Simplifying (4w⁻⁶x³)⁻³ A Comprehensive Guide With Positive Exponents

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In the realm of algebra, simplifying expressions involving exponents is a fundamental skill. This article delves into the process of simplifying the expression (4w⁻⁶x³)⁻³, focusing on the crucial aspect of expressing the final answer using only positive exponents. We'll break down the steps involved, providing a comprehensive understanding of the rules of exponents and their application in simplifying complex expressions. By mastering these techniques, you'll be well-equipped to tackle a wide range of algebraic simplification problems.

The initial expression we're tackling is (4w⁻⁶x³)⁻³. This expression contains a product raised to a power, as well as negative exponents, making it a prime candidate for simplification. The key to simplifying this expression lies in understanding and applying the rules of exponents, especially the power of a product rule and the rule for negative exponents. Let's embark on this simplification journey step by step.

The first step in simplifying (4w⁻⁶x³)⁻³ is to apply the power of a product rule. This rule states that (ab)ⁿ = aⁿbⁿ, where a and b are any non-zero numbers or variables, and n is any integer. Applying this rule to our expression, we distribute the exponent -3 to each factor within the parentheses:

(4w⁻⁶x³)⁻³ = 4⁻³ * (w⁻⁶)⁻³ * (x³)⁻³

Now we have each factor raised to the power of -3. Next, we'll address the exponents individually. The constant term 4⁻³ can be simplified directly using the definition of negative exponents. The terms (w⁻⁶)⁻³ and (x³)⁻³ involve powers raised to powers, which require another exponent rule.

Having distributed the exponent, we now encounter terms where a power is raised to another power. This situation calls for the power of a power rule, which states that (aᵐ)ⁿ = aᵐⁿ, where a is any non-zero number or variable, and m and n are any integers. This rule is essential for simplifying expressions with nested exponents.

Applying the power of a power rule to the w and x terms in our expression, we get:

  • (w⁻⁶)⁻³ = w(⁻⁶ * ⁻³) = w¹⁸
  • (x³)⁻³ = x(³ * ⁻³) = x⁻⁹

Notice how multiplying the exponents simplifies the expression. The negative exponent in (w⁻⁶)⁻³ becomes positive after multiplication, while the positive exponent in (x³)⁻³ becomes negative. Now, let's consider the constant term 4⁻³.

To simplify 4⁻³, we use the rule for negative exponents, which states that a⁻ⁿ = 1/aⁿ, where a is any non-zero number or variable, and n is any integer. Applying this rule, we get:

4⁻³ = 1/4³ = 1/64

Now we have simplified each factor individually. Let's combine these simplified factors back into a single expression.

After applying the power of a product rule and the power of a power rule, we have the following simplified factors:

  • 4⁻³ = 1/64
  • (w⁻⁶)⁻³ = w¹⁸
  • (x³)⁻³ = x⁻⁹

Combining these factors, we get:

4⁻³ * (w⁻⁶)⁻³ * (x³)⁻³ = (1/64) * w¹⁸ * x⁻⁹

This expression is much simpler than our original expression, but it still contains a negative exponent (x⁻⁹). Our goal is to express the final answer using only positive exponents. To achieve this, we need to apply the rule for negative exponents once more.

Recall that the rule for negative exponents states that a⁻ⁿ = 1/aⁿ. Applying this rule to x⁻⁹, we get:

x⁻⁹ = 1/x⁹

Now we can substitute this back into our expression:

(1/64) * w¹⁸ * x⁻⁹ = (1/64) * w¹⁸ * (1/x⁹)

Finally, we can combine the terms into a single fraction:

(1/64) * w¹⁸ * (1/x⁹) = w¹⁸ / (64x⁹)

This is our fully simplified expression, with all exponents being positive.

After meticulously applying the rules of exponents, we have successfully simplified the expression (4w⁻⁶x³)⁻³ and expressed the result using only positive exponents. The final simplified expression is:

w¹⁸ / (64x⁹)

This result demonstrates the power of understanding and applying the rules of exponents. By breaking down the problem into smaller steps and applying the appropriate rules, we can simplify even complex expressions.

Simplifying expressions with exponents is a crucial skill in algebra. This process relies heavily on the rules of exponents, which dictate how to manipulate expressions involving powers. Let's recap the key takeaways from our simplification journey:

  1. Power of a Product Rule: (ab)ⁿ = aⁿbⁿ. This rule allows us to distribute an exponent over a product.
  2. Power of a Power Rule: (aᵐ)ⁿ = aᵐⁿ. This rule simplifies expressions where a power is raised to another power.
  3. Negative Exponent Rule: a⁻ⁿ = 1/aⁿ. This rule enables us to express negative exponents as positive exponents by taking the reciprocal of the base.

By mastering these rules, you can confidently simplify a wide range of expressions involving exponents. Remember to break down complex expressions into smaller, manageable steps, and carefully apply the appropriate rules. With practice, you'll become proficient in simplifying expressions and manipulating exponents with ease.

In conclusion, simplifying algebraic expressions, particularly those involving exponents, is a cornerstone of mathematical proficiency. The ability to navigate and manipulate exponents not only enhances problem-solving skills but also lays a solid foundation for more advanced mathematical concepts. This detailed exploration of simplifying (4w⁻⁶x³)⁻³ serves as a practical guide to mastering exponent rules, ensuring clarity and precision in algebraic manipulations. The journey from the initial expression to the final simplified form, w¹⁸ / (64x⁹), underscores the importance of methodical application of rules and the transformative power of algebraic simplification. By internalizing these techniques and principles, learners can approach complex algebraic challenges with confidence and achieve accurate results. The emphasis on expressing answers with positive exponents aligns with standard mathematical conventions, fostering clarity and consistency in mathematical communication. This comprehensive approach to simplifying exponents equips students and practitioners alike with the necessary tools to excel in algebra and beyond, reinforcing the significance of mastering fundamental mathematical operations.