Simplify The Expression:$\sqrt[3]{\frac{1}{8}}$

Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently and accurately. One of the most common types of expressions that require simplification is radicals, particularly cube roots. In this article, we will focus on simplifying the expression $\sqrt[3]{\frac{1}{8}}$ using various mathematical techniques.

Understanding Cube Roots

Before we dive into simplifying the expression, let's first understand what cube roots are. A cube root of a number is a value that, when multiplied by itself twice, gives the original number. In mathematical notation, this is represented as $\sqrt[3]{x} = y$, where $y$ is the cube root of $x$. For example, $\sqrt[3]{8} = 2$ because $2 \times 2 \times 2 = 8$.

Simplifying the Expression

To simplify the expression $\sqrt[3]{\frac{1}{8}}$, we need to find the cube root of $\frac{1}{8}$. One way to approach this is to rewrite $\frac{1}{8}$ as a product of prime factors. We can write $\frac{1}{8}$ as $\frac{1}{2^3}$, which means that the cube root of $\frac{1}{8}$ is equivalent to the cube root of $\frac{1}{2^3}$.

Using Prime Factorization

Using prime factorization, we can rewrite $\frac{1}{2^3}$ as $\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}$. Now, we can take the cube root of each factor separately. The cube root of $\frac{1}{2}$ is $\frac{1}{\sqrt[3]{2}}$, and the cube root of $\frac{1}{2}$ is also $\frac{1}{\sqrt[3]{2}}$. Therefore, the cube root of $\frac{1}{2^3}$ is $\frac{1}{\sqrt[3]{2}} \times \frac{1}{\sqrt[3]{2}} \times \frac{1}{\sqrt[3]{2}}$.

Simplifying Further

We can simplify the expression further by combining the cube roots. Using the property of cube roots that $\sqrt[3]{a} \times \sqrt[3]{b} = \sqrt[3]{ab}$, we can rewrite the expression as $\frac{1}{\sqrt[3]{2^3}}$. Now, we can simplify the cube root of $2^3$ as $\sqrt[3]{2^3} = 2$, and therefore, the expression becomes $\frac{1}{2}$.

Conclusion

In this article, we simplified the expression $\sqrt[3]{\frac{1}{8}}$ using various mathematical techniques, including prime factorization and the properties of cube roots. We showed that the cube root of $\frac{1}{8}$ is equivalent to the cube root of $\frac{1}{2^3}$, and by taking the cube root of each factor separately, we arrived at the simplified expression $\frac{1}{2}$. This example demonstrates the importance of simplifying expressions in mathematics and how it can help us solve problems more efficiently and accurately.

Frequently Asked Questions

• What is the cube root of $\frac{1}{8}$?
• How do we simplify the expression $\sqrt[3]{\frac{1}{8}}$?
• What is the relationship between the cube root of $\frac{1}{8}$ and the cube root of $\frac{1}{2^3}$?

Final Answer

The final answer is $\boxed{\frac{1}{2}}$.

Introduction

In our previous article, we simplified the expression $\sqrt[3]{\frac{1}{8}}$ using various mathematical techniques, including prime factorization and the properties of cube roots. In this article, we will answer some of the most frequently asked questions related to this topic.

Q&A

Q: What is the cube root of $\frac{1}{8}$?

A: The cube root of $\frac{1}{8}$ is equivalent to the cube root of $\frac{1}{2^3}$, which is $\frac{1}{\sqrt[3]{2^3}}$. Simplifying further, we get $\frac{1}{2}$.

Q: How do we simplify the expression $\sqrt[3]{\frac{1}{8}}$?

A: To simplify the expression $\sqrt[3]{\frac{1}{8}}$, we need to find the cube root of $\frac{1}{8}$. One way to approach this is to rewrite $\frac{1}{8}$ as a product of prime factors. We can write $\frac{1}{8}$ as $\frac{1}{2^3}$, which means that the cube root of $\frac{1}{8}$ is equivalent to the cube root of $\frac{1}{2^3}$.

Q: What is the relationship between the cube root of $\frac{1}{8}$ and the cube root of $\frac{1}{2^3}$?

A: The cube root of $\frac{1}{8}$ is equivalent to the cube root of $\frac{1}{2^3}$. This is because $\frac{1}{8}$ can be written as $\frac{1}{2^3}$, and the cube root of a product is equal to the product of the cube roots.

Q: Can we simplify the expression $\sqrt[3]{\frac{1}{8}}$ further?

A: Yes, we can simplify the expression $\sqrt[3]{\frac{1}{8}}$ further by combining the cube roots. Using the property of cube roots that $\sqrt[3]{a} \times \sqrt[3]{b} = \sqrt[3]{ab}$, we can rewrite the expression as $\frac{1}{\sqrt[3]{2^3}}$. Now, we can simplify the cube root of $2^3$ as $\sqrt[3]{2^3} = 2$, and therefore, the expression becomes $\frac{1}{2}$.

Q: What is the final answer to the expression $\sqrt[3]{\frac{1}{8}}$?

A: The final answer to the expression $\sqrt[3]{\frac{1}{8}}$ is $\boxed{\frac{1}{2}}$.

Conclusion

In this article, we answered some of the most frequently asked questions related to the expression $\sqrt[3]{\frac{1}{8}}$. We showed that the cube root of $\frac{1}{8}$ is equivalent to the cube root of $\frac{1}{2^3}$, and by taking the cube root of each factor separately, we arrived at the simplified expression $\frac{1}{2}$. This example demonstrates the importance of simplifying expressions in mathematics and how it can help us solve problems more efficiently and accurately.

Final Answer

The final answer is $\boxed{\frac{1}{2}}$.