Simplifying Cube Roots A Step By Step Guide To $\sqrt[3]{5 A B^2} \cdot \sqrt[3]{25 A B}$

$$

This article delves into the simplification of expressions involving cube roots, focusing on the specific problem: Which expression is equivalent to $\sqrt[3]{5 a b^2} \cdot \sqrt[3]{25 a b}$? We will break down the step-by-step solution, explore the underlying mathematical principles, and discuss common pitfalls to avoid. This exploration aims to provide a comprehensive understanding of how to manipulate and simplify radical expressions, particularly those involving cube roots.

Understanding Cube Roots and Their Properties

Before diving into the problem, let's revisit the concept of cube roots and their fundamental properties. A cube root of a number x is a value that, when multiplied by itself three times, equals x. Mathematically, this is represented as $\sqrt[3]{x}$. For example, the cube root of 8 is 2, because $2 \cdot 2 \cdot 2 = 8$. Understanding this basic definition is crucial for simplifying expressions with cube roots.

One of the key properties we will use is the product rule for radicals, which states that the n-th root of a product is equal to the product of the n-th roots. In mathematical notation, this is expressed as $\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$. This property allows us to combine or separate radicals, making it a powerful tool for simplification. In our case, we are dealing with cube roots (where n = 3), so this rule becomes $\sqrt[3]{ab} = \sqrt[3]{a} \cdot \sqrt[3]{b}$.

Furthermore, understanding how to factor numbers and variables within the cube root is essential. The goal is to identify perfect cubes – numbers or expressions that can be written as the cube of another number or expression. For example, $8 = 2^3$, $a^3$ is a perfect cube, and so on. By extracting perfect cubes from within the radical, we can simplify the expression significantly.

In summary, to effectively work with cube roots, it is vital to grasp the definition of a cube root, the product rule for radicals, and the concept of perfect cubes. These foundational principles will be instrumental in simplifying the given expression and arriving at the correct answer. The ability to identify and manipulate these concepts will not only help in solving this particular problem but also in tackling a wide range of mathematical problems involving radicals and exponents.

Step-by-Step Solution

To solve the problem, we will simplify the expression $\sqrt[3]{5 a b^2} \cdot \sqrt[3]{25 a b}$ step by step, using the properties of cube roots discussed earlier.

Step 1: Combining the Radicals

First, we apply the product rule for radicals, which allows us to combine the two cube roots into a single cube root. This gives us:

$\sqrt[3]{5 a b^2} \cdot \sqrt[3]{25 a b} = \sqrt[3]{(5 a b^2) \cdot (25 a b)}$

This step is crucial as it consolidates the expression, making it easier to identify and simplify like terms. By multiplying the terms inside the cube root, we prepare the expression for further simplification.

Step 2: Multiplying the Terms Inside the Cube Root

Next, we multiply the terms inside the cube root:

$\sqrt[3]{(5 a b^2) \cdot (25 a b)} = \sqrt[3]{125 a^2 b^3}$

Here, we multiply the coefficients (5 and 25) to get 125. We also multiply the variables using the rule of exponents, which states that $x^m \cdot x^n = x^{m+n}$. Thus, $a \cdot a = a^{1+1} = a^2$ and $b^2 \cdot b = b^{2+1} = b^3$. This multiplication step is vital to group similar terms together, making it easier to identify perfect cubes in the next step.

Step 3: Identifying and Extracting Perfect Cubes

Now, we look for perfect cubes within the cube root. We can rewrite 125 as $5^3$, and we already have $b^3$. The expression becomes:

$\sqrt[3]{125 a^2 b^3} = \sqrt[3]{5^3 a^2 b^3}$

Identifying perfect cubes is the key to simplifying radical expressions. Recognizing that 125 is $5^3$ and that $b^3$ is a perfect cube allows us to extract these terms from the cube root. The term $a^2$ is not a perfect cube, so it will remain inside the cube root.

Step 4: Simplifying the Cube Root

We can now extract the perfect cubes from the cube root:

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

Here, we use the property $\sqrt[3]{x^3} = x$ to simplify $\sqrt[3]{5^3}$ to 5 and $\sqrt[3]{b^3}$ to b. The term $\sqrt[3]{a^2}$ cannot be simplified further, so it remains under the cube root. This step demonstrates the power of extracting perfect cubes to simplify complex radical expressions.

Conclusion

Therefore, the expression $\sqrt[3]{5 a b^2} \cdot \sqrt[3]{25 a b}$ simplifies to $5 b \sqrt[3]{a^2}$, which corresponds to option B. By systematically applying the properties of cube roots and identifying perfect cubes, we can effectively simplify complex radical expressions. This step-by-step approach not only helps in solving this particular problem but also in developing a deeper understanding of radical simplification techniques.

Analyzing the Options

Now, let's examine the given options in the context of our solution to understand why the correct answer is B and why the other options are incorrect. This analysis will reinforce our understanding of cube root simplification and highlight common errors to avoid.

Option A: $5 a b$

Option A, $5 a b$, is incorrect because it completely removes the radical, which is not justified by the simplification process. When we simplify $\sqrt[3]{5 a b^2} \cdot \sqrt[3]{25 a b}$, we obtain $5 b \sqrt[3]{a^2}$. The term $a^2$ remains under the cube root because it is not a perfect cube. Therefore, removing the radical entirely is a mistake. This option reflects a misunderstanding of how to handle non-perfect cube terms within a radical expression.

Option B: $5 b \sqrt[3]{a^2}$

Option B, $5 b \sqrt[3]{a^2}$, is the correct answer. As demonstrated in the step-by-step solution, this is the simplified form of the original expression. We correctly combined the radicals, identified and extracted the perfect cubes ($5^3$ and $b^3$), and left the non-perfect cube term ($a^2$) under the cube root. This option demonstrates a thorough understanding of the properties of cube roots and the process of simplification.

Option C: $5 a b \sqrt{5 b}$

Option C, $5 a b \sqrt{5 b}$, is incorrect because it introduces a square root instead of a cube root. The original expression involves cube roots, so the simplified form should also maintain the cube root unless all terms under the radical are perfect cubes. This option likely results from a confusion between square roots and cube roots or a misapplication of the product rule for radicals. The presence of a square root where a cube root is expected indicates a fundamental error in the simplification process.

Option D: $5 a b \sqrt[3]{5 b}$

Option D, $5 a b \sqrt[3]{5 b}$, is also incorrect. While it correctly retains the cube root, the terms outside and inside the radical are not accurate. In our simplification, we found that the term outside the radical should be $5b$, and the term inside the cube root should be $a^2$. The presence of $5b$ under the cube root in this option suggests an error in identifying and extracting perfect cubes or in the multiplication of terms inside the radical. This option highlights the importance of carefully tracking each step in the simplification process to avoid errors.

Conclusion

In summary, by analyzing each option, we can see why Option B is the only correct answer. The other options either incorrectly remove the radical, introduce the wrong type of radical (square root instead of cube root), or misplace terms inside and outside the radical. This analysis reinforces the importance of a systematic and careful approach to simplifying radical expressions, ensuring that each step is justified by the properties of radicals and exponents. Understanding why incorrect options are wrong is just as important as understanding why the correct option is right, as it helps prevent similar errors in future problems.

Common Mistakes to Avoid

When simplifying expressions with cube roots, several common mistakes can lead to incorrect answers. Being aware of these pitfalls is crucial for developing accurate problem-solving skills. Let's discuss some of these common errors and how to avoid them.

Mistake 1: Incorrectly Applying the Product Rule for Radicals

One frequent mistake is misapplying the product rule for radicals. While it is true that $\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$, this rule only applies when the radicals have the same index (the small number indicating the type of root, such as 3 for cube root). Trying to combine a square root and a cube root under the same radical is a common error. In our case, we are dealing with cube roots, so we can apply the rule, but it's essential to ensure that all radicals have the same index before combining them.

How to Avoid: Always check that the radicals have the same index before applying the product rule. If they don't, you cannot directly combine them under a single radical.

Mistake 2: Failing to Identify Perfect Cubes

Another common mistake is failing to recognize perfect cubes within the radical. A perfect cube is a number or expression that can be written as the cube of another number or expression (e.g., 8 = $2^3$, $a^3$, $27b^3$). If you don't identify these perfect cubes, you won't be able to simplify the expression fully. In our problem, recognizing that 125 is $5^3$ was crucial for simplifying the expression.

How to Avoid: Practice identifying perfect cubes. Familiarize yourself with the cubes of common numbers (1, 8, 27, 64, 125, etc.) and be on the lookout for expressions with exponents that are multiples of 3 (e.g., $a^3$, $b^6$, $c^9$).

Mistake 3: Incorrectly Extracting Terms from the Cube Root

Even if you identify perfect cubes, you might make a mistake when extracting them from the radical. Remember that when you take the cube root of a perfect cube, you are essentially “undoing” the cubing operation. For example, $\sqrt[3]{5^3} = 5$ and $\sqrt[3]{b^3} = b$. A common error is to forget this step or to perform it incorrectly.

How to Avoid: Double-check your work when extracting terms from the cube root. Make sure you are taking the cube root of the perfect cube and not just removing the exponent. Write out the steps clearly to avoid confusion.

Mistake 4: Not Simplifying Completely

Sometimes, students may simplify the expression partially but fail to simplify it completely. This can happen if they miss a perfect cube or stop simplifying prematurely. In our problem, it was essential to extract both $5^3$ and $b^3$ to arrive at the fully simplified form.

How to Avoid: After each step, ask yourself if the expression can be simplified further. Look for any remaining perfect cubes or any other opportunities to apply the properties of radicals. Practice and attention to detail are key.

Mistake 5: Confusing Cube Roots with Square Roots

A fundamental mistake is confusing cube roots with square roots. This can lead to incorrect simplification and the application of the wrong rules. Remember that a cube root is the inverse operation of cubing, while a square root is the inverse operation of squaring. They are distinct operations with different properties.

How to Avoid: Pay close attention to the index of the radical. If there is no index written, it is assumed to be a square root (index of 2). If the index is 3, it is a cube root. Always be mindful of this distinction and use the appropriate rules and techniques.

Conclusion

By being aware of these common mistakes and actively working to avoid them, you can significantly improve your accuracy and confidence in simplifying expressions with cube roots. Consistent practice and a thorough understanding of the underlying principles are essential for mastering this skill. Understanding these common mistakes is crucial for students to avoid errors and enhance their problem-solving abilities in mathematics.

Practice Problems

To solidify your understanding of simplifying expressions with cube roots, working through practice problems is essential. These problems will give you the opportunity to apply the concepts and techniques we've discussed, identify areas where you may need further clarification, and build confidence in your problem-solving skills.

Practice Problem 1

Simplify the expression: $\sqrt[3]{8 x^4 y^6}$

Solution:

1. Identify perfect cubes: 8 is $2^3$, and $y^6$ is $(y^2)^3$. We can rewrite $x^4$ as $x^3 \cdot x$.
2. Rewrite the expression: $\sqrt[3]{8 x^4 y^6} = \sqrt[3]{2^3 x^3 x (y^2)^3}$
3. Extract perfect cubes: $\sqrt[3]{2^3 x^3 (y^2)^3} \cdot \sqrt[3]{x} = 2 x y^2 \sqrt[3]{x}$

Therefore, the simplified expression is $2 x y^2 \sqrt[3]{x}$.

Practice Problem 2

Simplify the expression: $\sqrt[3]{27 a^5 b^2} \cdot \sqrt[3]{a b^4}$

Solution:

1. Combine the radicals: $\sqrt[3]{(27 a^5 b^2) \cdot (a b^4)} = \sqrt[3]{27 a^6 b^6}$
2. Identify perfect cubes: 27 is $3^3$, $a^6$ is $(a^2)^3$, and $b^6$ is $(b^2)^3$.
3. Rewrite the expression: $\sqrt[3]{3^3 (a^2)^3 (b^2)^3}$
4. Extract perfect cubes: $\sqrt[3]{3^3} \cdot \sqrt[3]{(a^2)^3} \cdot \sqrt[3]{(b^2)^3} = 3 a^2 b^2$

Therefore, the simplified expression is $3 a^2 b^2$.

Practice Problem 3

Simplify the expression: $\sqrt[3]{-64 c^7 d^3}$

Solution:

1. Identify perfect cubes: -64 is $(-4)^3$, and $d^3$ is a perfect cube. We can rewrite $c^7$ as $c^6 \cdot c$, where $c^6$ is $(c^2)^3$.
2. Rewrite the expression: $\sqrt[3]{(-4)^3 c^6 c d^3} = \sqrt[3]{(-4)^3 (c^2)^3 d^3 c}$
3. Extract perfect cubes: $\sqrt[3]{(-4)^3} \cdot \sqrt[3]{(c^2)^3} \cdot \sqrt[3]{d^3} \cdot \sqrt[3]{c} = -4 c^2 d \sqrt[3]{c}$

Therefore, the simplified expression is $-4 c^2 d \sqrt[3]{c}$.

Additional Practice

Here are a few more problems to try on your own:

• Simplify: $\sqrt[3]{125 x^9 y^3}$
• Simplify: $\sqrt[3]{16 a^4 b^5}$
• Simplify: $\sqrt[3]{-216 m^8 n^6}$

By working through these practice problems, you will reinforce your understanding of the concepts and techniques involved in simplifying expressions with cube roots. Remember to carefully apply the properties of radicals, identify perfect cubes, and double-check your work to avoid common mistakes. Consistent practice is the key to mastering this skill.

Conclusion

In conclusion, simplifying expressions with cube roots involves a systematic approach using the properties of radicals and the identification of perfect cubes. The problem $\sqrt[3]{5 a b^2} \cdot \sqrt[3]{25 a b}$ serves as an excellent example to illustrate these principles, with the correct simplified form being $5 b \sqrt[3]{a^2}$. By understanding the underlying concepts, avoiding common mistakes, and engaging in consistent practice, you can confidently tackle a wide range of problems involving cube roots. This skill is not only crucial for success in mathematics but also for various applications in science and engineering. Remember, the key to mastering any mathematical concept is a combination of understanding the theory and applying it through practice. Keep exploring, keep practicing, and you will continue to enhance your mathematical abilities.