Problem 22 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

For Problems 22 through 24, simplify as much as possible. $$ \frac{a^{x+y}-a^{2 x}}{a^{x}} $$

Step-by-Step Solution

Answer

The simplified form of the original expression is \(a^y - a^x\).

Step 1: Use the first property to rewrite the numerator

In the numerator of the given expression, the exponents are added together. The first property of exponent, \(a^{m+n} =a^m \cdot a^n\), will be used here to simplify the numerator. So, the given expression, \(\frac{a^{x+y}-a^{2x}}{a^{x}}\) becomes \(\frac{a^x \cdot a^y - a^{2x}}{a^x}\).

Step 2: Simplify the result further

After step 1, you can see that each term in the numerator has a common factor of \(a^x\). Factor out \(a^x\) from the numerator. So, the expression becomes \( \frac{a^x(a^y - a^x)}{a^x}\).

Step 3: Cancel common factors

Now, cancel the common factor of \(a^x\) from the numerator and the denominator. This results in the final simplified expression \(a^y - a^x\).

Key Concepts

Properties of Exponents

Understanding the properties of exponents is essential for simplifying exponential expressions. Some fundamental properties include the Product of Powers, which states that when multiplying two exponents with the same base, you can add the exponents: \( a^m \times a^n = a^{m+n} \). There's also the Power of a Power, where \( (a^m)^n = a^{mn} \) and the Quotient of Powers, indicating that when dividing exponents with the same base, you can subtract the exponents: \( a^m \/ a^n = a^{m-n} \).
Applying these properties allows us to handle complex expressions more effectively. For instance, if you encounter a term like \( a^{x+y} \), you can break it down using the Product of Powers property to \( a^x \times a^y \). Similarly, when simplifying expressions like the exercise in question, recognizing and factoring out common factors relies on these exponent rules.

When dealing with subtraction or addition of exponential terms, these properties may not directly apply, but factoring can help in identifying common bases and simplifying from there.

Factoring Exponents

Factoring exponents is a crucial step to simplify expressions, especially when common factors exist. In the given solution, the expression \( a^{x+y} - a^{2x} \) is factored by recognizing \( a^x \) as a common factor. This method mirrors regular factoring in algebra, where you would take out a common term to simplify the expression.
In this case, by dividing each term by the common factor \( a^x \) before performing any operations, you streamline the expression to \( a^x(a^y - a^x) \) which is easier to manage. This technique underscores how exponents can be manipulated in ways that maintain the underlying mathematical principles while making the expressions more accessible.

Remember, the goal is to make the expression as simple as possible. Factoring doesn't change the value of the expression; it simply rearranges it to reveal a more straightforward structure for further simplification.

Simplifying Algebraic Fractions

Simplifying algebraic fractions often requires a combination of the previous concepts along with identifying and canceling out common factors in the numerator and the denominator. The given expression showcases this process—after factoring out the common exponent of \( a^x \) in the numerator, it becomes apparent that it can be canceled out with the \( a^x \) in the denominator, just like simplifying \( \frac{1}{x} \times \frac{x}{1} = 1 \).
The end goal is to strip down the expression to its most basic form, which reveals \( a^y - a^x \) as the simplified version in this case. Simplifying algebraic fractions can dramatically reduce the complexity of an expression, making it easier to evaluate or use in subsequent calculations.

It's important to proceed cautiously and ensure that terms are indeed similar and can be canceled correctly. Misidentifying terms as common factors when they are not can lead to errors, so attention to detail is paramount.