Problem 97 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The simplified form of the algebraic expression is \((x+3)^{1/2}(1-x).\

Step 1: Identify Common Factors

In the two terms of the equation \((x+3)^{1/2}\) and \((x+3)^{3/2}\), it can be seen that \((x+3)^{1/2}\) is a common factor. This common factor can be used for factoring out.

Step 2: Recognize the Greater Exponent

Rewriting the greater exponent \((x+3)^{3/2}\) as \((x+3)^{1/2}(x+3)\). Rewriting the entire expression in these terms results in the following: \((x+3)^{1/2}-(x+3)^{1/2}(x+3)\).

Step 3: Factor Out the Common Term

Once the terms are determined, the common factor, \((x+3)^{1/2}\) can be factored out: \((x+3)^{1/2}(1-(x+3))\).

Step 4: Simplify the Expression

The term in the brackets can be simplified by subtracting 3 from 1 and replacing the expression \(x+3\) with \(x\) to find the final solution: \((x+3)^{1/2}(1-x).\

Key Concepts

Simplify Algebraic Expressions

Simplifying algebraic expressions is a fundamental skill in algebra. It involves reducing an expression to its simplest form, making it easier to understand and work with. To simplify an expression, you combine like terms and use arithmetic operations. Like terms are terms that have exactly the same variable factors, regardless of their coefficients.

For instance, in the problem \( (x+3)^{\frac{1}{2}} - (x+3)^{\frac{3}{2}} \), simplification starts by noticing that the terms have common factors, and it becomes crucial to recognize and extract these common factors in order to simplify the expression effectively. After factoring, remaining terms must also be simplified, which often involves performing arithmetic operations. In this example, after factoring, we're left with \((x + 3)^{\frac{1}{2}} (1 - (x + 3))\), which then needs to be simplified by distributing the factor across the terms inside the parentheses.

Exponent Rules

Exponent rules are the guidelines for performing operations involving powers. Understanding exponent rules is key to simplifying expressions with exponents properly. One of the most basic rules is the Product of Powers rule, which states that when multiplying two exponents with the same base, you add the exponents: \(a^m \times a^n = a^{m+n}\).

In the given algebraic expression, we deal with fractional exponents. The exponent \(\frac{3}{2}\) can be understood as \(a^{1 + \frac{1}{2}}\), which follows the rule of adding exponents since \((x+3)^{\frac{3}{2}}\) is the same as \((x+3)^{\frac{1}{2}} \times (x+3)\). The understanding and application of exponent rules are fundamental to identify the common factors and simplify the terms correctly in the expression.

Common Factors in Algebra

Identifying common factors plays a pivotal role in factoring algebraic expressions. A common factor is a factor that is shared by all terms in an expression. When you factor out a common factor, you are essentially dividing each term by that factor, which simplifies the original expression.

For the exercise given, \((x+3)^{\frac{1}{2}}\) is identified as the common factor between the terms. By factoring it out, the expression is rewritten as \((x+3)^{\frac{1}{2}}(1-(x+3))\). In practice, the distributive property is often used to find common factors, which states that \(a(b + c) = ab + ac\). Here, \((x+3)^{\frac{1}{2}}\) is distributed across the terms inside the subsequent parentheses. This method significantly streamlines expressions, making further operations much less complex.