Select All The Correct Answers.Which Expressions Are Equivalent To The Given Expression?$\sqrt{80}$A. $8 \sqrt{5}$B. $4 \sqrt{5}$C. $4 \sqrt{10}$D. $80^{\frac{1}{3}}$E. $160^{\frac{1}{2}}$

Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will explore the concept of equivalent expressions and learn how to simplify radical expressions. We will focus on the given expression $\sqrt{80}$ and determine which of the provided options are equivalent to it.

Understanding Radical Expressions

A radical expression is a mathematical expression that contains a root or a radical sign. The most common radical sign is the square root sign, denoted by $\sqrt{}$. 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 $4 \times 4 = 16$.

Breaking Down the Given Expression

The given expression is $\sqrt{80}$. To simplify this expression, we need to find the largest perfect square that divides 80. 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 $4 \times 4$.

Finding the Largest Perfect Square

To find the largest perfect square that divides 80, we can start by listing the factors of 80:

• 1
• 2
• 4
• 5
• 8
• 10
• 16
• 20
• 40
• 80

From this list, we can see that 16 is the largest perfect square that divides 80. Therefore, we can rewrite the given expression as:

$$\sqrt{80} = \sqrt{16 \times 5}$$

Simplifying the Expression

Now that we have broken down the given expression into its prime factors, we can simplify it further. We can rewrite the expression as:

$$\sqrt{16 \times 5} = \sqrt{16} \times \sqrt{5}$$

Since $\sqrt{16} = 4$, we can simplify the expression to:

$$\sqrt{16} \times \sqrt{5} = 4 \times \sqrt{5}$$

Evaluating the Options

Now that we have simplified the given expression, we can evaluate the options provided:

A. $8 \sqrt{5}$ B. $4 \sqrt{5}$ C. $4 \sqrt{10}$ D. $80^{\frac{1}{3}}$ E. $160^{\frac{1}{2}}$

From our simplification, we can see that option B is equivalent to the given expression $\sqrt{80}$.

Conclusion

In this article, we learned how to simplify radical expressions by breaking them down into their prime factors and identifying the largest perfect square that divides the expression. We applied this concept to the given expression $\sqrt{80}$ and determined that option B, $4 \sqrt{5}$, is equivalent to it. This skill is essential in mathematics, and we hope that this article has provided a clear and concise guide to simplifying radical expressions.

Additional Tips and Examples

• When simplifying radical expressions, always look for the largest perfect square that divides the expression.
• Use the prime factorization of the expression to break it down into its simplest form.
• Be careful when simplifying expressions with multiple radicals, as the order of operations may affect the final result.

Common Mistakes to Avoid

• Failing to identify the largest perfect square that divides the expression.
• Not using the prime factorization of the expression to break it down into its simplest form.
• Ignoring the order of operations when simplifying expressions with multiple radicals.

Real-World Applications

• Simplifying radical expressions is essential in mathematics, particularly in algebra and geometry.
• It is also used in physics and engineering to simplify complex mathematical expressions.
• In finance, simplifying radical expressions can help to calculate interest rates and investment returns.

Final Thoughts

Introduction

In our previous article, we explored the concept of simplifying radical expressions and learned how to break down complex expressions into their simplest form. In this article, we will provide a Q&A guide to help you understand and apply the concepts of simplifying radical expressions.

Q: What is a radical expression?

A: A radical expression is a mathematical expression that contains a root or a radical sign. The most common radical sign is the square root sign, denoted by $\sqrt{}$. 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 expression. 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 $4 \times 4$.

Q: What is the largest perfect square that divides 80?

A: The largest perfect square that divides 80 is 16. This is because 16 is a perfect square that can be expressed as $4 \times 4$, and 80 can be expressed as $16 \times 5$.

Q: How do I simplify the expression $\sqrt{80}$?

A: To simplify the expression $\sqrt{80}$, you need to break it down into its prime factors. The prime factors of 80 are 2, 2, 2, 2, 5. Since 16 is a perfect square that can be expressed as $4 \times 4$, you can rewrite the expression as:

$$\sqrt{80} = \sqrt{16 \times 5}$$

Q: How do I simplify the expression $\sqrt{16 \times 5}$?

A: To simplify the expression $\sqrt{16 \times 5}$, you need to use the prime factorization of the expression. Since 16 is a perfect square that can be expressed as $4 \times 4$, you can rewrite the expression as:

$$\sqrt{16 \times 5} = \sqrt{16} \times \sqrt{5}$$

Q: How do I simplify the expression $\sqrt{16} \times \sqrt{5}$?

A: To simplify the expression $\sqrt{16} \times \sqrt{5}$, you need to evaluate the square root of 16. Since $\sqrt{16} = 4$, you can simplify the expression to:

$$\sqrt{16} \times \sqrt{5} = 4 \times \sqrt{5}$$

Q: What is the final simplified form of the expression $\sqrt{80}$?

A: The final simplified form of the expression $\sqrt{80}$ is $4 \sqrt{5}$.

Q: How do I know if an expression is a perfect square?

A: To determine if an expression is a perfect square, you need to check if it can be expressed as the product of an integer with itself. For example, 16 is a perfect square because it can be expressed as $4 \times 4$.

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

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 $4 \times 4$. A perfect cube is a number that can be expressed as the product of an integer with itself three times. For example, 27 is a perfect cube because it can be expressed as $3 \times 3 \times 3$.

Q: How do I simplify an expression with multiple radicals?

A: To simplify an expression with multiple radicals, you need to follow the order of operations. First, simplify the radicals inside the expression, and then simplify the expression as a whole.

Conclusion

In this Q&A guide, we have provided answers to common questions about simplifying radical expressions. We have also provided examples and explanations to help you understand and apply the concepts of simplifying radical expressions. Remember to always follow the order of operations and to simplify radicals inside the expression before simplifying the expression as a whole. With practice and dedication, you can become proficient in simplifying radical expressions and tackle even the most complex mathematical problems.