Select The Correct Answer.Select The Simplification That Accurately Explains The Following Statement: 2 4 = 2 1 2 \sqrt[4]{2}=2^{\frac{1}{2}} 4 2 ​ = 2 2 1 ​ A. $\left(2 {\frac{1}{2}}\right) 4=2^{\frac{1}{2}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot

Introduction

In mathematics, there are various ways to represent the same value. One such representation is the relationship between roots and exponents. The given statement $\sqrt[4]{2}=2^{\frac{1}{2}}$ is a classic example of this relationship. In this article, we will delve into the world of roots and exponents, exploring the correct simplification of the given statement.

What are Roots and Exponents?

Before we dive into the relationship between roots and exponents, let's briefly discuss what they are. A root is a value that, when raised to a certain power, gives a specified number. For example, the fourth root of 2, denoted as $\sqrt[4]{2}$, is a value that, when raised to the power of 4, gives 2. On the other hand, an exponent is a value that is raised to a certain power. For instance, $2^{\frac{1}{2}}$ represents 2 raised to the power of $\frac{1}{2}$.

The Relationship Between Roots and Exponents

Now that we have a basic understanding of roots and exponents, let's explore the relationship between them. The given statement $\sqrt[4]{2}=2^{\frac{1}{2}}$ can be rewritten as $\left(2^{\frac{1}{2}}\right)^4=2^{\frac{1}{2}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}}$. This is because when we raise a power to another power, we multiply the exponents.

Simplifying the Expression

To simplify the expression, we can use the properties of exponents. When we multiply powers with the same base, we add the exponents. Therefore, $2^{\frac{1}{2}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}}$ can be simplified to $2^{\frac{1}{2}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}}$. This simplifies to $2^{\frac{1}{2}+\frac{3}{4}}$, which is equal to $2^{\frac{2}{4}+\frac{3}{4}}$, and finally simplifies to $2^{\frac{5}{4}}$.

Conclusion

In conclusion, the correct simplification of the given statement $\sqrt[4]{2}=2^{\frac{1}{2}}$ is $\left(2^{\frac{1}{2}}\right)^4=2^{\frac{1}{2}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}}$, which simplifies to $2^{\frac{5}{4}}$. This demonstrates the relationship between roots and exponents, and how we can use the properties of exponents to simplify expressions.

Frequently Asked Questions

Q: What is the relationship between roots and exponents?

A: The relationship between roots and exponents is that a root can be represented as an exponent. For example, the fourth root of 2 can be represented as $2^{\frac{1}{4}}$.

Q: How do we simplify expressions involving roots and exponents?

A: We can simplify expressions involving roots and exponents by using the properties of exponents. When we multiply powers with the same base, we add the exponents.

Q: What is the correct simplification of the given statement $\sqrt[4]{2}=2^{\frac{1}{2}}$?

A: The correct simplification of the given statement $\sqrt[4]{2}=2^{\frac{1}{2}}$ is $\left(2^{\frac{1}{2}}\right)^4=2^{\frac{1}{2}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}}$, which simplifies to $2^{\frac{5}{4}}$.

Final Answer

The final answer is: $\boxed{2^{\frac{5}{4}}}$

Introduction

In our previous article, we explored the relationship between roots and exponents, and how we can use the properties of exponents to simplify expressions. In this article, we will continue to delve into the world of roots and exponents, answering some of the most frequently asked questions in this topic.

Q&A: Roots and Exponents

Q: What is the difference between a root and an exponent?

A: A root is a value that, when raised to a certain power, gives a specified number. For example, the fourth root of 2, denoted as $\sqrt[4]{2}$, is a value that, when raised to the power of 4, gives 2. On the other hand, an exponent is a value that is raised to a certain power. For instance, $2^{\frac{1}{2}}$ represents 2 raised to the power of $\frac{1}{2}$.

Q: How do we represent a root as an exponent?

A: We can represent a root as an exponent by using the property that $a^{\frac{1}{n}}=\sqrt[n]{a}$. For example, the fourth root of 2 can be represented as $2^{\frac{1}{4}}$.

Q: What is the relationship between the nth root and the nth power?

A: The nth root and the nth power are related by the property that $\sqrt[n]{a}=a^{\frac{1}{n}}$. This means that the nth root of a number can be represented as the nth power of that number.

Q: How do we simplify expressions involving roots and exponents?

A: We can simplify expressions involving roots and exponents by using the properties of exponents. When we multiply powers with the same base, we add the exponents. For example, $2^{\frac{1}{2}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}}$ can be simplified to $2^{\frac{5}{4}}$.

Q: What is the correct simplification of the given statement $\sqrt[4]{2}=2^{\frac{1}{2}}$?

A: The correct simplification of the given statement $\sqrt[4]{2}=2^{\frac{1}{2}}$ is $\left(2^{\frac{1}{2}}\right)^4=2^{\frac{1}{2}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}}$, which simplifies to $2^{\frac{5}{4}}$.

Q: How do we handle negative exponents?

A: When we have a negative exponent, we can handle it by using the property that $a^{-n}=\frac{1}{a^n}$. For example, $2^{-\frac{1}{2}}$ can be simplified to $\frac{1}{2^{\frac{1}{2}}}$.

Q: What is the relationship between the reciprocal of a root and the reciprocal of an exponent?

A: The reciprocal of a root and the reciprocal of an exponent are related by the property that $\frac{1}{\sqrt[n]{a}}=\sqrt[n]{\frac{1}{a}}$. This means that the reciprocal of the nth root of a number is equal to the nth root of the reciprocal of that number.

Conclusion

In conclusion, roots and exponents are two fundamental concepts in mathematics that are closely related. By understanding the properties of exponents and how to simplify expressions involving roots and exponents, we can solve a wide range of mathematical problems. We hope that this article has provided you with a comprehensive guide to roots and exponents, and that you will find it helpful in your future mathematical endeavors.

Frequently Asked Questions

Q: What is the difference between a root and an exponent?

A: A root is a value that, when raised to a certain power, gives a specified number. For example, the fourth root of 2, denoted as $\sqrt[4]{2}$, is a value that, when raised to the power of 4, gives 2. On the other hand, an exponent is a value that is raised to a certain power. For instance, $2^{\frac{1}{2}}$ represents 2 raised to the power of $\frac{1}{2}$.

Q: How do we represent a root as an exponent?

A: We can represent a root as an exponent by using the property that $a^{\frac{1}{n}}=\sqrt[n]{a}$. For example, the fourth root of 2 can be represented as $2^{\frac{1}{4}}$.

Q: What is the relationship between the nth root and the nth power?

A: The nth root and the nth power are related by the property that $\sqrt[n]{a}=a^{\frac{1}{n}}$. This means that the nth root of a number can be represented as the nth power of that number.

Final Answer

The final answer is: $\boxed{2^{\frac{5}{4}}}$