Problem 99 - College Algebra Essentials

Question

Factor and simplify each algebraic expression. $$(x+5)^{-\frac{1}{2}}-(x+5)^{-\frac{3}{2}}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \((x+5)^{-\frac{3}{2}}(x+4)\).

Step 1: Identifying the common factor

Firstly, you can identify that both terms share a common expression \((x+5)\). But also, you can notice that \((x+5)^{-\frac{1}{2}}\) is part of the expression \((x+5)^{-\frac{3}{2}}\). Since \((x+5)^{-\frac{3}{2}} = (x+5)^{-\frac{1}{2}}\cdot (x+5)^{-\frac{1}{2}}\), we can see that the common factor is \((x+5)^{-\frac{3}{2}}\).

Step 2: Factoring the common factor

Now that the common factor has been identified, we can factor it out. So we express the given expression as a product of the common factor and other factors. So, we rewrite it as: \((x+5)^{-\frac{3}{2}}((x+5) - 1)\), here \((x+5) - 1\) is obtained because factoring out \((x+5)^{-\frac{3}{2}}\) from \((x+5)^{-\frac{1}{2}}\) leaves us with \((x+5)\), and factoring out \((x+5)^{-\frac{3}{2}}\) from \((x+5)^{-\frac{3}{2}}\) leaves us with \(1\), that's why it is written as \((x+5) - 1\).

Step 3: Simplifying the obtained expression

We can further simplify the obtained expression to get the final simplified expression which is \((x+5)^{-\frac{3}{2}}(x+4)\). This is gotten by simplifying \((x+5) - 1\) to \(x+4\).

Key Concepts

Common Factor

Understanding the concept of a common factor is crucial when factoring algebraic expressions. A common factor is a term that is shared by all terms in an expression. In the case of the given expression, \((x+5)^{-rac{1}{2}}-(x+5)^{-rac{3}{2}}\), both terms share the expression \((x+5)\). This means that when we factor the expression, we look for the highest power of \((x+5)\) that is common to both terms.

• Identify parts of the expression that are the same in each term.
• Look for powers or exponents that appear in common.

In this exercise, notice how \((x+5)^{-rac{1}{2}}\) is a factor of \((x+5)^{-rac{3}{2}}\). This allows us to factor out \((x+5)^{-rac{3}{2}}\) as the common term. Identifying the common factor helps simplify and organize the expression, making it easier for further manipulation and reduction.

Negative Exponents

Working with negative exponents might seem tricky at first. However, they follow straightforward rules that make simplifying expressions easier. A negative exponent indicates division, essentially representing the reciprocal of the base raised to the corresponding positive exponent. For example, \((x+5)^{-rac{1}{2}}\) means \(\frac{1}{(x+5)^{\frac{1}{2}}}\).

• Multiple negative exponents within a term suggest repeated division by the base.
• Importantly, these can often be simplified by using properties of exponents, like adding or subtracting them when bases are the same.

In the provided expression, you see how negative exponents allow us to factor \((x+5)^{-rac{3}{2}}\), which breaks down into \((x+5)^{-rac{1}{2}}\times(x+5)^{-rac{1}{2}}\). This use of negative exponents simplifies to repeated multiplication or division by a base expression, elegantly restructuring our algebraic terms.Mastering negative exponents facilitates a better understanding of algebraic manipulations and leads to cleaner, more comprehensible solutions.

Simplifying Expressions

Simplifying expressions is the process of reducing them to their simplest form while maintaining equivalence. Starting with the factored expression \((x+5)^{-rac{3}{2}}((x+5) - 1)\), we can perform operations to make the expression simpler and easier to interpret, which is an essential skill in algebra.

• Combine like terms wherever possible, such as simplifying \((x+5) - 1\) to \(x+4\).
• Use properties of exponents to reduce expressions, especially when dealing with negative or fractional exponents.

Combining gives us \[(x+5)^{-rac{3}{2}}(x+4)\], which is the fully simplified version.This approach shows not only the power of understanding algebraic principles like factoring and exponent rules but also the elegance of simplifying complicated expressions into concise, compact forms.Simplifying expressions is not only a cleaner way to present solutions but also enhances computational efficiency in more complex problems.