What Is $\sqrt[3]{125 X^{12}}$?A. $5 X^2$B. $5 X^4$C. $25 X^2$D. $25 X^4$

Understanding the Problem

The given problem involves finding the cube root of a variable expression, which is a fundamental concept in algebra. We are required to simplify the expression $\sqrt[3]{125 x^{12}}$ and determine the correct answer from the given options.

Breaking Down the Expression

To simplify the expression, we need to break it down into its prime factors. The expression can be rewritten as $\sqrt[3]{125} \times \sqrt[3]{x^{12}}$. Now, let's focus on simplifying each part separately.

Simplifying the Cube Root of 125

The cube root of 125 can be found by identifying its prime factors. We know that $125 = 5^3$. Therefore, the cube root of 125 is simply 5, as $\sqrt[3]{5^3} = 5$.

**Simplifying the Cube Root of $x^{12}$

Next, let's simplify the cube root of $x^{12}$. We can rewrite $x^{12}$ as $(x^4)^3$. Now, we can take the cube root of both the expression and the exponent, which gives us $\sqrt[3]{(x^4)^3} = x^4$.

Combining the Simplified Expressions

Now that we have simplified both parts of the expression, we can combine them to get the final result. We have $\sqrt[3]{125} \times \sqrt[3]{x^{12}} = 5 \times x^4 = 5x^4$.

Conclusion

Based on our simplification, we can conclude that the correct answer is $5x^4$. This is the final result of the given expression $\sqrt[3]{125 x^{12}}$.

Answer

The correct answer is B. $5 x^4$.

Why is this Important?

Understanding how to simplify cube roots of variable expressions is crucial in algebra and mathematics. It helps us to solve complex equations and expressions, and it's a fundamental concept in many mathematical operations.

Real-World Applications

Simplifying cube roots of variable expressions has many real-world applications, such as:

• Physics and Engineering: When dealing with complex equations and expressions, simplifying cube roots is essential to solve problems related to motion, forces, and energies.
• Computer Science: In computer science, simplifying cube roots is used in algorithms and data structures to optimize performance and solve complex problems.
• Economics: In economics, simplifying cube roots is used to model and analyze complex economic systems, such as supply and demand curves.

Tips and Tricks

Here are some tips and tricks to help you simplify cube roots of variable expressions:

• Identify Prime Factors: When simplifying cube roots, identify the prime factors of the expression to simplify it.
• Use Exponent Rules: Use exponent rules to simplify the expression and make it easier to work with.
• Practice, Practice, Practice: The more you practice simplifying cube roots, the more comfortable you'll become with the process.

Conclusion

Frequently Asked Questions

Q: What is the cube root of a variable expression?

A: The cube root of a variable expression is a mathematical operation that involves finding the value that, when multiplied by itself three times, gives the original expression.

Q: How do I simplify the cube root of a variable expression?

A: To simplify the cube root of a variable expression, you need to identify the prime factors of the expression and use exponent rules to simplify it.

Q: What are some common mistakes to avoid when simplifying cube roots?

A: Some common mistakes to avoid when simplifying cube roots include:

• Not identifying prime factors: Failing to identify the prime factors of the expression can lead to incorrect simplification.
• Not using exponent rules: Not using exponent rules can make the simplification process more complicated than it needs to be.
• Not checking the final result: Not checking the final result can lead to incorrect answers.

Q: How do I check my work when simplifying cube roots?

A: To check your work when simplifying cube roots, you need to:

• Verify the prime factors: Verify that you have identified the correct prime factors of the expression.
• Apply exponent rules correctly: Apply exponent rules correctly to simplify the expression.
• Check the final result: Check the final result to ensure that it is correct.

Q: What are some real-world applications of simplifying cube roots?

A: Simplifying cube roots has many real-world applications, including:

• Physics and Engineering: Simplifying cube roots is used to solve complex equations and expressions related to motion, forces, and energies.
• Computer Science: Simplifying cube roots is used in algorithms and data structures to optimize performance and solve complex problems.
• Economics: Simplifying cube roots is used to model and analyze complex economic systems, such as supply and demand curves.

Q: How can I practice simplifying cube roots?

A: You can practice simplifying cube roots by:

• Working through examples: Work through examples of simplifying cube roots to practice the process.
• Using online resources: Use online resources, such as calculators and worksheets, to practice simplifying cube roots.
• Seeking help: Seek help from a teacher or tutor if you are struggling to simplify cube roots.

Q: What are some common expressions that involve cube roots?

A: Some common expressions that involve cube roots include:

• $\sqrt[3]{x^3}$: This expression simplifies to $x$.
• $\sqrt[3]{x^6}$: This expression simplifies to $x^2$.
• $\sqrt[3]{x^9}$: This expression simplifies to $x^3$.

Q: How do I simplify cube roots with negative numbers?

A: To simplify cube roots with negative numbers, you need to:

• Identify the prime factors: Identify the prime factors of the expression, including any negative numbers.
• Apply exponent rules: Apply exponent rules correctly to simplify the expression.
• Check the final result: Check the final result to ensure that it is correct.

Conclusion

In conclusion, simplifying cube roots of variable expressions is a fundamental concept in algebra and mathematics. By understanding how to simplify these expressions, we can solve complex equations and expressions, and it has many real-world applications. With practice and patience, you can become proficient in simplifying cube roots and tackle complex mathematical problems with confidence.