Problem 1 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

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

Step-by-Step Solution

Answer

The simplified form of the expression is \(z^2 + xz\).

Step 1: Understanding the problem

Firstly, rewrite the given equation replacing negative exponents by bringing them to the opposite part of the fraction. So \(x^-1\) would be replaced with \(\frac{1}{x}\) and \(x^-2\) with \(\frac{1}{x^2}\). Thus, the problem becomes \(\frac{\frac{1}{x} + \frac{1}{z}}{xz^{-2}}\).

Step 2: Simplify the denominator

Rewrite the denominator to reflect the reversed exponent and move it to the numerator. The denominator simplifies to \(z^{-2}x\) which is \(\frac{x}{z^2}\). The entire equation becomes \(\frac{(\frac{1}{x} + \frac{1}{z})}{(\frac{x}{z^2})}\).

Step 3: Eliminate the complex fraction

Multiply the numerator and the denominator by \(xz^2\) which is the least common multiple (LCM) of the denominators in the numerator. Thus, the equation becomes \(\frac{xz^2}{x} + \frac{xz^2}{z} \div \frac{xz^2}{z^2}x\).

Step 4: Final simplification

Now, continue simplification by eliminating alike terms. This yields \(z^2 + xz \div x\). Finally, cancel out the \(x\) in the denominator and numerator to get to the final answer.

Key Concepts

Negative Exponents

Understanding negative exponents is integral in tackling this problem. Negative exponents mean that the base is on the wrong side of a fraction.
Move the base and make the exponent positive. For example, with a term like \(x^{-1}\), you can convert this to \(\frac{1}{x}\). Here, moving \(x\) to the denominator changes the sign of the exponent.
This technique helps transform expressions into a simpler form, making them easier to solve.

• \(x^{-1} = \frac{1}{x}\)
• \(z^{-1} = \frac{1}{z}\)

Such transformations let you proceed with solving the expression without any negative exponents left.

Complex Fractions

Complex fractions are fractions where the numerator, denominator, or both are also fractions. This exercise contains a complex fraction in its initial form.
To simplify complex fractions, you often need to find a common denominator within the smaller fractions. Once you get rid of the fractions in the numerator and denominator, it becomes easier to work with.
Here’s why the LCM method helps:

• Identify common denominators.
• Multiply every part (numerator and denominator) by this common value.

In this exercise, transforming the complex fraction is done by multiplying through by \(xz^2\), eliminating smaller fractions and simplifying the problem.

Simplification Techniques

Effective simplification techniques are vital when solving algebraic expressions. Simplification makes an expression more workable and understandable.
Some key techniques include:

• Using common denominators
• Cancelling like terms
• Rewriting with positive exponents

In the solution, you see the chosen LCM \(xz^2\) playing a crucial role, as multiplying the entire expression by \(xz^2\) helps eliminate the complex fraction.The simplification actions make it possible to later cancel similar terms, arriving at a cleaner, final expression.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They can be as simple as a single term or as intricate as several terms combined.
The expression in this exercise initially appears challenging due to its elements like negative exponents and complex fractions.

• Identify like terms or similar components.
• Throughout the simplication, maintain algebraic balance.

Our problem is reduced to the expression \(z^2 + xz \div x\) after simplification.
Algebraic rules guide the process to ensure we arrive at an expression that succinctly conveys the same information as the original problem.