Problem 141 - College Algebra

Question

Factor completely. $$ (x-5)^{-\frac{1}{2}}(x+5)^{-\frac{1}{2}}-(x+5)^{\frac{1}{2}}(x-5)^{-\frac{3}{2}} $$

Step-by-Step Solution

Answer

The completely factored form of the given expression is \( (x-5)^{-\frac{1}{2}}(x+5)^{-\frac{1}{2}} (-x - 4) \)

Step 1: Identify the Common Factors

First, it's crucial to identify the common factors in the given expression. Notice that both terms in the expression have a factor of \((x-5)^{-\frac{1}{2}}\) and \((x+5)^{-\frac{1}{2}}\)

Step 2: Factor out the Common Factors

Next, factor out these common factors \((x-5)^{-\frac{1}{2}}\) and \((x+5)^{-\frac{1}{2}}\). The expression becomes: \[ (x-5)^{-\frac{1}{2}}(x+5)^{-\frac{1}{2}} \left[ 1 - (x+5) \] \]

Step 3: Simplify the Expression Inside the Brackets

We then look at the expression inside the brackets and simplify where possible. We thus achieve: \[ (x-5)^{-\frac{1}{2}}(x+5)^{-\frac{1}{2}} \left[1 - x - 5 \right] \]

Step 4: Continue Simplifying

Continuing to simplify, we get: \[ =(x-5)^{-\frac{1}{2}}(x+5)^{-\frac{1}{2}} \left[ -x - 4 \right] \]

Key Concepts

Common Factors

Identifying common factors is a fundamental step when factoring algebraic expressions. In essence, a common factor is a term that is present in each term of an expression.

Let’s say we have an expression like \( a \times b + a \times c \). Here, \( a \) is a common factor because it is multiplied with other terms in both parts of the expression. By extracting \( a \) and placing it outside of the parentheses, the expression simplifies to \( a(b + c) \). This is analogous to reducing a fraction; just as you would divide both the numerator and denominator by their common factor to simplify a fraction, factoring out common terms simplifies an algebraic expression.

When applying this concept to the exercise \( (x-5)^{-\frac{1}{2}}(x+5)^{-\frac{1}{2}}-(x+5)^{\frac{1}{2}}(x-5)^{-\frac{3}{2}} \), we recognize that both terms share factors with negative exponents. Prioritizing these makes additional steps like simplifying the expression much more effortless.

Simplifying Expressions

Once common factors are factored out, simplifying the expression becomes the next task. This involves combining like terms, reducing fractions, and carrying out arithmetic operations to condense the expression to its simplest form.

The key is to perform operations that result in fewer terms or lower exponents, ultimately making the expression more manageable. In the context of our initial expression, after extracting the common factors, we get a bracketed term \(1 - (x+5)\). Here, simplification means expanding and combining like terms which eventually streamlines the inside of the brackets to \( -x - 4 \).

Remember, when simplifying, always look for opportunities to reduce the expression further without altering its value. It’s like de-cluttering a room – combining objects that belong together and getting rid of redundancy, so you're left with the most organized and simplified space – or in this case, a mathematical expression.

Negative Exponents

Negative exponents in an algebraic expression can be intimidating, but understanding their role is crucial for simplification. A negative exponent indicates the reciprocal of the base raised to the opposite positive power. For instance, \( a^{-n} = \frac{1}{a^n} \).

Looking at our exercise, the presence of negative exponents \( (x-5)^{-\frac{1}{2}} \) and \( (x+5)^{-\frac{1}{2}} \) means we are actually dealing with the reciprocals of these expressions raised to the positive one-half power. To simplify an expression with negative exponents, you may first want to express them as fractions, which then can be simplified further in subsequent steps.

However, while factoring, it is sometimes more practical to keep the negative exponents until common factors are factored out and simplification is performed within the bracket, as seen in the step-by-step solution for the exercise. This helps maintain a clearer view of the expression's structure.