Evaluate The Radical. 8 3 \sqrt[3]{8} 3 8 ​ Enter Your Answer In The Box. □ \square □

Introduction

Radicals, also known as roots, are mathematical operations that involve finding the value of a number that, when raised to a certain power, equals a given value. In this article, we will focus on evaluating radicals, specifically the cube root of 8. We will break down the process into simple steps and provide examples to help you understand the concept.

What is a Radical?

A radical is a mathematical operation that involves finding the value of a number that, when raised to a certain power, equals a given value. The general form of a radical is:

$$\sqrt[n]{x}$$

where $n$ is the index of the radical and $x$ is the radicand.

Evaluating the Cube Root of 8

The cube root of 8 is denoted by $\sqrt[3]{8}$. To evaluate this radical, we need to find the value of $x$ such that $x^3 = 8$.

Step 1: Factorize the Radicand

The first step in evaluating the cube root of 8 is to factorize the radicand. We can write 8 as:

$$8 = 2^3$$

Step 2: Simplify the Radical

Now that we have factorized the radicand, we can simplify the radical. Since $8 = 2^3$, we can write:

$$\sqrt[3]{8} = \sqrt[3]{2^3}$$

Using the property of radicals that $\sqrt[n]{a^n} = a$, we can simplify the radical further:

$$\sqrt[3]{2^3} = 2$$

Conclusion

In this article, we evaluated the cube root of 8 using the steps outlined above. We factorized the radicand, simplified the radical, and arrived at the final answer of 2. This process can be applied to any radical, and with practice, you will become proficient in evaluating radicals.

Common Mistakes to Avoid

When evaluating radicals, there are several common mistakes to avoid:

• Not factorizing the radicand: Failing to factorize the radicand can lead to incorrect simplification of the radical.
• Not using the correct property of radicals: Using the wrong property of radicals can lead to incorrect simplification of the radical.
• Not checking the final answer: Failing to check the final answer can lead to incorrect solutions.

Practice Problems

To practice evaluating radicals, try the following problems:

• $\sqrt[3]{27}$
• $\sqrt[3]{64}$
• $\sqrt[3]{125}$

Solutions

• $\sqrt[3]{27} = 3$
• $\sqrt[3]{64} = 4$
• $\sqrt[3]{125} = 5$

Conclusion

Introduction

In our previous article, we discussed the basics of evaluating radicals, specifically the cube root of 8. In this article, we will provide a Q&A guide to help you better understand the concept of evaluating radicals.

Q: What is a radical?

A: A radical is a mathematical operation that involves finding the value of a number that, when raised to a certain power, equals a given value. The general form of a radical is:

$$\sqrt[n]{x}$$

where $n$ is the index of the radical and $x$ is the radicand.

Q: How do I evaluate a radical?

A: To evaluate a radical, you need to follow these steps:

1. Factorize the radicand: Factorize the radicand to simplify the radical.
2. Simplify the radical: Use the property of radicals that $\sqrt[n]{a^n} = a$ to simplify the radical.
3. Check the final answer: Check the final answer to ensure accuracy.

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

A: A square root is a radical with an index of 2, while a cube root is a radical with an index of 3. The square root of a number is the value that, when squared, equals the given number, while the cube root of a number is the value that, when cubed, equals the given number.

Q: How do I evaluate a radical with a negative index?

A: To evaluate a radical with a negative index, you need to follow these steps:

1. Change the sign of the radicand: Change the sign of the radicand to make it positive.
2. Evaluate the radical: Evaluate the radical using the steps outlined above.
3. Change the sign of the result: Change the sign of the result to make it negative.

Q: Can I simplify a radical with a variable?

A: Yes, you can simplify a radical with a variable. To simplify a radical with a variable, you need to follow these steps:

1. Factorize the radicand: Factorize the radicand to simplify the radical.
2. Use the property of radicals: Use the property of radicals that $\sqrt[n]{a^n} = a$ to simplify the radical.
3. Check the final answer: Check the final answer to ensure accuracy.

Q: How do I evaluate a radical with a decimal?

A: To evaluate a radical with a decimal, you need to follow these steps:

1. Change the decimal to a fraction: Change the decimal to a fraction to make it easier to evaluate.
2. Evaluate the radical: Evaluate the radical using the steps outlined above.
3. Check the final answer: Check the final answer to ensure accuracy.

Q: Can I use a calculator to evaluate a radical?

A: Yes, you can use a calculator to evaluate a radical. However, it's always a good idea to check the final answer to ensure accuracy.

Conclusion

Evaluating radicals is an essential skill in mathematics, and with practice, you will become proficient in evaluating radicals. Remember to factorize the radicand, simplify the radical using the correct property of radicals, and check the final answer to ensure accuracy. If you have any further questions, feel free to ask.

Practice Problems

To practice evaluating radicals, try the following problems:

• $\sqrt[3]{27}$
• $\sqrt[3]{64}$
• $\sqrt[3]{125}$
• $\sqrt[2]{16}$
• $\sqrt[2]{25}$

Solutions

• $\sqrt[3]{27} = 3$
• $\sqrt[3]{64} = 4$
• $\sqrt[3]{125} = 5$
• $\sqrt[2]{16} = 4$
• $\sqrt[2]{25} = 5$