Simplifying (2ab / (a^-5 B^2))^-3 A Step-by-Step Guide

This article provides a step-by-step guide on simplifying the algebraic expression (2ab / (a^-5 b²⁾⁾-3, assuming that a and b are non-zero. We will delve into the fundamental exponent rules and algebraic manipulations required to arrive at the most simplified form. This detailed explanation will help students and anyone interested in algebra to understand the process thoroughly.

Breaking Down the Expression: Understanding the Fundamentals

To effectively simplify the given expression, (2ab / (a^-5 b²⁾⁾-3, we need to understand and apply several key exponent rules. These rules are the building blocks for manipulating algebraic expressions and are essential for achieving simplification. First and foremost, let's acknowledge the negative exponent rule, which states that x^-n = 1/x^n. This rule dictates how we handle negative exponents by converting them into their reciprocal form. Then, there's the quotient rule for exponents, which tells us that when dividing exponents with the same base, we subtract the powers: x^m / x^n = x^(m-n). This is crucial for simplifying fractions within expressions. The power of a quotient rule states that (x/y)^n = x^n / y^n, meaning that an exponent outside parentheses applies to both the numerator and the denominator. Finally, the power of a product rule says that (xy)^n = x^n y^n, indicating that an exponent outside parentheses applies to each factor inside. Grasping these rules is not just about memorization; it’s about understanding how they interact to transform expressions. By applying these rules strategically, we can break down complex algebraic expressions into simpler, more manageable forms. In our specific expression, we'll see how these rules work in tandem to eliminate negative exponents, combine like terms, and ultimately arrive at the most simplified version of (2ab / (a^-5 b²⁾⁾-3. The journey to simplification is a methodical process, and understanding these core principles is the first, vital step.

Step-by-Step Simplification: A Detailed Walkthrough

Let's embark on the step-by-step simplification of the expression (2ab / (a^-5 b²⁾⁾-3. This journey will illustrate how to apply the exponent rules discussed earlier. The first critical step in simplifying this expression is to address the negative exponent outside the parentheses. Recall the negative exponent rule, x^-n = 1/x^n. Applying this, we invert the fraction inside the parentheses and change the sign of the exponent: (2ab / (a^-5 b²⁾⁾-3 becomes ((a^-5 b^2) / (2ab))^3. This maneuver is crucial because it transforms the expression into a form where we can apply the power of a quotient rule more easily. Next, we invoke the power of a quotient rule, which states that (x/y)^n = x^n / y^n. Applying this rule to our expression, we distribute the exponent of 3 to both the numerator and the denominator: ((a^-5 b^2) / (2ab))^3 becomes (a^-5 b²⁾3 / (2ab)^3. This step separates the expression into two parts, each of which can be simplified independently. Now, let's focus on the numerator, (a^-5 b²⁾3. We apply the power of a product rule, (xy)^n = x^n y^n, and the power of a power rule, (x^m)n = x^(mn). This means we raise each factor within the parentheses to the power of 3: (a^-5 b²⁾3 becomes **a^(-53) b^(2*3)**, which simplifies to a^-15 b^6. Similarly, we simplify the denominator, (2ab)^3, by applying the power of a product rule: (2ab)^3 becomes 2^3 a^3 b^3, which simplifies to 8a^3 b^3. By meticulously applying these rules, we've transformed the original complex expression into a more manageable form. The next step will involve combining these simplified numerator and denominator terms, further reducing the expression to its simplest possible form. This systematic approach, guided by the fundamental exponent rules, is key to successfully navigating algebraic simplifications.

Combining Terms and Final Simplification

Having simplified the numerator and denominator separately, we now combine them to achieve the final simplification of our expression. We've reached the stage where our expression looks like this: (a^-15 b^6) / (8a^3 b^3). At this point, the quotient rule for exponents, x^m / x^n = x^(m-n), becomes our primary tool. This rule allows us to simplify terms with the same base by subtracting their exponents. Applying the quotient rule to the a terms, we have a^-15 / a^3, which simplifies to a^(-15-3) or a^-18. For the b terms, we have b^6 / b^3, which simplifies to b^(6-3) or b^3. The expression now looks like this: (a^-18 b^3) / 8. We still have a negative exponent to address. Again, we employ the negative exponent rule, x^-n = 1/x^n, to rewrite a^-18 as 1/a^18. Substituting this back into our expression, we get (1/a^18 * b^3) / 8. To tidy up the expression, we can rewrite it as a single fraction. This involves recognizing that dividing by 8 is the same as multiplying by 1/8. Thus, our expression becomes (1 * b^3) / (8 * a^18), which simplifies to b^3 / (8a^18). This is the fully simplified form of the original expression, (2ab / (a^-5 b²⁾⁾-3. Through careful application of exponent rules and step-by-step simplification, we've successfully transformed a complex expression into its most basic form. This process highlights the importance of understanding and correctly applying fundamental algebraic principles.

Final Answer

Therefore, the simplified form of (2ab / (a^-5 b²⁾⁾-3 is b^3 / (8a^18).

Common Mistakes to Avoid

When simplifying algebraic expressions like (2ab / (a^-5 b²⁾⁾-3, there are several common pitfalls that students often encounter. Recognizing and avoiding these mistakes is crucial for achieving accurate results. One of the most frequent errors is misapplying the negative exponent rule. It’s essential to remember that a negative exponent means taking the reciprocal of the base, not simply changing the sign of the exponent. For example, x^-n is equal to 1/x^n, not -x^n. Confusing these two can lead to incorrect simplification right from the start. Another common mistake involves incorrectly applying the quotient rule. The quotient rule, x^m / x^n = x^(m-n), requires careful attention to the order of subtraction. Students sometimes subtract the exponent in the numerator from the exponent in the denominator, which is the reverse of what should be done. Remember, it's the exponent in the numerator minus the exponent in the denominator. Errors in distributing exponents are also prevalent, especially when dealing with the power of a product or quotient. For instance, (xy)^n is x^n y^n, and each factor inside the parentheses must be raised to the power n. Neglecting to apply the exponent to all factors can result in an incorrect simplification. Additionally, students sometimes overlook the coefficient when applying the power of a product rule. For example, (2x)^2 is 4x^2, not 2x^2. The coefficient must also be raised to the power. Another mistake is combining unlike terms inappropriately. Only terms with the same base can have their exponents added or subtracted. Trying to combine terms like a^2 and b^2 will lead to errors. Lastly, students may make arithmetic errors when simplifying exponents. Double-checking calculations, especially when dealing with negative numbers, is always a good practice. By being aware of these common mistakes and taking the time to carefully apply each rule, you can significantly improve your accuracy in simplifying algebraic expressions. Attention to detail and a thorough understanding of the exponent rules are the keys to success.

Practice Problems: Sharpen Your Skills

To solidify your understanding of simplifying algebraic expressions, working through practice problems is essential. These exercises allow you to apply the concepts discussed and identify areas where you may need further clarification. Let's explore some practice problems that mirror the complexity of the example we’ve tackled.

Practice Problem 1: Simplify (3x^2 y / (x^-1 y³⁾⁾-2

This problem challenges you to apply the negative exponent rule, power of a quotient rule, and quotient rule, similar to the main example. Begin by addressing the negative exponent outside the parentheses, then distribute the exponent, and finally, simplify using the quotient rule. Pay close attention to the coefficients and negative exponents.

Practice Problem 2: Simplify ((4a^-3 b^2) / (2a^2 b^-1))3

This problem adds another layer of complexity with negative exponents in both the numerator and the denominator. Remember to simplify the expression inside the parentheses first before applying the outer exponent. This will help manage the negative exponents and make the simplification process smoother.

Practice Problem 3: Simplify (5p^4 q^-2)2 / (p^-1 q^3)

Here, you’ll need to apply the power of a product rule in the numerator and then use the quotient rule to combine terms. Be mindful of the order of operations and the rules for handling negative exponents. This problem reinforces the importance of systematic simplification.

Practice Problem 4: Simplify ((x^2 y^-3) / (x^-2 y²⁾⁾-1

This problem is designed to test your understanding of the negative exponent rule and the quotient rule. The negative exponent outside the parentheses will require you to invert the fraction, and then you’ll need to simplify the resulting expression. Watch out for the signs when applying the quotient rule.

Practice Problem 5: Simplify (2m^-1 n⁴⁾3 / (4m^2 n^-2)

This problem combines several concepts, including the power of a product rule, negative exponents, and the quotient rule. Remember to apply the power of a product rule carefully, ensuring that the coefficient is also raised to the power. Then, simplify using the quotient rule, paying attention to the signs of the exponents.

By working through these practice problems, you’ll gain confidence in your ability to simplify algebraic expressions. Each problem offers a unique challenge, allowing you to hone your skills and deepen your understanding of exponent rules and algebraic manipulations. Remember to break down each problem into smaller steps, apply the rules methodically, and double-check your work to avoid common mistakes. Practice makes perfect, and with consistent effort, you’ll master the art of simplifying algebraic expressions.

Conclusion

In conclusion, simplifying algebraic expressions like (2ab / (a^-5 b²⁾⁾-3 requires a solid grasp of exponent rules and a methodical approach. By understanding and correctly applying rules such as the negative exponent rule, quotient rule, and power of a product rule, complex expressions can be transformed into simpler, more manageable forms. Common mistakes, such as misapplying the negative exponent rule or incorrectly distributing exponents, can be avoided with careful attention to detail and a step-by-step approach. Practice problems are invaluable for reinforcing these concepts and building confidence in your skills. The simplified form of (2ab / (a^-5 b²⁾⁾-3 is b^3 / (8a^18), achieved through the systematic application of algebraic principles. Mastering these skills is fundamental for success in algebra and beyond.