Select All The Correct Answers.Which Expressions Are Equivalent To The Given Expression 40 \sqrt{40} 40 ​ ?A. 40 1 2 40^{\frac{1}{2}} 4 0 2 1 ​ B. 4 10 4 \sqrt{10} 4 10 ​ C. 2 10 2 \sqrt{10} 2 10 ​ D. 160 1 2 160^{\frac{1}{2}} 16 0 2 1 ​ E. 5 8 5 \sqrt{8} 5 8 ​

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 equivalent expressions and how to simplify radical expressions, specifically focusing on the given expression $\sqrt{40}$. We will examine each option and determine which expressions are equivalent to the given expression.

Understanding Radical Expressions

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

Simplifying Radical Expressions

To simplify a radical expression, we need to find the largest perfect square that divides the number inside the radical sign. We can then rewrite the expression as the product of the perfect square and the remaining factor. For example, to simplify $\sqrt{40}$, we can rewrite it as $\sqrt{4 \times 10}$, which is equal to $2 \sqrt{10}$.

Analyzing the Options

Now, let's analyze each option and determine which expressions are equivalent to the given expression $\sqrt{40}$.

Option A: $40^{\frac{1}{2}}$

This option is equivalent to the given expression $\sqrt{40}$, because $40^{\frac{1}{2}}$ is the same as $\sqrt{40}$. However, this option is not in the simplest form, because we can simplify it further by rewriting it as $2 \sqrt{10}$.

Option B: $4 \sqrt{10}$

This option is not equivalent to the given expression $\sqrt{40}$, because it is a different expression. However, we can simplify it further by rewriting it as $2 \sqrt{10}$, which is equivalent to the given expression.

Option C: $2 \sqrt{10}$

This option is equivalent to the given expression $\sqrt{40}$, because we can rewrite it as $\sqrt{4 \times 10}$, which is equal to $\sqrt{40}$. This option is in the simplest form, because we cannot simplify it further.

Option D: $160^{\frac{1}{2}}$

This option is not equivalent to the given expression $\sqrt{40}$, because it is a different expression. However, we can simplify it further by rewriting it as $12 \sqrt{10}$, which is not equivalent to the given expression.

Option E: $5 \sqrt{8}$

Q&A: 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 expression is the square root, denoted by $\sqrt{x}$. 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, we need to find the largest perfect square that divides the number inside the radical sign. We can then rewrite the expression as the product of the perfect square and the remaining factor. For example, to simplify $\sqrt{40}$, we can rewrite it as $\sqrt{4 \times 10}$, which is equal to $2 \sqrt{10}$.

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 and 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 and 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 a radical expression with a perfect cube?

A: To simplify a radical expression with a perfect cube, we need to find the largest perfect cube that divides the number inside the radical sign. We can then rewrite the expression as the product of the perfect cube and the remaining factor. For example, to simplify $\sqrt[3]{64}$, we can rewrite it as $\sqrt[3]{4 \times 4 \times 4}$, which is equal to $4 \sqrt[3]{4}$.

Q: What is the difference between a rational and an irrational number?

A: A rational number is a number that can be expressed as the ratio of two integers. For example, 3/4 is a rational number because it can be expressed as the ratio of 3 and 4. An irrational number is a number that cannot be expressed as the ratio of two integers. For example, the square root of 2 is an irrational number because it cannot be expressed as the ratio of two integers.

Q: How do I simplify a radical expression with an irrational number?

A: To simplify a radical expression with an irrational number, we need to find the largest perfect square that divides the number inside the radical sign. We can then rewrite the expression as the product of the perfect square and the remaining factor. For example, to simplify $\sqrt{2}$, we can rewrite it as $\sqrt{1 \times 2}$, which is equal to $\sqrt{1} \times \sqrt{2}$, which is equal to $1 \times \sqrt{2}$, which is equal to $\sqrt{2}$.

Q: What is the difference between a real and an imaginary number?

A: A real number is a number that can be expressed as a rational or irrational number. For example, 3 and $\sqrt{2}$ are real numbers because they can be expressed as rational or irrational numbers. An imaginary number is a number that can be expressed as the square root of a negative number. For example, $i$ is an imaginary number because it can be expressed as the square root of -1.

Q: How do I simplify a radical expression with an imaginary number?

A: To simplify a radical expression with an imaginary number, we need to find the largest perfect square that divides the number inside the radical sign. We can then rewrite the expression as the product of the perfect square and the remaining factor. For example, to simplify $\sqrt{-4}$, we can rewrite it as $\sqrt{-1 \times 4}$, which is equal to $\sqrt{-1} \times \sqrt{4}$, which is equal to $i \times 2$, which is equal to $2i$.

Conclusion

Simplifying radical expressions is a crucial skill for students and professionals alike. By understanding the concept of equivalent expressions and how to simplify radical expressions, we can solve complex mathematical problems with ease. In this article, we have explored the concept of radical expressions, simplified radical expressions, and answered common questions related to simplifying radical expressions. We hope that this article has provided valuable information and insights for our readers.