Problem 46 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Perform the indicated operations and simplify. \(\frac{\frac{1}{(x+h)^{2}}-\frac{1}{x^{2}}}{h}\)

Step-by-Step Solution

Answer

The simplified expression for the given complex fraction is \(\frac{-2xh-h^2}{x^2(x+h)^2 \cdot h}\).

Step 1: Identify the main fraction components

The given expression is a complex fraction, which consists of numerator and denominator: Numerator: \(\frac{1}{(x+h)^{2}}-\frac{1}{x^{2}}\) Denominator: \(h\)

Step 2: Combine the terms in the numerator

First, we will focus on the numerator part \(\frac{1}{(x+h)^{2}}-\frac{1}{x^{2}}\). In order to subtract two fractions, we need to have a common denominator. The common denominator for \((x+h)^{2}\) and \(x^{2}\) is their product, i.e., \((x+h)^{2} \cdot x^{2}\). Now, we need to rewrite both fractions with this common denominator: \(\frac{1}{(x+h)^{2}}\cdot \frac{x^2}{x^2} = \frac{x^2}{x^2(x+h)^2}\) \(\frac{1}{x^{2}}\cdot \frac{(x+h)^2}{(x+h)^2} = \frac{(x+h)^2}{(x+h)^2x^2}\) Now we can subtract them: \(\frac{x^2-(x+h)^2}{x^2(x+h)^2}\)

Step 3: Expand the numerator and simplify

In the numerator part, we need to expand the \((x+h)^2\) and simplify the expression: \((x+h)^2 = x^2 +2xh + h^2\) Now, substitute back into the fraction: \(\frac{x^2-(x^2 +2xh + h^2)}{x^2(x+h)^2}\) Simplify by subtracting the terms in the numerator: \(\frac{-2xh-h^2}{x^2(x+h)^2}\)

Step 4: Simplify the complex fraction

Now we have the simplified numerator, we can get back to the main expression and rewrite it as a single fraction: \(\frac{\frac{-2xh-h^2}{x^2(x+h)^2}}{h}\) To simplify a complex fraction, we multiply the numerator and the denominator by the reciprocal of the denominator. Here we just need to multiply by 1/h: \(\frac{-2xh-h^2}{x^2(x+h)^2} \cdot \frac{1}{h}\) This gives: \(\frac{-2xh-h^2}{x^2(x+h)^2 \cdot h}\) And this is the final, simplified expression for the given complex fraction.

Key Concepts

Fraction Simplification

Fraction simplification is all about reducing fractions to their simplest form. Understanding this process is essential whether we are dealing with regular fractions, complex fractions, or rational expressions. When you have a complex fraction like the one in the original exercise, simplifying involves breaking down the components to the simplest possible terms. The key is to find a common denominator for any terms within the fraction so that you can effectively manipulate and simplify the expression. Additionally, with complex fractions, once you simplify the individual components, you'll often need to multiply by the reciprocal of the denominator—something shown as multiplying by 1/(denominator)—to simplify the entire expression into one coherent fraction. This helps in balancing the numerator and the denominator without changing the value of the expression.

Rational Expressions

Rational expressions are essentially fractions where the numerator and the denominator are polynomials. These can often seem challenging at first due to the algebraic content. However, with steady practice, simplifying them becomes intuitive. In dealing with rational expressions, the foundational rule is similar to fractions: Always look for a common denominator before adding or subtracting. For instance, in the exercise, the numerators in the complex fraction needed to be expressed over a common denominator to enable subtraction. When working with these expressions, remember:

• Identify common factors in the numerator and the denominator.
• Cancel out these factors if possible, always respecting the rules of algebra.
• Recheck your work to ensure no simple factor can be canceled out, as this means the expression isn’t fully simplified yet.

Polynomial Algebra

Polynomial algebra involves expressions that include variables raised to various powers. Understanding how to expand and simplify such expressions is critical for successfully manipulating rational expressions.In the original problem, expanding \( (x+h)^2 \) into \( x^2 + 2xh + h^2 \) is a crucial step. This expansion allowed us to subtract the polynomial terms properly in the numerator.Important aspects of working with polynomials include:

• Learning to recognize polynomial identities and expansions like square expansions.
• Practicing how to efficiently distribute and collect like terms for simplifying purposes.
• Be mindful of common polynomial division and factoring techniques to simplify expressions further.

Once you master polynomial manipulations, you'll find solving and simplifying complex fractions comes more naturally, improving overall problem-solving skills in algebra.