Problem 4 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

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

Step-by-Step Solution

Answer

The simplified expression is \( \frac{z}{x + z} \).

Step 1: Rewrite the expression using properties of exponents

The expression \( \frac{x^{-1}}{z x^{-1}+z^{-1}} \) can be rewritten using the property \( x^{-n} = 1/x^{n} \). This gives \( \frac{1/x}{z/ x + 1/z} \).

Step 2: Remove the complex fraction

The goal is to remove the complex fraction. That is, to deal with the fractions in the numerator and in the denominator. This can be done by multiplying the entire expression by \(xz\) (which is the product of the denominators of the fractions in the expression). So the expression becomes \( \frac{1/x \cdot xz}{z/x \cdot xz + 1/z \cdot xz} = \frac{z}{x + z} \).

Step 3: Final expression

So, the simplified form of the given expression is \( \frac{z}{x + z}. \)

Key Concepts

Understanding the Properties of Exponents

Exponents represent repeated multiplication of a base number. When you encounter expressions with exponents, certain rules and properties can be used to simplify them. One such rule is the negative exponent, which is the reciprocal of the base raised to the positive exponent.

For example, if you have an expression like \( x^{-1} \), this means \( \frac{1}{x} \). Essentially, the negative sign moves the base to the denominator. When you simplify expressions with exponents, always keep in mind these basic properties.

• Product of powers: \( a^m \cdot a^n = a^{m+n} \)
• Power of a power: \( (a^m)^n = a^{mn} \)
• Quotient of powers: \( \frac{a^m}{a^n} = a^{m-n} \), for \( a eq 0 \)
• Zero exponent: \( a^0 = 1 \), for \( a eq 0 \)

Understanding these properties makes it easier to manipulate and simplify expressions in calculus.

Simplifying Expressions: Step-by-Step Guide

To simplify an expression means to rewrite it in a form that is easier to interpret or more compact. The goal is often to make it as simple as possible. For algebraic expressions involving exponents, this can be achieved by applying properties of exponents.

In our exercise, the initial expression \( \frac{x^{-1}}{z x^{-1}+z^{-1}} \) is first rewritten using exponent properties. The negative exponents are transformed using \( x^{-n} = \frac{1}{x^n} \), leading to \( \frac{1/x}{z/x + 1/z} \).

• Firstly, convert all negative exponents to their reciprocal forms.
• Next, try to combine terms with the same base or factor terms where possible.
• Finally, simplify complex fractions if they exist, to make the expression cleaner.

Following these steps helps achieve a simpler form that is often required in calculus.

Tackling Complex Fractions

A complex fraction is one that contains a fraction in its numerator, denominator, or both. Simplifying complex fractions requires a keen eye and some algebraic manipulations. The initial step often involves eliminating the smaller fractions within the larger fraction.

In our initial problem, we remove the complex fraction \( \frac{1/x}{z/x + 1/z} \) by multiplying the numerator and the denominator by the product of the smaller denominators, here, \( xz \). This kind of multiplication eliminates the small fractions by effectively canceling out the denominators.

• Identify the least common multiple of denominators.
• Multiply the entire fraction by this common multiple to eliminate smaller fractions.
• Simplify the resulting expression to your final answer, \( \frac{z}{x + z} \) in our case.

Understanding how to manipulate these fractions is crucial for better grasping calculus concepts.