Problem 5 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

For Problems 1 through 9, simplify the following expressions. $$ \frac{z^{0} x^{-1} y^{-2}}{z^{-2} x^{-1} y^{2}} $$

Step-by-Step Solution

Answer

The simplified form of the expression \(\frac{z^{0} x^{-1} y^{-2}}{z^{-2} x^{-1} y^{2}}\) is \(z^{2}/ y^{4}\).

Step 1: Understand exponent rules

Based on exponent rules, any number or variable to the power of zero equals one \(a^{0} = 1\). Inversely, if the exponent is negative, you can take the reciprocal of the base to make the exponent positive. Thus, when we have \(a^{-n}\), it's equivalent to \(\frac{1}{a^{n}}\) where \(a\) is the base and \(n\) is the number of times the base is multiplied.

Step 2: Apply the rules to the expression.

We have the expression \(\frac{z^{0} x^{-1} y^{-2}}{z^{-2} x^{-1} y^{2}}\) . Because the exponent of \(z\) in the numerator is 0, \(z^{0}\) equals 1. So, our expression reduces to \(\frac{1 * x^{-1} y^{-2}}{z^{-2} x^{-1} y^{2}}\) which further simplifies to \(\frac{x^{-1} y^{-2}}{z^{-2} x^{-1} y^{2}}\).

Step 3: Simplify the fraction.

We apply the rules of exponents to simplify. When dividing terms with the same base, we subtract the exponents. The x terms cancel out as their exponents are equal. \(y^{-2} / y^{2} = y^{-2-2} = y^{-4}\) and \(1/ z^{-2} = z^{2}\). So, our expression simplifies to \(z^{2} * y^{-4}\) which is also equal to \(z^{2}/ y^{4}\) according to the rule which states that \(a^{-n} = 1/ a^{n}\).

Key Concepts

Negative Exponents

Negative exponents can initially seem a bit tricky, but understanding them is quite simple. When you see a negative exponent, it indicates that you should take the reciprocal of the base and change the sign of the exponent to positive. In mathematical terms, this means:

• For any base a, \( a^{-n} = \frac{1}{a^{n}} \).

This is a crucial rule because it allows us to move terms from the numerator to the denominator, or vice versa, converting "negative tension" into more manageable expressions. This property is especially useful in algebraic fraction simplification, as it helps to adjust and align exponents across a fraction quickly.
Negative exponents can be powerful tools in calculus and beyond, allowing for more precise manipulation of expressions, especially those that involve rate-based calculations.

Simplification of Expressions

Simplifying expressions is an essential skill in algebra, aimed at transforming complicated expressions into simpler, equivalent forms. The primary goal of simplification is to make expressions easier to understand, compare, and compute.

• Key techniques involve:
• Combining like terms, which often share the same base and exponent.
• Applying exponent rules, such as the power of a product, power of a power, and product of powers.

For example, in the given expression \( \frac{x^{-1} y^{-2}}{z^{-2} x^{-1} y^{2}} \), recognizing and cleverly interacting with negative exponents allows us to adjust terms, eliminate redundancy and arrive at a simplified power structure. As a result, we achieve a more manageable form that is easier to work with for further calculations or applications.

Algebraic Fraction Simplification

Algebraic fraction simplification is crucial when dealing with polynomial expressions, especially to reduce them to their most concise forms. This process involves employing both arithmetic and algebraic manipulations:

• Identifying common factors in the numerator and denominator, and canceling them out.
• Applying exponent rules to subtract exponents, particularly when fractions share bases.

In the case of the given problem \( \frac{z^{0} x^{-1} y^{-2}}{z^{-2} x^{-1} y^{2}} \), different terms with like bases were manipulated by subtracting the exponents of like terms, leading to a much simpler expression: \( z^{2} / y^{4} \). Simplification like this not only aids in making calculations easier but also ensures that expressions are presented in their simplest and most elegant form.