Simplify: $\sqrt[3]{125 X^{21} Y^{33}}$

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Understanding the Problem

When dealing with radicals, it's essential to simplify the expression by breaking down the radicand into its prime factors. In this case, we're given the expression $\sqrt[3]{125 x^{21} y^{33}}$, and our goal is to simplify it.

Breaking Down the Radicand

To simplify the expression, we need to break down the radicand into its prime factors. We can start by factoring 125, which is equal to $5^3$. So, we can rewrite the expression as $\sqrt[3]{5^3 x^{21} y^{33}}$.

Simplifying the Radicand

Now that we have the radicand broken down into its prime factors, we can simplify it further. We know that $\sqrt[3]{5^3} = 5$, so we can rewrite the expression as $5\sqrt[3]{x^{21} y^{33}}$.

Simplifying the Radicals

Next, we need to simplify the radicals inside the expression. We can start by factoring out the largest perfect cube from the radicand. In this case, we can factor out $x^6$ and $y^9$ from the radicand.

Factoring Out Perfect Cubes

We can rewrite the expression as $5\sqrt[3]{x^6 x^3 y^9 y^3}$. Now, we can simplify the radicals inside the expression by taking the cube root of the perfect cubes.

Simplifying the Radicals

Taking the cube root of the perfect cubes, we get $5x^2 y^3 \sqrt[3]{x^3 y^3}$.

Final Simplification

Now that we have simplified the radicals, we can simplify the expression further. We can rewrite the expression as $5x^2 y^3 \sqrt[3]{x^3 y^3}$.

Conclusion

In conclusion, we have simplified the expression $\sqrt[3]{125 x^{21} y^{33}}$ by breaking down the radicand into its prime factors, simplifying the radicand, and simplifying the radicals. The final simplified expression is $5x^2 y^3 \sqrt[3]{x^3 y^3}$.

Key Takeaways

• To simplify a radical expression, we need to break down the radicand into its prime factors.
• We can simplify the radicand by factoring out the largest perfect cube.
• We can simplify the radicals by taking the cube root of the perfect cubes.
• The final simplified expression is $5x^2 y^3 \sqrt[3]{x^3 y^3}$.

Common Mistakes to Avoid

• Not breaking down the radicand into its prime factors.
• Not factoring out the largest perfect cube.
• Not taking the cube root of the perfect cubes.

Real-World Applications

• Simplifying radical expressions is essential in many real-world applications, such as engineering, physics, and computer science.
• Understanding how to simplify radical expressions can help us solve complex problems in these fields.

Final Thoughts

Simplifying radical expressions is a crucial skill in mathematics, and it requires a deep understanding of the underlying concepts. By breaking down the radicand into its prime factors, simplifying the radicand, and simplifying the radicals, we can simplify complex radical expressions and solve problems in various fields.

Additional Resources

• For more information on simplifying radical expressions, check out the following resources:
• Khan Academy: Simplifying Radical Expressions
• Mathway: Simplifying Radical Expressions
• Wolfram Alpha: Simplifying Radical Expressions

Frequently Asked Questions

• Q: How do I simplify a radical expression? A: To simplify a radical expression, you need to break down the radicand into its prime factors, simplify the radicand, and simplify the radicals.
• Q: What is the largest perfect cube? A: The largest perfect cube is the largest number that can be expressed as the product of three equal integers.
• Q: How do I take the cube root of a perfect cube? A: To take the cube root of a perfect cube, you need to divide the number by 3 and then take the square root of the result.
  

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Understanding the Problem

When dealing with radicals, it's essential to simplify the expression by breaking down the radicand into its prime factors. In this case, we're given the expression $\sqrt[3]{125 x^{21} y^{33}}$, and our goal is to simplify it.

Q&A

Q: What is the first step in simplifying a radical expression?

A: The first step in simplifying a radical expression is to break down the radicand into its prime factors.

Q: How do I break down the radicand into its prime factors?

A: To break down the radicand into its prime factors, you need to factor out the largest perfect square or perfect cube from the radicand.

Q: What is the largest perfect square or perfect cube?

A: The largest perfect square or perfect cube is the largest number that can be expressed as the product of two or three equal integers.

Q: How do I simplify the radicand?

A: To simplify the radicand, you need to take the square root of the perfect square and the cube root of the perfect cube.

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

A: A perfect square is a number that can be expressed as the product of two equal integers, while a perfect cube is a number that can be expressed as the product of three equal integers.

Q: How do I simplify the radicals?

A: To simplify the radicals, you need to take the cube root of the perfect cubes and simplify the resulting expression.

Q: What is the final simplified expression?

A: The final simplified expression is $5x^2 y^3 \sqrt[3]{x^3 y^3}$.

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

A: Some common mistakes to avoid when simplifying radical expressions include not breaking down the radicand into its prime factors, not factoring out the largest perfect square or perfect cube, and not taking the square root of the perfect square and the cube root of the perfect cube.

Q: What are some real-world applications of simplifying radical expressions?

A: Simplifying radical expressions is essential in many real-world applications, such as engineering, physics, and computer science. Understanding how to simplify radical expressions can help us solve complex problems in these fields.

Q: Where can I find more information on simplifying radical expressions?

A: For more information on simplifying radical expressions, check out the following resources:

• Khan Academy: Simplifying Radical Expressions
• Mathway: Simplifying Radical Expressions
• Wolfram Alpha: Simplifying Radical Expressions

Additional Tips and Tricks

• When simplifying radical expressions, it's essential to break down the radicand into its prime factors.
• Factoring out the largest perfect square or perfect cube can help simplify the radicand.
• Taking the square root of the perfect square and the cube root of the perfect cube can help simplify the radicals.
• Simplifying radical expressions is essential in many real-world applications, such as engineering, physics, and computer science.

Conclusion

In conclusion, simplifying radical expressions is a crucial skill in mathematics, and it requires a deep understanding of the underlying concepts. By breaking down the radicand into its prime factors, simplifying the radicand, and simplifying the radicals, we can simplify complex radical expressions and solve problems in various fields.

Final Thoughts

Simplifying radical expressions is a complex topic, and it requires a lot of practice and patience. However, with the right resources and guidance, anyone can learn how to simplify radical expressions and become proficient in this skill.

Additional Resources

• For more information on simplifying radical expressions, check out the following resources:
• Khan Academy: Simplifying Radical Expressions
• Mathway: Simplifying Radical Expressions
• Wolfram Alpha: Simplifying Radical Expressions

Frequently Asked Questions

• Q: How do I simplify a radical expression? A: To simplify a radical expression, you need to break down the radicand into its prime factors, simplify the radicand, and simplify the radicals.
• Q: What is the largest perfect square or perfect cube? A: The largest perfect square or perfect cube is the largest number that can be expressed as the product of two or three equal integers.
• Q: How do I take the square root of a perfect square and the cube root of a perfect cube? A: To take the square root of a perfect square and the cube root of a perfect cube, you need to divide the number by 2 and then take the square root of the result, and divide the number by 3 and then take the square root of the result.