Problem 54 - Algebra A Combined Function

Question

Simplify each of the following. First, write each expression with positive exponents. Then simplify the complex fraction. The first step has been completed for Exercise \(51 .\) $$ \frac{4+x^{-1}}{3+x^{-1}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{4x + 1}{3x + 1} \).

Step 1: Write Expression with Positive Exponents

The given expression is \( \frac{4 + x^{-1}}{3 + x^{-1}} \). To rewrite the terms with positive exponents, recall that \( x^{-1} = \frac{1}{x} \). Substitute this in the expression: \( \frac{4 + \frac{1}{x}}{3 + \frac{1}{x}} \).

Step 2: Clear the Fraction in Numerator and Denominator

To simplify, multiply both the numerator and the denominator by \( x \) to eliminate the fractions. This gives \( \frac{x(4 + \frac{1}{x})}{x(3 + \frac{1}{x})} = \frac{4x + 1}{3x + 1} \).

Step 3: Check for Further Simplification

The expression \( \frac{4x + 1}{3x + 1} \) cannot be simplified further since 4 and 3, as well as the coefficient of \( x \), are in simplest terms with no common factors to be simplified.

Key Concepts

Positive Exponents

When dealing with algebraic expressions, it's important to express all terms using positive exponents. This makes the expression easier to simplify and understand. Remember:

• A negative exponent like \(x^{-1}\) is equal to \(\frac{1}{x}\). This rule applies generally: \(a^{-n} = \frac{1}{a^n}\).
• Converting negative exponents to positive ones provides clarity in both the process and the solution.

In the exercise, the term \(x^{-1}\) is rewritten as \(\frac{1}{x}\). Writing everything with positive exponents is the first step before tackling complex fractions, allowing us to apply usual algebraic operations comfortably.

Complex Fractions

A complex fraction has fractions in the numerator, the denominator, or both. Simplifying these involves clearing the fraction-by-fraction structure.Here's how to simplify a complex fraction:

• Find a common denominator for the small fractions, if needed, to combine them.
• Multiply the numerator and denominator by a factor that will eliminate the smaller fractions. This is often the least common multiple of the denominators.

In our solution:- The complex fraction \(\frac{4 + \frac{1}{x}}{3 + \frac{1}{x}}\) is handled by multiplying the whole expression by \(x\). - This step effectively clears the smaller fractions, transforming the expression into a form that's easier to simplify: \(\frac{4x + 1}{3x + 1}\). - Notice how both \(\frac{1}{x}\) terms disappear, simplifying the complexity considerably.

Algebraic Fractions

Algebraic fractions are fractions where the numerator and/or the denominator contain algebraic expressions. Simplifying these fractions involves similar techniques used in numerical fractions, focusing on reducing to the simplest form.To simplify:

• Factor expressions in the numerator and the denominator if possible.
• Cancel out any common factors.

In this example, the expression \(\frac{4x + 1}{3x + 1}\) appears as an algebraic fraction post simplification. Despite being simplified from a complex fraction, it cannot be simplified further here, as there are no common factors between \("4x + 1"\) and \("3x + 1"\). Thus, this expression remains in its most reduced form, demonstrating a core principle: always check, but simplify only when feasible.