Simplifying Radical Expressions Understanding The Difference Of 2ab(∛192ab²) And 5(∛81a⁴b⁵)

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    At its core, this mathematical problem challenges us to simplify an algebraic expression involving cube roots. Specifically, we are tasked with finding the difference between 2ab(³√192ab²) and 5(³√81a⁴b⁵). This requires a solid understanding of radical simplification, exponent rules, and algebraic manipulation. Let's break down the process step-by-step to ensure clarity and accuracy.

    The key to tackling this problem lies in simplifying the radicals. Simplifying radicals means expressing them in their simplest form, where the radicand (the expression under the radical) has no perfect cube factors other than 1. To achieve this, we need to identify and extract any perfect cube factors from the radicands 192ab² and 81a⁴b⁵.

    Let's start with the first term, 2ab(³√192ab²). We need to factor 192ab² into its prime factors. The prime factorization of 192 is 2⁶ × 3. Therefore, 192ab² can be written as 2⁶ × 3 × a × b². Now, we look for factors that are perfect cubes. A perfect cube is a number or expression that can be obtained by cubing an integer or algebraic term. In this case, 2⁶ is a perfect cube because it can be written as (2²)³, which is 4³. We can rewrite the expression inside the cube root as (2²)³ × 3 × a × b². Now, we can extract the perfect cube factor:

    ³√192ab² = ³√((2²)³ × 3 × a × b²) = ³√(4³ × 3 × a × b²) = 4³√(3ab²)

    So, the first term becomes 2ab(4³√(3ab²)) = 8ab³√(3ab²). Now, let's move on to the second term, 5(³√81a⁴b⁵). We factor 81a⁴b⁵ into its prime factors. The prime factorization of 81 is 3⁴. Thus, 81a⁴b⁵ can be written as 3⁴ × a⁴ × b⁵. We look for perfect cube factors. We can rewrite this as 3³ × 3 × a³ × a × b³ × b². Now, we extract the perfect cube factors:

    ³√81a⁴b⁵ = ³√(3³ × 3 × a³ × a × b³ × b²) = ³√((3a b)³ × 3ab²) = 3ab³√(3ab²)

    So, the second term becomes 5(3ab³√(3ab²)) = 15ab³√(3ab²). Now that we have simplified both terms, we can find the difference:

    8ab³√(3ab²) - 15ab³√(3ab²)

    Now that we've simplified the radicals, the next step is to combine like terms. In this context, like terms are terms that have the same radical part. Examining our simplified expression, we have:

    8ab³√(3ab²) - 15ab³√(3ab²)

    Notice that both terms contain the same radical part, which is ³√(3ab²). This allows us to combine these terms by simply subtracting their coefficients. The coefficients are the numerical factors multiplying the radical part. In our expression, the coefficients are 8ab and -15ab. To combine these like terms, we perform the subtraction:

    (8ab - 15ab)³√(3ab²)

    Now, we subtract the coefficients:

    8ab - 15ab = -7ab

    So, our expression becomes:

    -7ab³√(3ab²)

    This is the simplified form of the original expression. By identifying and extracting perfect cube factors, and then combining like terms, we've reduced the expression to its most basic form. This process highlights the importance of understanding prime factorization, exponent rules, and the properties of radicals in simplifying algebraic expressions.

    To effectively simplify radical expressions, it's crucial to have a firm grasp of exponent rules and the properties of radicals. These mathematical principles provide the foundation for manipulating and simplifying expressions involving roots and powers. Let's delve into some of the key rules and properties that are particularly relevant to our problem.

    Exponent Rules

    1. Product of Powers Rule: This rule states that when multiplying powers with the same base, you add the exponents. Mathematically, it's expressed as aᵐ × aⁿ = a^(m+n). This rule is essential when combining terms with the same variable raised to different powers.

    2. Power of a Power Rule: This rule states that when raising a power to another power, you multiply the exponents. Mathematically, it's expressed as (aᵐ)ⁿ = a^(m×n). This rule is useful when simplifying expressions where a power is raised to another power.

    3. Power of a Product Rule: This rule states that when raising a product to a power, you raise each factor to that power. Mathematically, it's expressed as (ab)ⁿ = aⁿbⁿ. This rule is crucial when dealing with radicals containing products.

    Properties of Radicals

    1. Product Property of Radicals: This property states that the nth root of a product is equal to the product of the nth roots. Mathematically, it's expressed as ⁿ√(ab) = ⁿ√a × ⁿ√b. This property allows us to separate radicals into simpler components, which is vital for simplification.

    2. Quotient Property of Radicals: This property states that the nth root of a quotient is equal to the quotient of the nth roots. Mathematically, it's expressed as ⁿ√(a/b) = ⁿ√a / ⁿ√b. This property is useful when dealing with radicals containing fractions.

    3. Simplifying Radicals: To simplify a radical, we look for perfect nth power factors within the radicand. If we find a perfect nth power factor, we can extract it from the radical. For example, if we have ³√a³, we can simplify it to a because a³ is a perfect cube.

    These exponent rules and properties of radicals are the tools we use to dissect and simplify complex expressions. By understanding and applying these principles, we can efficiently manipulate radicals and express them in their simplest forms.

    To solidify our understanding, let's walk through the step-by-step solution of the problem. This will reinforce the concepts we've discussed and provide a clear roadmap for tackling similar problems.

    Step 1: Simplify the First Term 2ab(³√192ab²)

    1. Factor the Radicand: Begin by factoring 192ab² into its prime factors. We find that 192 = 2⁶ × 3. So, 192ab² = 2⁶ × 3 × a × b².

    2. Identify Perfect Cube Factors: Look for factors that are perfect cubes. In this case, 2⁶ is a perfect cube since it can be written as (2²)³ = 4³. Thus, we can rewrite the radicand as 4³ × 3 × a × b².

    3. Extract Perfect Cube Factors: Apply the product property of radicals to separate the perfect cube factor:³√192ab² = ³√(4³ × 3 × a × b²) = ³√4³ × ³√(3ab²) = 4³√(3ab²)

    4. Substitute Back: Substitute the simplified radical back into the first term:2ab(³√192ab²) = 2ab(4³√(3ab²)) = 8ab³√(3ab²)

    Step 2: Simplify the Second Term 5(³√81a⁴b⁵)

    1. Factor the Radicand: Factor 81a⁴b⁵ into its prime factors. We know 81 = 3⁴. So, 81a⁴b⁵ = 3⁴ × a⁴ × b⁵.

    2. Identify Perfect Cube Factors: Look for factors that are perfect cubes. We can rewrite the radicand as 3³ × 3 × a³ × a × b³ × b².

    3. Extract Perfect Cube Factors: Apply the product property of radicals:³√81a⁴b⁵ = ³√(3³ × 3 × a³ × a × b³ × b²) = ³√(3³a³b³ × 3ab²) = 3ab³√(3ab²)

    4. Substitute Back: Substitute the simplified radical back into the second term:5(³√81a⁴b⁵) = 5(3ab³√(3ab²)) = 15ab³√(3ab²)

    Step 3: Combine Like Terms

    1. Write the Expression: Write the original expression with the simplified terms:8ab³√(3ab²) - 15ab³√(3ab²)

    2. Identify Like Terms: Notice that both terms have the same radical part, ³√(3ab²), which means they are like terms.

    3. Combine Coefficients: Subtract the coefficients of the like terms:(8ab - 15ab)³√(3ab²) = -7ab³√(3ab²)

    Step 4: Final Simplified Expression

    The final simplified expression is -7ab³√(3ab²).

    By following these steps, we've successfully simplified the given expression. This detailed walkthrough demonstrates the application of exponent rules, properties of radicals, and the process of combining like terms to achieve a simplified result.

    Simplifying radical expressions can be tricky, and it's easy to make mistakes if you're not careful. Let's discuss some common mistakes and how to avoid them to ensure accurate results.

    1. Incorrectly Factoring the Radicand

    Mistake: Failing to completely factor the radicand or making errors in the factorization process. For example, incorrectly factoring 192 as 2⁵ × 3 instead of 2⁶ × 3.

    How to Avoid: Always double-check your prime factorization. Write out the prime factors explicitly to ensure you haven't missed any factors or made any errors. Use a factor tree if necessary.

    2. Failing to Identify All Perfect Cube Factors

    Mistake: Overlooking perfect cube factors within the radicand. For example, not recognizing that 2⁶ contains a perfect cube factor of 2³ (or 4³).

    How to Avoid: Systematically examine the factored radicand for perfect cubes. Remember that any exponent that is a multiple of 3 indicates a perfect cube factor. For variables, ensure that you're looking for exponents that are multiples of 3 (e.g., a³, b⁶).

    3. Incorrectly Applying the Product Property of Radicals

    Mistake: Misapplying the rule ³√(ab) = ³√a × ³√b. For example, incorrectly separating the radical or extracting factors that are not perfect cubes.

    How to Avoid: Ensure that you are only extracting factors that are perfect cubes. Double-check that the index of the radical (in this case, 3) matches the exponent of the factor you are extracting. For example, you can extract ³√2³ but not ³√2².

    4. Not Combining Like Terms Correctly

    Mistake: Failing to recognize like terms or making errors when combining them. For example, incorrectly adding or subtracting the coefficients of the radical terms.

    How to Avoid: Make sure that the radicals are identical before combining terms. Only combine terms that have the same radicand and the same index. Then, carefully add or subtract the coefficients, paying attention to the signs.

    5. Forgetting to Simplify Completely

    Mistake: Stopping the simplification process prematurely, leaving some factors within the radical that can be further simplified.

    How to Avoid: Always review your final expression to ensure that there are no remaining perfect cube factors within the radical. If you find any, continue the simplification process until the radical is in its simplest form.

    By being aware of these common mistakes and implementing the strategies to avoid them, you can improve your accuracy and confidence in simplifying radical expressions. Remember, practice and attention to detail are key to mastering this skill.

    In conclusion, simplifying the expression 2ab(³√192ab²) - 5(³√81a⁴b⁵) involves a systematic application of mathematical principles, including prime factorization, exponent rules, and properties of radicals. By breaking down the problem into manageable steps—factoring radicands, identifying and extracting perfect cube factors, and combining like terms—we can arrive at the simplified form, -7ab³√(3ab²). Throughout this process, it's crucial to maintain accuracy and attention to detail to avoid common pitfalls.

    Mastering the simplification of radical expressions is not just about solving a specific problem; it's about developing a deeper understanding of algebraic manipulation and mathematical reasoning. The skills acquired in this process are transferable to a wide range of mathematical contexts, making it a valuable asset for anyone pursuing further studies in mathematics or related fields. By understanding the underlying principles and practicing consistently, you can confidently tackle even the most complex radical expressions.

    To further enhance your understanding, consider working through additional practice problems and exploring more advanced topics in algebra. The journey of mathematical learning is continuous, and each problem solved is a step towards greater proficiency and mastery. Remember, the key to success lies in a combination of theoretical knowledge, practical application, and diligent practice.

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