Simplifying Exponential Expressions A Comprehensive Guide To (x⁻⁶/x²)³

In the realm of mathematics, particularly when dealing with algebra, simplifying expressions is a fundamental skill. This article delves into the process of identifying an expression equivalent to (x⁻⁶/x²)³. This involves applying the rules of exponents to manipulate and simplify the given expression into one of the provided options. Let's embark on a comprehensive exploration of this problem, ensuring each step is meticulously explained for clarity and understanding. We will break down the problem, explore the relevant exponent rules, and arrive at the correct solution, solidifying your grasp of this essential mathematical concept.

Breaking Down the Problem

To tackle the problem (x⁻⁶/x²)³, we need to understand the order of operations and the fundamental rules governing exponents. The expression involves a fraction raised to a power, and within the fraction, we have exponents and division. Remember, exponents indicate repeated multiplication, and negative exponents signify reciprocals. The base, in this case, is x, and the exponents are -6, 2, and 3. Our goal is to simplify this expression by first addressing the division within the parentheses and then applying the outer exponent. The simplification process involves using the quotient of powers rule and the power of a power rule. Each step taken will be explained in detail to ensure a clear understanding of the process. Understanding the properties of exponents is crucial for simplifying algebraic expressions and solving equations. This problem not only tests our knowledge of these properties but also our ability to apply them strategically to arrive at the correct answer. Let's start by understanding the core concepts.

Exponent Rules: The Key to Simplification

The bedrock of simplifying expressions like (x⁻⁶/x²)³ lies in the mastery of exponent rules. These rules provide a systematic way to manipulate expressions involving exponents, making complex calculations manageable. We will focus on two key rules: the quotient of powers rule and the power of a power rule.

1. Quotient of Powers Rule: This rule states that when dividing exponential expressions with the same base, you subtract the exponents. Mathematically, it's expressed as xᵃ / xᵇ = xᵃ⁻ᵇ. This rule allows us to simplify fractions where the numerator and denominator share a common base. For example, if we have x⁵ / x², the quotient of powers rule tells us this simplifies to x⁵⁻² = x³. Understanding this rule is crucial for dealing with expressions involving division of exponents.

2. Power of a Power Rule: This rule dictates how to simplify an expression where a power is raised to another power. The rule states that you multiply the exponents. Mathematically, it's expressed as (xᵃ)ᵇ = xᵃᵇ. This rule is particularly useful when dealing with expressions enclosed in parentheses and raised to an exponent. For instance, (x²)³ simplifies to x²*³ = x⁶. The power of a power rule helps to further consolidate and simplify expressions.

These two rules are the primary tools we will use to simplify the given expression. By applying them correctly and in the appropriate order, we can transform (x⁻⁶/x²)³ into a simpler, equivalent form. Let's apply these rules step-by-step to our problem.

Step-by-Step Simplification of (x⁻⁶/x²)³

Now, let's apply the exponent rules to simplify the expression (x⁻⁶/x²)³ step-by-step, ensuring clarity at each stage.

Step 1: Simplify the Expression Inside the Parentheses

We begin by simplifying the fraction inside the parentheses, x⁻⁶/x². Here, we apply the quotient of powers rule, which states that xᵃ / xᵇ = xᵃ⁻ᵇ. In our case, a = -6 and b = 2. Therefore, we have:

x⁻⁶/x² = x⁻⁶⁻² = x⁻⁸

This step consolidates the fraction into a single term with a negative exponent. Understanding how negative exponents work is essential here. A negative exponent indicates a reciprocal, meaning x⁻⁸ is the same as 1/x⁸. This intermediate step sets the stage for applying the next exponent rule.

Step 2: Apply the Power of a Power Rule

Now that we've simplified the inside of the parentheses to x⁻⁸, we need to apply the outer exponent, which is 3. This means we have (x⁻⁸)³. The power of a power rule states that (xᵃ)ᵇ = xᵃᵇ. Applying this rule, we multiply the exponents:

(x⁻⁸)³ = x⁻⁸*³ = x⁻²⁴

This step further simplifies the expression by combining the exponents. The result, x⁻²⁴, is a simplified form, but it's often preferable to express exponents with positive values, especially when comparing against answer choices.

Step 3: Convert the Negative Exponent to a Positive Exponent

To express x⁻²⁴ with a positive exponent, we use the property that x⁻ᵃ = 1/xᵃ. Therefore:

x⁻²⁴ = 1/x²⁴

This final step transforms the expression into its simplest form with a positive exponent. We've successfully simplified the original expression using the exponent rules. Now, let's compare our simplified expression with the given options to find the equivalent one.

Identifying the Equivalent Expression

After simplifying the expression (x⁻⁶/x²)³, we arrived at the equivalent expression 1/x²⁴. Now, let's compare this result with the options provided:

A. 1/x B. 1/x⁵ C. 1/x⁹ D. 1/x²⁴

By direct comparison, we can see that our simplified expression, 1/x²⁴, matches option D. Therefore, option D is the correct answer. This process of simplification and comparison is crucial in mathematics for solving equations and understanding the relationships between different expressions. The correct answer highlights the importance of accurately applying exponent rules and simplifying expressions to their most basic form.

Conclusion: Mastering Exponent Rules

In conclusion, the expression equivalent to (x⁻⁶/x²)³ is 1/x²⁴, which corresponds to option D. This problem serves as an excellent example of how mastering exponent rules is essential for simplifying algebraic expressions. By understanding and correctly applying the quotient of powers rule and the power of a power rule, we can efficiently solve complex mathematical problems. Remember, the key is to break down the problem into manageable steps, apply the appropriate rules, and simplify systematically. This approach not only helps in finding the correct answer but also enhances your understanding of the underlying mathematical principles. Keep practicing, and you'll become more proficient in simplifying expressions and solving algebraic problems. Understanding exponents is not just about solving problems; it's about building a solid foundation for more advanced mathematical concepts. So, continue to explore, practice, and deepen your understanding of exponents and their applications.