{12c^2}$

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 concept of simplifying radical expressions and identify which of the given expressions are listed in the simplest form.

What is a Simplified Radical Expression?

A simplified radical expression is a radical expression that has been reduced to its simplest form, where the radicand (the number inside the square root) is expressed in terms of its prime factors. A simplified radical expression has no perfect square factors in the radicand.

Simplifying Radical Expressions

To simplify a radical expression, we need to identify the perfect square factors in the radicand and take their square roots. We can do this by factoring the radicand into its prime factors and then taking the square root of each factor.

Example 1: Simplifying $5 \sqrt{3b}$

Let's simplify the expression $5 \sqrt{3b}$. We can start by factoring the radicand into its prime factors:

$$3b = 3 \cdot b$$

Since 3 is a perfect square factor, we can take its square root:

$$\sqrt{3} = \sqrt{3 \cdot 1} = \sqrt{3} \cdot \sqrt{1} = \sqrt{3}$$

So, the simplified expression is:

$$5 \sqrt{3b} = 5 \sqrt{3} \sqrt{b}$$

Example 2: Simplifying $2 \sqrt{21}$

Let's simplify the expression $2 \sqrt{21}$. We can start by factoring the radicand into its prime factors:

$$21 = 3 \cdot 7$$

Since neither 3 nor 7 is a perfect square factor, we cannot simplify the expression further.

Example 3: Simplifying $x \sqrt{8}$

Let's simplify the expression $x \sqrt{8}$. We can start by factoring the radicand into its prime factors:

$$8 = 2 \cdot 2 \cdot 2 = 2^3$$

Since 2 is a perfect square factor, we can take its square root:

$$\sqrt{8} = \sqrt{2^3} = \sqrt{2^2} \cdot \sqrt{2} = 2 \sqrt{2}$$

So, the simplified expression is:

$$x \sqrt{8} = x \cdot 2 \sqrt{2} = 2x \sqrt{2}$$

Example 4: Simplifying $2y \sqrt{36}$

Let's simplify the expression $2y \sqrt{36}$. We can start by factoring the radicand into its prime factors:

$$36 = 6 \cdot 6 = 6^2$$

Since 6 is a perfect square factor, we can take its square root:

$$\sqrt{36} = \sqrt{6^2} = 6$$

So, the simplified expression is:

$$2y \sqrt{36} = 2y \cdot 6 = 12y$$

Example 5: Simplifying $\sqrt{5}$

The expression $\sqrt{5}$ is already in its simplest form, since 5 is a prime number and has no perfect square factors.

Example 6: Simplifying $c \sqrt{12c^2}$

Let's simplify the expression $c \sqrt{12c^2}$. We can start by factoring the radicand into its prime factors:

$$12c^2 = 4 \cdot 3 \cdot c^2$$

Since 4 is a perfect square factor, we can take its square root:

$$\sqrt{12c^2} = \sqrt{4 \cdot 3 \cdot c^2} = \sqrt{4} \cdot \sqrt{3} \cdot \sqrt{c^2} = 2 \sqrt{3} \cdot c$$

So, the simplified expression is:

$$c \sqrt{12c^2} = c \cdot 2 \sqrt{3} \cdot c = 2c^2 \sqrt{3}$$

Conclusion

In conclusion, the expressions that are listed in the simplest form are:

• $2 \sqrt{21}$
• $\sqrt{5}$
• $c \sqrt{12c^2}$

The other expressions can be simplified further by factoring the radicand into its prime factors and taking the square root of each factor.

Final Answer

The final answer is:

• $2 \sqrt{21}$
• $\sqrt{5}$
• $c \sqrt{12c^2}$
  Simplifying Radical Expressions: A Comprehensive Guide ===========================================================

Q&A: Simplifying Radical Expressions

Q: What is a simplified radical expression?

A: A simplified radical expression is a radical expression that has been reduced to its simplest form, where the radicand (the number inside the square root) is expressed in terms of its prime factors. A simplified radical expression has no perfect square factors in the radicand.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to identify the perfect square factors in the radicand and take their square roots. You can do this by factoring the radicand into its prime factors and then taking the square root of each factor.

Q: What is the difference between a simplified radical expression and a non-simplified radical expression?

A: A simplified radical expression has no perfect square factors in the radicand, while a non-simplified radical expression has perfect square factors in the radicand.

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

A: Yes, you can simplify a radical expression with a negative radicand. To do this, you need to factor the radicand into its prime factors and then take the square root of each factor.

Q: How do I simplify a radical expression with a variable radicand?

A: To simplify a radical expression with a variable radicand, you need to factor the radicand into its prime factors and then take the square root of each factor.

Q: Can I simplify a radical expression with a rational radicand?

A: Yes, you can simplify a radical expression with a rational radicand. To do this, you need to factor the radicand into its prime factors and then take the square root of each factor.

Q: What is the rule for simplifying radical expressions with multiple factors?

A: The rule for simplifying radical expressions with multiple factors is to take the square root of each factor and then multiply the results.

Q: Can I simplify a radical expression with a complex radicand?

A: Yes, you can simplify a radical expression with a complex radicand. To do this, you need to factor the radicand into its prime factors and then take the square root of each factor.

Q: How do I check if a radical expression is simplified?

A: To check if a radical expression is simplified, you need to make sure that there are no perfect square factors in the radicand.

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

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

• Not factoring the radicand into its prime factors
• Not taking the square root of each factor
• Not multiplying the results when simplifying radical expressions with multiple factors

Q: How do I practice simplifying radical expressions?

A: To practice simplifying radical expressions, you can try simplifying different types of radical expressions, such as:

• Simplifying radical expressions with perfect square factors
• Simplifying radical expressions with variable radicands
• Simplifying radical expressions with rational radicands
• Simplifying radical expressions with complex radicands

Conclusion

In conclusion, simplifying radical expressions is an important skill in mathematics that requires attention to detail and practice. By following the rules and guidelines outlined in this article, you can simplify radical expressions with ease and confidence.

Final Answer

The final answer is:

• Simplify radical expressions by factoring the radicand into its prime factors and taking the square root of each factor.
• Check if a radical expression is simplified by making sure that there are no perfect square factors in the radicand.
• Practice simplifying different types of radical expressions to become proficient in simplifying radical expressions.