Simplifying Cube Roots Equivalent Expression For $\sqrt[3]{64 A^6 B^7 C^9}$

Hey math enthusiasts! Let's dive into a fascinating algebraic problem today. We're going to break down the expression $\sqrt[3]{64 a^6 b^7 c^9}$ and figure out which of the given options is its equivalent. This kind of problem is super common in algebra, and mastering it will definitely boost your math skills. So, let's get started, guys!

Understanding Cube Roots and Simplification

Before we jump into the solution, let's quickly recap what cube roots are all about. A cube root of a number is a value that, when multiplied by itself three times, gives you the original number. For instance, the cube root of 8 is 2 because 2 * 2 * 2 = 8. When we deal with variables and exponents under a cube root, we're looking for factors that are perfect cubes – meaning their exponents are divisible by 3. This understanding is crucial for simplifying expressions like the one we have.

Our main goal here is to extract as many perfect cubes as we can from under the radical (the cube root symbol). We'll do this by breaking down each term (the constant, the variables, and their exponents) and identifying the largest perfect cube factors. This process often involves a bit of trial and error, but with practice, you'll become a pro at spotting these perfect cubes. Remember, we're aiming to make the expression inside the cube root as simple as possible.

For example, consider the term $a^6$. Since 6 is divisible by 3, $a^6$ is a perfect cube. Its cube root is $a^{6/3} = a^2$. But what about $b^7$? Well, 7 isn't divisible by 3, but we can rewrite $b^7$ as $b^6 * b$, where $b^6$ is a perfect cube. This little trick of breaking down the exponents is key to simplifying these expressions. We'll apply this same logic to all the terms under the cube root in our problem.

So, keep in mind, we need to identify perfect cube factors, extract them from the cube root, and then see what's left behind. This is the essence of simplifying cube roots, and it's a technique you'll use again and again in algebra. Let's apply this to our specific problem and see how it all works out.

Breaking Down the Expression $\sqrt[3]{64 a^6 b^7 c^9}$

Now, let's tackle the expression $\sqrt[3]{64 a^6 b^7 c^9}$ step by step. We'll break it down into its components and simplify each one individually. This approach makes the whole process much more manageable and less intimidating. Think of it as dissecting a puzzle – piece by piece, we'll reveal the solution.

First, let's look at the constant term, 64. We need to find the cube root of 64. Ask yourself: what number, when multiplied by itself three times, equals 64? The answer is 4, since 4 * 4 * 4 = 64. So, $\sqrt[3]{64} = 4$. This is our first piece of the puzzle!

Next, let's consider the variable terms. We have $a^6$, $b^7$, and $c^9$. Remember, we're looking for exponents that are divisible by 3. For $a^6$, the exponent 6 is divisible by 3, so $\sqrt[3]{a^6} = a^{6/3} = a^2$. Easy peasy!

Now, let's tackle $b^7$. As we discussed earlier, 7 isn't divisible by 3, so we need to rewrite $b^7$ as a product of a perfect cube and a remaining factor. We can write $b^7$ as $b^6 * b$. The cube root of $b^6$ is $b^{6/3} = b^2$, and we'll leave the remaining 'b' under the cube root. This is where things get a little more interesting!

Finally, we have $c^9$. The exponent 9 is divisible by 3, so $\sqrt[3]{c^9} = c^{9/3} = c^3$. Another straightforward simplification!

So, to recap, we've found that $\sqrt[3]{64} = 4$, $\sqrt[3]{a^6} = a^2$, $\sqrt[3]{b^7} = b^2 \sqrt[3]{b}$, and $\sqrt[3]{c^9} = c^3$. Now, the fun part: putting it all back together!

Combining the Simplified Terms

Alright, guys, we've simplified each part of the expression $\sqrt[3]{64 a^6 b^7 c^9}$. Now comes the exciting part: putting those pieces back together to get our final simplified expression. This is where we'll combine the cube roots we found for each term and see which of the answer options matches our result. Think of it as the grand finale of our simplification journey!

We found that $\sqrt[3]{64} = 4$, $\sqrt[3]{a^6} = a^2$, $\sqrt[3]{b^7} = b^2 \sqrt[3]{b}$, and $\sqrt[3]{c^9} = c^3$. Now, let's multiply these together:

$4 * a^2 * b^2 \sqrt[3]{b} * c^3$

This can be rewritten as:

$4 a^2 b^2 c^3 \sqrt[3]{b}$

And there we have it! Our simplified expression is $4 a^2 b^2 c^3 \sqrt[3]{b}$. Now, let's compare this to the answer options given in the problem. Remember, we're looking for the option that is exactly equivalent to our simplified form.

Looking back at the options:

A. $2 a b c^2(\sqrt[3]{4 a^2 b^3 c})$ B. $4 a^2 b^2 c^3(\sqrt[3]{b})$ C. $8 a^3 b^3 c^4(\sqrt[3]{b c})$ D. $8 a^2 b^2 c^3(\sqrt[3]{b})$

We can clearly see that option B, $4 a^2 b^2 c^3(\sqrt[3]{b})$, matches our simplified expression perfectly! So, we've found our answer. This process of breaking down, simplifying, and then combining is a powerful technique for tackling complex algebraic expressions.

The Final Answer and Key Takeaways

So, after our deep dive into the expression $\sqrt[3]{64 a^6 b^7 c^9}$, we've successfully simplified it to $4 a^2 b^2 c^3 \sqrt[3]{b}$. This means that option B is the correct answer. Woohoo! Give yourselves a pat on the back for making it through this algebraic adventure.

But beyond just getting the right answer, what have we learned along the way? Let's recap some key takeaways that will help you in future simplification problems:

1. Understand Cube Roots: Remember that a cube root of a number is a value that, when multiplied by itself three times, gives you the original number. This fundamental understanding is crucial.
2. Identify Perfect Cubes: When simplifying expressions with variables and exponents, look for perfect cubes – terms where the exponent is divisible by 3. This is the cornerstone of simplification.
3. Break Down and Conquer: Don't be intimidated by complex expressions. Break them down into smaller, manageable parts. Simplify each part individually, and then combine the results.
4. Rewrite Exponents: If an exponent isn't divisible by 3, rewrite the term as a product of a perfect cube and a remaining factor. This trick is essential for handling variables like $b^7$.
5. Double-Check Your Work: Always compare your simplified expression to the answer options to ensure you've arrived at the correct solution.

Simplifying radical expressions like this one is a fundamental skill in algebra. By mastering these techniques, you'll be well-equipped to tackle a wide range of algebraic problems. Keep practicing, and you'll become a simplification superstar in no time! You got this, guys!

Understanding the Problem

The original question asks us to identify which expression is equivalent to $\sqrt[3]{64 a^6 b^7 c^9}$. This requires a solid understanding of cube roots and how to simplify expressions with exponents and variables under a radical. Let's break down why this is an important concept in algebra.

Simplifying radical expressions is a fundamental skill that appears frequently in various mathematical contexts. It's not just about finding the answer to this specific question; it's about developing a broader understanding of how to manipulate algebraic expressions. This skill is crucial for higher-level math courses and even real-world applications where you need to work with complex formulas and equations.

When we simplify a radical expression, we're essentially trying to rewrite it in its most basic form. This often involves extracting perfect cube factors from under the radical symbol. A perfect cube is a number or variable that can be expressed as the cube of another number or variable (e.g., 8 is a perfect cube because 2 * 2 * 2 = 8, and $a^6$ is a perfect cube because $a^2 * a^2 * a^2 = a^6$).

The process of simplification makes the expression easier to work with and understand. It can also reveal hidden relationships and patterns that might not be obvious in the original form. So, by mastering this skill, you're not just solving problems; you're developing a deeper understanding of mathematical structures.

In this particular case, we have a cube root expression, which means we're looking for factors that appear three times under the radical. The challenge lies in identifying these factors and extracting them correctly while leaving the remaining factors inside the radical. This involves a combination of numerical understanding (knowing the cube roots of common numbers) and algebraic manipulation (applying the rules of exponents). So, let's dive in and see how we can unravel this expression!

Step-by-Step Solution

The options given are:

A. $2 a b c^2(\sqrt[3]{4 a^2 b^3 c})$ B. $4 a^2 b^2 c^3(\sqrt[3]{b})$ C. $8 a^3 b^3 c^4(\sqrt[3]{b c})$ D. $8 a^2 b^2 c^3(\sqrt[3]{b})$

Let’s analyze the original expression $\sqrt[3]{64 a^6 b^7 c^9}$ step by step to determine its simplified form:

1. Simplify the constant:
   • $\sqrt[3]{64}$ = 4, because 4 * 4 * 4 = 64.
2. Simplify the variable $a^6$:
   • $\sqrt[3]{a^6}$ = $a^{6/3}$ = $a^2$
3. Simplify the variable $b^7$:
   • We can rewrite $b^7$ as $b^6 * b$ because 6 is the largest multiple of 3 less than 7.
   • $\sqrt[3]{b^7}$ = $\sqrt[3]{b^6 * b}$ = $\sqrt[3]{b^6} * \sqrt[3]{b}$ = $b^{6/3} * \sqrt[3]{b}$ = $b^2(\sqrt[3]{b})$
4. Simplify the variable $c^9$:
   • $\sqrt[3]{c^9}$ = $c^{9/3}$ = $c^3$

Now, let's combine these simplified terms:

$4 * a^2 * b^2(\sqrt[3]{b}) * c^3$ = $4 a^2 b^2 c^3(\sqrt[3]{b})$

Comparing our simplified expression, $4 a^2 b^2 c^3(\sqrt[3]{b})$, with the given options, we can see that option B matches exactly.

Therefore, the correct answer is B. $4 a^2 b^2 c^3(\sqrt[3]{b})$

Tips for Similar Problems

Solving problems like this can become easier with practice and a few key strategies in mind. Here are some tips to help you tackle similar cube root simplification questions:

• Master Perfect Cubes: Memorize the cube roots of common numbers like 8, 27, 64, 125, and so on. This will save you time and effort when simplifying constants under the radical.
• Exponent Rules are Your Friend: Remember the rules of exponents, especially when dealing with radicals. The rule $\sqrt[n]{x^m} = x^{m/n}$ is fundamental for simplifying expressions with exponents under radicals.
• Break It Down: When you encounter a variable with an exponent that's not divisible by the index of the radical (in this case, 3 for a cube root), break it down into a product of a perfect cube and a remaining factor. For example, $b^7$ becomes $b^6 * b$.
• Combine Like Terms: After simplifying individual terms, remember to multiply them together correctly. Pay attention to the order of operations and ensure you're combining the coefficients and variables appropriately.
• Check Your Answer: If possible, mentally check your answer by plugging in simple values for the variables. This can help you catch any errors in your simplification process.
• Practice, Practice, Practice: The more you practice simplifying radical expressions, the more comfortable and confident you'll become. Work through a variety of problems to solidify your understanding.

By keeping these tips in mind and practicing regularly, you'll be well-prepared to handle cube root simplification problems and other algebraic challenges. Remember, math is a journey, and each problem you solve is a step forward!

Keywords: cube roots, simplifying expressions, algebraic manipulation, perfect cubes, exponents, radicals, math problem solving