Simplify: $\sqrt[3]{250 A^{11}}$A. $5 \sqrt[3]{2 A^{11}}$ B. $15 A^5 \sqrt{10 A}$ C. $5 A^3 \sqrt[3]{2 A^2}$ D. $5 A^5 \sqrt{10 A}$

Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will focus on simplifying the expression $\sqrt[3]{250 a^{11}}$. We will break down the process into manageable steps, using the properties of exponents and radicals to simplify the expression.

Understanding the Properties of Exponents and Radicals

Before we dive into the simplification process, let's review the properties of exponents and radicals that we will use.

• Product of Powers Property: When multiplying two powers with the same base, we add their exponents. For example, $a^m \cdot a^n = a^{m+n}$.
• Power of a Power Property: When raising a power to another power, we multiply their exponents. For example, $(a^m)^n = a^{m \cdot n}$.
• Cube Root Property: The cube root of a number is the number raised to the power of $\frac{1}{3}$. For example, $\sqrt[3]{a} = a^{\frac{1}{3}}$.

Step 1: Factor the Radicand

The first step in simplifying the expression is to factor the radicand, which is the number inside the radical sign. In this case, the radicand is $250 a^{11}$.

250 a^{11} = 2 \cdot 5^3 \cdot a^{11}

Step 2: Simplify the Cube Root

Now that we have factored the radicand, we can simplify the cube root. We will use the property of cube roots to rewrite the expression.

\sqrt[3]{250 a^{11}} = \sqrt[3]{2 \cdot 5^3 \cdot a^{11}}

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

\sqrt[3]{250 a^{11}} = 5 \sqrt[3]{2 \cdot a^{11}}

Step 3: Simplify the Expression Inside the Radical

Now that we have simplified the cube root, we can focus on simplifying the expression inside the radical. We will use the property of exponents to rewrite the expression.

\sqrt[3]{2 \cdot a^{11}} = \sqrt[3]{2 \cdot a^9 \cdot a^2}

Using the property of exponents, we can rewrite the expression as:

\sqrt[3]{2 \cdot a^{11}} = a^3 \sqrt[3]{2 \cdot a^2}

Step 4: Simplify the Final Expression

Now that we have simplified the expression inside the radical, we can simplify the final expression. We will use the property of exponents to rewrite the expression.

5 \sqrt[3]{2 \cdot a^9 \cdot a^2} = 5 a^3 \sqrt[3]{2 \cdot a^2}

Conclusion

In this article, we have simplified the expression $\sqrt[3]{250 a^{11}}$ using the properties of exponents and radicals. We have broken down the process into manageable steps, using the product of powers property, power of a power property, and cube root property to simplify the expression.

The final simplified expression is:

5 a^3 \sqrt[3]{2 \cdot a^2}

This expression cannot be simplified further using the properties of exponents and radicals.

Answer

The correct answer is:

• C. $5 a^3 \sqrt[3]{2 a^2}$

Final Thoughts

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Introduction

In our previous article, we explored the process of simplifying radical expressions using the properties of exponents and radicals. In this article, we will provide a Q&A guide to help you better understand the concepts and apply them to real-world problems.

Q: What is a radical expression?

A: A radical expression is an expression that contains a radical sign, which is denoted by the symbol $\sqrt[n]{x}$. The radical sign indicates that the expression inside the sign is to be raised to the power of $\frac{1}{n}$.

Q: What are the properties of exponents and radicals?

A: The properties of exponents and radicals are:

• Product of Powers Property: When multiplying two powers with the same base, we add their exponents. For example, $a^m \cdot a^n = a^{m+n}$.
• Power of a Power Property: When raising a power to another power, we multiply their exponents. For example, $(a^m)^n = a^{m \cdot n}$.
• Cube Root Property: The cube root of a number is the number raised to the power of $\frac{1}{3}$. For example, $\sqrt[3]{a} = a^{\frac{1}{3}}$.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, follow these steps:

1. Factor the radicand: Factor the number inside the radical sign.
2. Simplify the cube root: Use the property of cube roots to rewrite the expression.
3. Simplify the expression inside the radical: Use the property of exponents to rewrite the expression.
4. Simplify the final expression: Use the property of exponents to rewrite the expression.

Q: What is the difference between a radical expression and a rational expression?

A: A radical expression is an expression that contains a radical sign, while a rational expression is an expression that contains a fraction. For example, $\sqrt[3]{x}$ is a radical expression, while $\frac{x}{y}$ is a rational expression.

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

A: Yes, you can simplify a radical expression with a negative exponent. To do this, follow these steps:

1. Change the negative exponent to a positive exponent: Change the negative exponent to a positive exponent by multiplying the expression by $-1$.
2. Simplify the expression: Simplify the expression using the properties of exponents and radicals.

Q: Can I simplify a radical expression with a variable exponent?

A: Yes, you can simplify a radical expression with a variable exponent. To do this, follow these steps:

1. Simplify the expression: Simplify the expression using the properties of exponents and radicals.
2. Use the variable exponent: Use the variable exponent to rewrite the expression.

Q: What are some common mistakes to avoid when simplifying radical expressions?

A: Some common mistakes to avoid when simplifying radical expressions include:

• Not factoring the radicand: Failing to factor the radicand can lead to incorrect simplification.
• Not using the property of cube roots: Failing to use the property of cube roots can lead to incorrect simplification.
• Not simplifying the expression inside the radical: Failing to simplify the expression inside the radical can lead to incorrect simplification.

Conclusion

In this article, we have provided a Q&A guide to help you better understand the concepts of simplifying radical expressions. We have covered topics such as the properties of exponents and radicals, simplifying radical expressions, and common mistakes to avoid. We hope that this article has provided a clear and concise guide to simplifying radical expressions.

Final Thoughts

Simplifying radical expressions is a crucial skill to master in mathematics. By understanding the properties of exponents and radicals, we can simplify complex expressions and arrive at the final answer. In this article, we have provided a Q&A guide to help you better understand the concepts and apply them to real-world problems. We hope that this article has provided a valuable resource for you to learn and grow.