Simplifying Cube Roots: A Step-by-Step Guide

Let's dive into simplifying the expression $\sqrt[3]{1024x^{15}y^{14}}$, given that both $x$ and $y$ are greater than zero. This problem involves understanding how to break down numbers and variables under a cube root to find perfect cube factors. By identifying these factors, we can pull them out of the radical, simplifying the overall expression. Let's begin!

Understanding the Problem

Before we start crunching numbers, it's crucial to understand what the problem is asking. We have a cube root, which means we're looking for factors that appear three times. For example, $\sqrt[3]{8}$ simplifies to 2 because $2 * 2 * 2 = 8$. Similarly, with variables, $\sqrt[3]{x^3}$ simplifies to $x$ because $x * x * x = x^3$. Our goal is to apply this concept to $\sqrt[3]{1024x^{15}y^{14}}$ and simplify it as much as possible.

Breaking Down the Numbers

The first step is to break down 1024 into its prime factors. Prime factorization helps us identify any factors that occur three times, which we can then take out of the cube root. 1024 can be written as $2^{10}$ (2 multiplied by itself ten times). Now we need to see how many groups of three 2's we can make from $2^{10}$. We can make three groups of three 2's ($2^3 * 2^3 * 2^3$), which equals $2^9$, leaving one 2 behind. So, we can rewrite $2^{10}$ as $2^9 * 2$, which is $(2^3)^3 * 2 = 8^3 * 2$. This means $\sqrt[3]{1024}$ can be simplified as $\sqrt[3]{8^3 * 2} = 8\sqrt[3]{2}$. Therefore, understanding the numerical component is key to simplifying the entire expression.

Simplifying the Variables

Next, let's tackle the variables $x^{15}$ and $y^{14}$. For $x^{15}$, we need to determine how many groups of three $x$'s we can make. Since $15$ is divisible by $3$, we can rewrite $x^{15}$ as $(x^5)^3$. This means $\sqrt[3]{x^{15}} = x^5$ because $x^5 * x^5 * x^5 = x^{15}$. For $y^{14}$, we can make four groups of three $y$'s, which equals $y^{12}$, leaving two $y$'s behind. So, we can rewrite $y^{14}$ as $y^{12} * y^2$, which is $(y^4)^3 * y^2$. This means $\sqrt[3]{y^{14}}$ can be simplified as $\sqrt[3]{(y^4)^3 * y^2} = y^4\sqrt[3]{y^2}$. Breaking down variable exponents into multiples of 3 allows us to simplify radical expressions effectively.

Putting It All Together

Now that we've simplified the numerical and variable components, let's put it all back together:

$\sqrt[3]{1024x^{15}y^{14}} = \sqrt[3]{8^3 * 2 * (x^5)^3 * (y^4)^3 * y^2}$

We can now take the cube root of each perfect cube:

$8 * x^5 * y^4 * \sqrt[3]{2y^2}$

So, the simplified expression is $8x^5y^4\sqrt[3]{2y^2}$.

Conclusion

By breaking down the number and variables into their prime factors and identifying groups of three, we successfully simplified the cube root expression. The final simplified expression is $8x^5y^4\sqrt[3]{2y^2}$. Always remember to look for perfect cubes when simplifying cube roots, and you'll be able to tackle these problems with confidence. This step-by-step approach ensures that you understand each part of the simplification process, making it easier to solve similar problems in the future.

Why This Answer Is Correct

The correct answer is $8 x^5 y^4 \sqrt[3]{2 y^2}$ because it accurately reflects the simplified form of the original expression. We arrived at this answer by:

• Prime Factorization: Breaking down 1024 into $2^{10}$ and identifying $2^9$ as a perfect cube ($8^3$).
• Variable Simplification: Recognizing $x^{15}$ as a perfect cube ($x^5)^3$ and simplifying $y^{14}$ into $(y^4)^3 * y^2$.
• Combining Terms: Pulling out the perfect cubes from under the radical and leaving the remaining factors inside.

This methodical approach ensures that each term is correctly simplified, leading to the accurate final expression.

Common Mistakes to Avoid

When simplifying cube roots, it's easy to make a few common mistakes. Here are some pitfalls to watch out for:

• Incorrect Prime Factorization: Ensure you correctly break down the number into its prime factors. A mistake here will throw off the entire simplification.
• Misunderstanding Cube Roots: Remember that you're looking for factors that appear three times, not just two (as in square roots).
• Forgetting to Simplify Variables: Always check if the variable exponents can be divided by 3. If not, separate the term into a perfect cube and a remaining factor.
• Incorrectly Combining Terms: Make sure you only combine terms that are outside the radical. Factors remaining inside the radical cannot be combined with those outside.

By avoiding these common mistakes, you can improve your accuracy and confidence in simplifying cube root expressions.

Additional Tips for Success

Here are some extra tips to help you master simplifying cube roots:

• Practice Regularly: The more you practice, the better you'll become at recognizing perfect cubes and simplifying expressions.
• Use Prime Factorization: Always break down numbers into their prime factors. This makes it easier to identify groups of three.
• Double-Check Your Work: Before finalizing your answer, double-check each step to ensure you haven't made any mistakes.
• Understand the Rules: Make sure you have a solid understanding of the rules of exponents and radicals.

With these tips, you'll be well-equipped to tackle any cube root simplification problem that comes your way!

More Practice Problems

Want to sharpen your skills further? Here are a few practice problems to try:

1. $\sqrt[3]{216a^6b^9}$
2. $\sqrt[3]{64x^{12}y^{15}}$
3. $\sqrt[3]{125p^3q^6}$

Work through these problems using the steps we've discussed, and check your answers to reinforce your understanding.

Conclusion

Simplifying cube roots might seem daunting at first, but with a systematic approach, it becomes much more manageable. Remember to break down numbers and variables into their prime factors, identify perfect cubes, and pull them out of the radical. By following these steps and avoiding common mistakes, you can confidently simplify complex expressions. Happy simplifying!

For further learning, visit Khan Academy's page on radicals to enhance your understanding.