Problem 38 - College Algebra and Calculus: An Applied Approach

Question

Simplify the expression.\(\frac{24(x-2)^{2}}{8(x-2)^{4}}\)

Step-by-Step Solution

Answer

The simplified form of \(\frac{24(x-2)^{2}}{8(x-2)^{4}}\) is \(3(x - 2)^{-2}\).

Step 1 - Reduce the coefficient

First simplify the fraction without the variables. This gives \( \frac{24}{8} = 3 \).

Step 2 - Apply the laws of exponents

For the \( (x - 2) \) part, use the rule of exponents \( a^{n} / a^{m} = a^{(n - m)} \). This gives \( (x - 2)^{2 - 4} = (x - 2)^{-2} \).

Step 3 - Write the final simplified expression

Combine the results from Step 1 and Step 2. The simplified expression is \( 3(x - 2)^{-2} \).

Key Concepts

Simplifying Expressions

Simplifying an expression involves reducing it to its most basic form. This can make equations easier to solve or understand. The process generally involves combining like terms, removing parentheses, and reducing fractions.

If you're dealing with fractions in an expression, you should first look at the numerical coefficients. In our exercise, we simplify \( \frac{24}{8} \) to 3. This step is crucial because it makes the expression easier to manage in subsequent steps.

Beyond coefficients, notice any terms that are multiplied together or any polynomials that could be simplified. While the main goal is to make the expression simpler, don't change the value of the expression.

Laws of Exponents

The laws of exponents are rules that guide how to handle expressions involving powers of the same base. These rules make it easier to simplify complex expressions. Here are the ones most commonly used:

• Product of powers: \( a^{m} \times a^{n} = a^{m+n} \)
• Quotient of powers: \( a^{m} / a^{n} = a^{m-n} \)
• Power of a power: \( (a^{m})^{n} = a^{m \cdot n} \)

In our problem, we applied the quotient of powers rule: \( (x - 2)^{2 - 4} \). This simplifies to \( (x - 2)^{-2} \), which is a compact way to rewrite without changing the value.

Realize that a negative exponent indicates reciprocal: \( a^{-n} = \frac{1}{a^{n}} \). It's a neat way to tuck away the excess powers within a fraction. Learning these laws simplifies many algebra problems.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. Simplifying them involves similar steps as simplifying numerical fractions but applied to algebraic terms.

To simplify the expression \( \frac{24(x-2)^{2}}{8(x-2)^{4}} \), start by reducing the numerical fraction or coefficient, which gives us a clue on reducing the rational part.

The polynomials in the expression follow the laws of exponents, as shown. After simplification, the rational expression is neatly condensed to \( 3(x-2)^{-2} \). By understanding these fundamental operations, you can tackle more complex rational expressions in the future.