Write Each Expression In Simplest Form.23. $\sqrt[3]{250}$

Introduction

Radical expressions, also known as roots, are a fundamental concept in mathematics. They are used to represent the value of a number that, when multiplied by itself a certain number of times, gives a specified value. In this article, we will focus on simplifying radical expressions, specifically cube roots. We will use the expression $\sqrt[3]{250}$ as an example to demonstrate the steps involved in simplifying radical expressions.

What is a Cube Root?

A cube root is a type of radical expression that represents the value of a number that, when multiplied by itself three times, gives a specified value. The cube root of a number $x$ is denoted by $\sqrt[3]{x}$. For example, $\sqrt[3]{8}$ represents the value of a number that, when multiplied by itself three times, gives 8.

Simplifying the Expression $\sqrt[3]{250}$

To simplify the expression $\sqrt[3]{250}$, we need to find the largest perfect cube that divides 250. A perfect cube is a number that can be expressed as the product of three equal integers. For example, 8 is a perfect cube because it can be expressed as $2^3$.

Step 1: Factorize 250

To simplify the expression $\sqrt[3]{250}$, we need to factorize 250 into its prime factors. The prime factorization of 250 is:

250 = 2 × 5 × 5 × 5

Step 2: Identify the Largest Perfect Cube

From the prime factorization of 250, we can see that 5 × 5 × 5 is a perfect cube. Therefore, we can rewrite 250 as:

250 = 5^3 × 2

Step 3: Simplify the Expression

Now that we have identified the largest perfect cube, we can simplify the expression $\sqrt[3]{250}$ by taking the cube root of the perfect cube:

$\sqrt[3]{250}$ = $\sqrt[3]{5^3 × 2}$

Using the property of cube roots, we can rewrite this expression as:

$\sqrt[3]{250}$ = 5 × $\sqrt[3]{2}$

Conclusion

In this article, we have demonstrated the steps involved in simplifying radical expressions, specifically cube roots. We used the expression $\sqrt[3]{250}$ as an example to show how to identify the largest perfect cube and simplify the expression. By following these steps, we can simplify any radical expression and find its simplest form.

Common Mistakes to Avoid

When simplifying radical expressions, there are several common mistakes to avoid. These include:

• Not identifying the largest perfect cube
• Not using the property of cube roots to simplify the expression
• Not checking the expression for any remaining factors that can be simplified

Tips and Tricks

Here are some tips and tricks to help you simplify radical expressions:

• Always factorize the number into its prime factors
• Identify the largest perfect cube and use the property of cube roots to simplify the expression
• Check the expression for any remaining factors that can be simplified
• Use a calculator to check your answer and ensure that it is correct

Practice Problems

Here are some practice problems to help you practice simplifying radical expressions:

• $\sqrt[3]{216}$
• $\sqrt[3]{343}$
• $\sqrt[3]{512}$

Answer Key

Here are the answers to the practice problems:

• $\sqrt[3]{216}$ = 6
• $\sqrt[3]{343}$ = 7
• $\sqrt[3]{512}$ = 8
  Simplifying Radical Expressions: A Q&A Guide =====================================================

Introduction

In our previous article, we discussed the steps involved in simplifying radical expressions, specifically cube roots. We used the expression $\sqrt[3]{250}$ as an example to demonstrate the process. In this article, we will answer some frequently asked questions about simplifying radical expressions.

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

A: A square root is a type of radical expression that represents the value of a number that, when multiplied by itself, gives a specified value. A cube root, on the other hand, represents the value of a number that, when multiplied by itself three times, gives a specified value.

Q: How do I simplify a radical expression with a variable?

A: To simplify a radical expression with a variable, you need to factorize the variable into its prime factors and identify the largest perfect power of the variable. Then, you can use the property of radical expressions to simplify the expression.

Q: Can I simplify a radical expression with a negative number?

A: Yes, you can simplify a radical expression with a negative number. However, you need to remember that the cube root of a negative number is also negative.

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

A: To simplify a radical expression with a decimal number, you need to convert the decimal number to a fraction and then simplify the expression.

Q: Can I simplify a radical expression with a fraction?

A: Yes, you can simplify a radical expression with a fraction. However, you need to remember that the cube root of a fraction is also a fraction.

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

A: To simplify a radical expression with a mixed number, you need to convert the mixed number to an improper fraction and then simplify the expression.

Q: Can I simplify a radical expression with a negative fraction?

A: Yes, you can simplify a radical expression with a negative fraction. However, you need to remember that the cube root of a negative fraction is also a negative fraction.

Q: How do I simplify a radical expression with a variable and a constant?

A: To simplify a radical expression with a variable and a constant, you need to factorize the variable into its prime factors and identify the largest perfect power of the variable. Then, you can use the property of radical expressions to simplify the expression.

Q: Can I simplify a radical expression with a variable and a negative number?

A: Yes, you can simplify a radical expression with a variable and a negative number. However, you need to remember that the cube root of a negative number is also negative.

Q: How do I simplify a radical expression with a variable and a decimal number?

A: To simplify a radical expression with a variable and a decimal number, you need to convert the decimal number to a fraction and then simplify the expression.

Q: Can I simplify a radical expression with a variable and a fraction?

A: Yes, you can simplify a radical expression with a variable and a fraction. However, you need to remember that the cube root of a fraction is also a fraction.

Conclusion

In this article, we have answered some frequently asked questions about simplifying radical expressions. We hope that this article has been helpful in clarifying any doubts you may have had about simplifying radical expressions.

Common Mistakes to Avoid

When simplifying radical expressions, there are several common mistakes to avoid. These include:

• Not identifying the largest perfect power of the variable
• Not using the property of radical expressions to simplify the expression
• Not checking the expression for any remaining factors that can be simplified

Tips and Tricks

Here are some tips and tricks to help you simplify radical expressions:

• Always factorize the variable into its prime factors
• Identify the largest perfect power of the variable and use the property of radical expressions to simplify the expression
• Check the expression for any remaining factors that can be simplified
• Use a calculator to check your answer and ensure that it is correct

Practice Problems

Here are some practice problems to help you practice simplifying radical expressions:

• $\sqrt[3]{216}$
• $\sqrt[3]{343}$
• $\sqrt[3]{512}$

Answer Key

Here are the answers to the practice problems:

• $\sqrt[3]{216}$ = 6
• $\sqrt[3]{343}$ = 7
• $\sqrt[3]{512}$ = 8