Problem 32 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Simplify as much as possible: (a) \(\frac{x^{2 y}+x^{y+2}}{x^{y}}\) (b) \(\frac{\frac{\sqrt{x}}{x^{-1 / 2} y}-1}{y-\frac{x^{2}}{y}}\) (c) \(\frac{A^{B+4}-A^{3 B}}{A^{B}\left(A^{2}-A^{B}\right)}\) (d) \(\frac{y^{3 w}-y^{w+4}}{y^{w}\left(y^{w}+y^{2}\right)}\)

Step-by-Step Solution

Answer

(a) \(x^{y} + x^{2}\), (b) \(\frac{y - y^2}{y^3 - x^2}\), (c) \(\frac{A^{4}-A^{2B}}{A^{2}-A^{B}} \) , (d) \(y^{2 w} - y^{4}\)

Step 1: Simplify expression (a) \(\frac{x^{2 y}+x^{y+2}}{x^{y}}\)

Use the property of exponents where the division of two exponentials with the same base is equivalent to subtracting their exponents. Simplified expression (a) is \(x^{2y-y} + x^{2}\) which becomes \(x^{y} + x^{2}\)

Step 2: Simplify expression (b) \(\frac{\frac{\sqrt{x}}{x^{-1 / 2} y}-1}{y-\frac{x^{2}}{y}}\)

Begin by clearing the fractions in the numerator and denominator, and then simplify the expression. Clearing the fractions, we have \(\frac{\sqrt{x}* y * x^{1/2} - y^2}{y^3- x^2}\). Once simplified, expression (b) becomes \(\frac{y - y^2}{y^3 - x^2}\)

Step 3: Simplify expression (c) \(\frac{A^{B+4}-A^{3 B}}{A^{B}\left(A^{2}-A^{B}\right)}\)

Similar to (a), use the property of exponents and simplify the expression. Simplified expression (c) is \(\frac{A^{4}-A^{2B}}{A^{2}-A^{B}} \)

Step 4: Simplify expression (d) \(\frac{y^{3 w}-y^{w+4}}{y^{w}\left(y^{w}+y^{2}\right)}\)

Apply the same principle of exponentials as in (a) and (c) to simplify the expression. Simplified expression (d) is \(y^{2 w} - y^{4}\)

Key Concepts

Properties of Exponents

Understanding the properties of exponents is essential when simplifying expressions. Here are some crucial properties:

• Product of Powers: When multiplying two powers with the same base, you add the exponents. For example, if you have\( x^a \times x^b = x^{a+b} \).


• Quotient of Powers: When dividing two powers with the same base, you subtract the exponents. For instance,\( \frac{x^a}{x^b} = x^{a-b} \).


• Power of a Power: If a power is raised to another power, multiply the exponents. In other words,\((x^a)^b = x^{a \cdot b}\).


• Negative Exponent: A negative exponent means you take the reciprocal of the base. This looks like\( x^{-a} = \frac{1}{x^a}\).

These rules can make complicated expressions easier to handle. They allow you to combine and rearrange terms to simplify them. Familiarity with these properties will make dealing with complex expressions, such as those presented in the exercise, much more manageable.

Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging algebraic expressions to simplify them. This often involves applying operations such as addition, subtraction, multiplication, and division in combination with exponent rules. In the exercise, for example, simplifying each expression involves:

• Cancelling Common Terms: Identify and eliminate terms numerator and denominator to make expressions simpler. This can occur through factoring.


• Distributing and Factoring: Use distribution to simplify expressions and factoring to break down complex expressions, making it easier to cancel out terms.


• Simplifying Fractions: Combine like terms and reduce fractions to their simplest form by ensuring there are no common factors between the numerator and denominator.

Understanding the proper steps and sequences in algebraic manipulation can solve high-complexity expressions easily and systematically.

Rational Expressions

Rational expressions are fractions in which the numerator and the denominator are both polynomials. Simplifying rational expressions involves manipulating these algebraic fractions according to specific rules and operations. In the given exercise, you simplify rational expressions by using the following principles:

• Factorization: Factor the numerator and denominator separately, turning them into simpler components that can be easily compared and reduced.


• Finding Common Denominators: It's crucial when adding or subtracting rational expressions, akin to dealing with ordinary fractions.


• Canceling Common Factors: Once factorized, cancel out any identical terms that appear in both the numerator and denominator.

Rational expressions require consistent practice to master, as they are common in algebra and calculus. The key is understanding the fundamental properties and operations of fractions and polynomials, allowing students to simplify complex algebraic expressions efficiently.