Problem 24 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Simplify as much as possible. $$ \frac{A^{4+p}-A^{5 p}}{A^{2+p}-A^{3 p}} $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \[\frac{A^{4} - A^{4p}}{A^{2}-A^{2p}}\].

Step 1: Rewrite the expression

Rewrite the given expression to highlight similarities in the terms: \[\frac{A^{4} \cdot A^{p}-A^{p}\cdot A^{5p}}{A^{2} \cdot A^{p} - A^{p} \cdot A^{3p}}\].

Step 2: Factorize the expressions

Both the numerator and the denominator have common terms that can be factored out: \[\frac{A^{p}(A^{4}-A^{4p}}{A^p(A^{2}-A^{2p}}\], we can then reduce A^p in the numerator and denominator.

Step3: Simplify the terms

Remove A^p from the numerator and denominator since they are common factors, ending with: \[\frac{A^{4} - A^{4p}}{A^{2}- A^{2p}}\].

Key Concepts

Exponents

Exponents are a way to express repeated multiplication of the same number. For example, when we see something like \(A^n\), it means 'A' multiplied by itself 'n' times. It’s an efficient shorthand for multiplication.
Understanding how to manipulate exponents is crucial in simplifying expressions. A few key rules simplify algebraic expressions:

• Product of Powers: \(A^m \cdot A^n = A^{m+n}\)
• Power of a Power: \((A^m)^n = A^{m \cdot n}\)
• Division of Powers: \(\frac{A^m}{A^n} = A^{m-n}\), provided \(A eq 0\).

When simplifying expressions with exponents, identify and apply these rules.
For example, in the original expression, rewritten as \(A^4 \cdot A^p\) and \(A^5p\) within the terms, we see repeated application of these rules to simplify terms. Understanding these rules helps make the manipulation of terms straightforward.

Factorization

Factorization involves breaking down complex expressions into simpler, multiplied factors. It’s like finding the building blocks of an expression. This technique is valuable because it reveals patterns and common terms that can be simplified or canceled.
In the provided example, we factor common elements in the numerator and the denominator: both share \(A^p\) as a common term. This insight allows us to extract \(A^p\) from each complex expression, simplifying our fraction.

• Finding Common Factors: Check each term for common bases or coefficients.
• Factor Out: Once identified, take common factors outside the parentheses.

After factorizing, complex expressions become easier to handle, as seen when removing \(A^p\) to concentrate on simplifying the remaining parts: \(A^4 - A^{4p}\) and \(A^2 - A^{2p}\). Factorization simplifies work and facilitates further simplification.

Algebraic Fractions

Algebraic fractions resemble numerical fractions but involve algebraic expressions in the numerator and denominator. Simplifying these fractions requires consistent and careful application of algebraic rules.
The goal is to make the fraction as simple as possible by applying factorization and reducing common terms. We approach this fraction like a numerical one by identifying common factors that can be canceled.

• Match Numerator and Denominator: Find and cancel out common factors, just as you would cancel duplicate numbers in regular fractions.
• Simplify: Once simplified, examine the new expression and continue reducing, if possible.

In our task, by identifying \(A^p\) as a common factor in the entire expression, we effectively simplified the fraction. This results in a clean, simplified algebraic fraction. The crux is in maintaining balance, reducing where possible, and consistently checking your work to ensure correctness.