Cracking the Code: Unveiling the Product of (3a)(2b)(4)(8ab)^3

The product of the expression (3a)(2b)(4)(8ab)^3 simplifies to 12288a^4b^4. This involves a combination of exponent rules, multiplication of constants, and careful attention to the order of operations. Let’s delve into the intricacies of how we arrive at this solution, and address frequently asked questions to solidify your understanding of similar algebraic manipulations.

Breaking Down the Expression: A Step-by-Step Guide

To understand how we arrived at the answer, let’s walk through the process systematically.

1. Focus on the exponent: The term (8ab)^3 needs to be expanded first. Remember the power of a product rule: (xy)^n = x^n * y^n. Applying this rule gives us 8^3 * a^3 * b^3, which simplifies to 512a^3b^3.

2. Rewriting the Expression: Now, we can rewrite the entire expression as (3a)(2b)(4)(512a^3b^3).

3. Multiplying the Constants: Next, multiply the constants together: 3 * 2 * 4 * 512 = 12288.

4. Multiplying the Variables: Now, multiply the ‘a’ terms: a * a^3 = a^(1+3) = a^4. Similarly, multiply the ‘b’ terms: b * b^3 = b^(1+3) = b^4.

5. Combining Everything: Finally, combine the multiplied constants and variables to get the final simplified expression: 12288a^4b^4.

Frequently Asked Questions (FAQs)

To further clarify the concepts and address potential stumbling blocks, let’s explore some frequently asked questions related to this type of algebraic manipulation.

H3 What is the Order of Operations (PEMDAS/BODMAS)?

The order of operations, often remembered by the acronyms PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction), dictates the sequence in which mathematical operations should be performed. In our case, the exponent was addressed before any multiplication.

H3 What is the Power of a Product Rule?

The power of a product rule states that when raising a product to a power, you must raise each factor within the product to that power. Mathematically, this is represented as (xy)^n = x^n * y^n. We used this extensively to simplify (8ab)^3.

H3 What is the Product of Powers Rule?

The product of powers rule states that when multiplying powers with the same base, you add the exponents. For example, x^m * x^n = x^(m+n). We used this when multiplying ‘a’ and ‘a^3’, and ‘b’ and ‘b^3’.

H3 How Do I Handle Negative Exponents?

A negative exponent indicates the reciprocal of the base raised to the positive exponent. For example, x^-n = 1/x^n. This wasn’t directly used in our original problem, but it’s a crucial concept in algebra.

H3 What Happens If There Were Negative Signs in the Expression?

If there were negative signs, you would need to pay close attention to the rules of multiplying and dividing signed numbers. Remember: a negative times a negative is a positive, and a negative times a positive is a negative.

H3 Can I Simplify This Expression Further?

The expression 12288a^4b^4 is simplified as much as possible, assuming ‘a’ and ‘b’ are distinct variables and there are no further constraints or given values for ‘a’ and ‘b’.

H3 What If ‘a’ and ‘b’ Represent the Same Value?

If ‘a’ and ‘b’ represented the same value (e.g., a = b), then you could further simplify the expression to 12288a^8 or 12288b^8. This assumes that the variables are interchangeable.

H3 What are Like Terms and Why Are They Important?

Like terms are terms that have the same variables raised to the same powers. We can only combine like terms by adding or subtracting their coefficients. In our problem, after expanding the exponent, we combined the ‘a’ terms and ‘b’ terms because they were like terms.

H3 What’s the Difference Between a Coefficient and an Exponent?

A coefficient is a numerical factor that multiplies a variable (e.g., 12288 in 12288a^4b^4). An exponent indicates the power to which a variable or number is raised (e.g., 4 in a^4). They serve different roles in determining the value of a term.

H3 How Does This Apply to Real-World Scenarios?

Algebraic expressions like these are fundamental in many real-world applications, including physics (calculating areas and volumes), engineering (designing structures), and computer science (optimizing algorithms). The ability to manipulate and simplify these expressions is crucial for solving complex problems.

H3 What’s the Best Way to Practice Simplifying Algebraic Expressions?

The best way to practice is to work through a variety of problems, starting with simpler ones and gradually increasing the complexity. Focus on understanding the underlying rules and principles, rather than just memorizing steps. Online resources, textbooks, and practice worksheets can be valuable tools.

H3 How Can I Check My Answer?

You can check your answer by substituting simple numerical values for ‘a’ and ‘b’ into both the original expression and the simplified expression. If both expressions yield the same result, your simplification is likely correct. For example, let a = 1 and b = 1.

Conclusion: Mastering Algebraic Manipulation

Simplifying algebraic expressions like (3a)(2b)(4)(8ab)^3 requires a solid understanding of exponent rules, the order of operations, and careful attention to detail. By mastering these concepts and practicing regularly, you can confidently tackle even more complex algebraic challenges. The key is to break down the problem into smaller, manageable steps and apply the appropriate rules consistently. Remember, the power of algebra lies in its ability to model and solve problems across a vast range of disciplines.