Equivalent Expression For (x^(1/2)y^(-1/4)z)^(-2) Explained

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In the realm of mathematics, simplifying expressions is a fundamental skill. This article delves into the process of finding an equivalent expression for the given algebraic term: (x(1/2)y(-1/4)z)^(-2). We will break down the expression step-by-step, applying the rules of exponents to arrive at the correct simplified form. This comprehensive exploration aims to enhance your understanding of exponent manipulation and algebraic simplification.

Deconstructing the Expression: (x(1/2)y(-1/4)z)^(-2)

The given expression is (x(1/2)y(-1/4)z)^(-2). To simplify this, we need to apply the power of a product rule, which states that (ab)^n = a^n * b^n. This rule allows us to distribute the exponent -2 to each term inside the parentheses. Let's break it down:

  1. Applying the Power of a Product Rule

    The power of a product rule dictates that when a product is raised to a power, each factor within the product is raised to that power. This is a core concept in exponent manipulation, allowing us to simplify complex expressions into manageable components. Applying this rule to our expression, (x(1/2)y(-1/4)z)^(-2), we distribute the exponent -2 to each term inside the parentheses:

    • (x(1/2))(-2)
    • (y(-1/4))(-2)
    • (z)^(-2)

    This step transforms the original expression into a series of simpler terms, each involving a single variable raised to a power. The subsequent steps will focus on simplifying these individual terms using the power of a power rule.

  2. Applying the Power of a Power Rule

    Now, we use the power of a power rule, which states that (am)n = a^(m*n). This rule is essential for simplifying expressions where an exponent is raised to another exponent. This step is critical in reducing the complexity of our expression and moving closer to the simplified form.

    • (x(1/2))(-2) = x^((1/2)*(-2)) = x^(-1)
    • (y(-1/4))(-2) = y^((-1/4)*(-2)) = y^(1/2)
    • (z)^(-2) = z^(-2)

    By applying this rule, we have further simplified each term, resulting in variables raised to single exponents. The next step involves dealing with negative exponents and expressing the result in a more conventional form.

  3. Dealing with Negative Exponents

    Negative exponents indicate the reciprocal of the base raised to the positive exponent. This concept is crucial for expressing our simplified terms in a standard format. Specifically, a^(-n) = 1/a^n. Recognizing and applying this rule allows us to rewrite terms with negative exponents as fractions, leading to a clearer and more widely accepted representation of the expression.

    • x^(-1) = 1/x
    • y^(1/2) remains as is since the exponent is positive.
    • z^(-2) = 1/z^2

    Now, we have rewritten the terms with negative exponents as fractions, setting the stage for combining these terms into a single simplified expression.

  4. Combining the Simplified Terms

    Now, let's combine the simplified terms:

    (1/x) * y^(1/2) * (1/z^2)

    This step involves bringing together the individual simplified components to form a cohesive expression. By multiplying these terms, we consolidate the expression into a single fraction, which represents the final simplified form. This process highlights the importance of keeping track of each term's transformation and correctly combining them to achieve the desired result.

    Combining these terms gives us:

    y^(1/2) / (x * z^2)

Identifying the Equivalent Expression

After simplifying (x(1/2)y(-1/4)z)^(-2), we have arrived at the expression y^(1/2) / (x * z^2). This simplified form allows us to directly compare it with the provided options and identify the correct equivalent expression. The process of simplification ensures that we have transformed the original expression into its most basic form, making it easier to recognize among the given choices. This step underscores the importance of accurate simplification in solving mathematical problems.

Comparing our simplified expression with the options:

  • A. x^(1/2) / (y * z^2)
  • B. x^(1/2) / (y^(1/4) * z^2)
  • C. y^(1/2) / (x * z^2)

Option C, y^(1/2) / (x * z^2), matches our simplified expression. Therefore, it is the equivalent expression.

Conclusion

In conclusion, by systematically applying the rules of exponents, we have successfully simplified the expression (x(1/2)y(-1/4)z)^(-2) and identified its equivalent form. The correct equivalent expression is y^(1/2) / (x * z^2). This exercise underscores the importance of understanding and applying exponent rules in algebraic simplification. The step-by-step approach, from distributing the exponent to handling negative exponents, provides a clear methodology for tackling similar problems. Mastering these techniques is crucial for success in algebra and beyond, enabling the simplification of complex expressions into manageable and understandable forms.