Problem 22 - Calculus with Concepts in Calculus

Question

Find the error. $$ \frac{e^{3}}{e^{\sqrt{2}} e^{\sqrt{2}}} \stackrel{?}{=} \frac{e^{3}}{e^{\sqrt{2}-\sqrt{2}}} \stackrel{?}{=} \frac{e^{3}}{e^{2}} \stackrel{?}{=} e $$

Step-by-Step Solution

Answer

The error is in the denominator simplification. It should be \( e^{3-2\sqrt{2}} \), not \( e \).

Step 1: Understand the Problem

We are given the expression \( \frac{e^{3}}{e^{\sqrt{2}} e^{\sqrt{2}}} \) and we need to check if it equates to \( e \) by simplifying it.

Step 2: Simplify the Denominator

The denominator \( e^{\sqrt{2}} e^{\sqrt{2}} \) can be simplified using the property \( e^a \cdot e^b = e^{a+b} \). Thus, it simplifies to \( e^{2\sqrt{2}} \).

Step 3: Simplify the Expression

Now the expression \( \frac{e^{3}}{e^{2\sqrt{2}}}\) can be further simplified by using the property \( \frac{e^a}{e^b} = e^{a-b} \). Therefore, it becomes \( e^{3-2\sqrt{2}} \).

Step 4: Find the Error

In the original problem, the calculation incorrectly equates \( \frac{e^{3}}{e^{\sqrt{2}} e^{\sqrt{2}}} \) directly to \( \frac{e^{3}}{e^{2}} \) by assuming \( e^{\sqrt{2} - \sqrt{2}} = e^{0} \) for the denominator, which is incorrect.

Step 5: Conclude

The error lies in the incorrect simplification of the denominator \( e^{\sqrt{2}} e^{\sqrt{2}} \). The correct simplified expression should be \( e^{3-2\sqrt{2}} \), not \( e \).

Key Concepts

Properties of Exponents

To understand exponential functions fully, we must be familiar with the important properties of exponents. These properties allow us to manipulate expressions, simplify them, and solve equations.

One crucial property is the product of powers property. This states that if you have the same base being multiplied, like in the expression \( e^a \cdot e^b \), you can combine the exponents by adding them together. Therefore, \( e^a \cdot e^b = e^{a+b} \). This makes calculations easier and automatically streamlines the expression.

Another key property is the quotient of powers property. This essentially follows the principle "divide and subtract." If we have an expression like \( \frac{e^a}{e^b} \), we simplify it by subtracting the exponents, resulting in \( e^{a-b} \). These properties are fundamental to grasping complex exponential relationships.

Simplification

Simplification is the process of reducing a complex expression into a simpler form without changing its value. Whenever we work with exponents, applying the properties we just talked about speeds up this process.

In the problem presented, we have a complex fraction \( \frac{e^3}{e^{\sqrt{2}} \cdot e^{\sqrt{2}}} \). To simplify, first recognize that the denominator can be combined into one term using the property \( e^a \cdot e^b = e^{a+b} \), becoming \( e^{2\sqrt{2}} \).
This reduction facilitates further simplification of the entire rational expression to \( \frac{e^3}{e^{2\sqrt{2}}} \).

At this juncture, apply the division property, transforming it into \( e^{3-2\sqrt{2}} \). With practice, simplification becomes a natural step in solving exponential equations effectively.

Mathematical Errors

When dealing with exponents, mathematical errors often stem from misunderstanding or misapplying properties. In the problem statement, we saw a common example where the properties of exponents were not correctly applied.

Instead of correctly combining the terms \( e^{\sqrt{2}} \cdot e^{\sqrt{2}} \) to \( e^{2\sqrt{2}} \), an incorrect operation led to a misunderstanding. The mistake again appeared in treating \( \frac{e^3}{e^{2\sqrt{2}}} \) as if it could simplify to \( e^3/e^2 \) just by confusing steps.

To avoid these errors, continually check and yield to exponent rules instead of jumping through steps. Confirm the logic of each transformation or calculation, and always remember that little errors can lead to drastically different outcomes. Being meticulous with every step ensures a strong grasp of exponential functions and a correct final answer. Such diligence forms a cornerstone of precise mathematical practice.