Identifying The Non-Applicable Rule In Simplifying (-1xz^2)^3(x^3)^2

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In the realm of mathematics, simplifying algebraic expressions is a fundamental skill. It allows us to take complex expressions and rewrite them in a more concise and manageable form. This process often involves applying various exponent rules, which are essentially mathematical shortcuts that help us manipulate expressions with powers. In this article, we will delve into the intricacies of simplifying algebraic expressions, focusing on a specific example: (-1xz2)3(x3)2. We will meticulously examine the exponent rules involved and identify the one rule that does not apply in this particular scenario.

Before we embark on simplifying the given expression, it is crucial to have a solid grasp of the exponent rules that govern the manipulation of powers. These rules provide the foundation for simplifying expressions and solving algebraic equations. Let's take a closer look at the relevant exponent rules:

  • Negative Rule: The negative rule states that any term raised to a negative exponent can be rewritten as its reciprocal with a positive exponent. Mathematically, this is expressed as x^-n = 1/x^n. This rule is particularly useful for dealing with expressions that involve negative exponents, allowing us to transform them into expressions with positive exponents, which are often easier to work with.
  • Product of a Power Rule: The product of a power rule comes into play when multiplying terms with the same base raised to different exponents. This rule states that to multiply such terms, we simply add the exponents while keeping the base the same. Mathematically, this is expressed as x^m * x^n = x^(m+n). This rule simplifies the multiplication of expressions with the same base, making it easier to combine terms and simplify the overall expression.
  • Power of a Power Rule: The power of a power rule applies when we raise a term with an exponent to another exponent. This rule states that to raise a power to another power, we multiply the exponents. Mathematically, this is expressed as (xm)n = x^(m*n). This rule is essential for simplifying expressions where a power is raised to another power, allowing us to reduce the expression to a single power.
  • Product Rule: The product rule, in the context of exponents, is closely related to the product of a power rule. It essentially extends the concept to situations where we have multiple terms inside parentheses raised to a power. This rule states that to raise a product to a power, we raise each factor in the product to that power. Mathematically, this is expressed as (xy)^n = x^n * y^n. This rule is crucial for distributing exponents over products, enabling us to simplify expressions with multiple factors raised to a power.

Now that we have reviewed the essential exponent rules, let's tackle the task of simplifying the expression (-1xz2)3(x3)2. We will proceed step-by-step, applying the appropriate rules at each stage to arrive at the simplest form of the expression.

  1. Applying the Product Rule:

    We begin by applying the product rule to the term (-1xz2)3. This involves raising each factor within the parentheses to the power of 3. This gives us:

    (-1)^3 * (x)^3 * (z2)3

  2. Simplifying Individual Terms:

    Next, we simplify each of the individual terms obtained in the previous step:

    • (-1)^3 = -1 (since a negative number raised to an odd power is negative)
    • (x)^3 = x^3
    • (z2)3 Here, we apply the power of a power rule, multiplying the exponents: (z2)3 = z^(2*3) = z^6

    Substituting these simplified terms back into our expression, we get:

    -1 * x^3 * z^6

  3. Applying the Power of a Power Rule Again:

    Now, let's turn our attention to the second part of the original expression, (x3)2. We apply the power of a power rule here, multiplying the exponents:

    (x3)2 = x^(3*2) = x^6

  4. Combining the Simplified Terms:

    We now have two simplified expressions: -1 * x^3 * z^6 and x^6. To complete the simplification, we multiply these two expressions together:

    (-1 * x^3 * z^6) * (x^6)

  5. Applying the Product of a Power Rule:

    To multiply the terms with the same base (x in this case), we apply the product of a power rule, adding the exponents:

    -1 * (x^3 * x^6) * z^6 = -1 * x^(3+6) * z^6 = -1 * x^9 * z^6

  6. Final Simplified Expression:

    Therefore, the simplified form of the expression (-1xz2)3(x3)2 is -x9z6.

Having successfully simplified the expression, let's now address the central question: Which exponent rule did we not apply during the simplification process?

Looking back at our step-by-step solution, we can see that we utilized the following rules:

  • Product Rule (in step 1)
  • Power of a Power Rule (in steps 2 and 3)
  • Product of a Power Rule (in step 5)

However, we did not need to apply the negative rule. The negative rule comes into play when we have terms raised to negative exponents. In our original expression, (-1xz2)3(x3)2, there were no negative exponents present. Therefore, the negative rule was not required to simplify the expression.

In conclusion, when simplifying the expression (-1xz2)3(x3)2, the exponent rule that does not apply is the negative rule. We successfully simplified the expression using the product rule, the power of a power rule, and the product of a power rule. This exercise highlights the importance of understanding the nuances of each exponent rule and recognizing when to apply them appropriately. By mastering these rules, we can confidently tackle complex algebraic expressions and simplify them to their most manageable forms. This ability is crucial for success in various mathematical disciplines and beyond.