In Exercises 93–102, factor and simplify each algebraic expression. (x+3)^1/2−(x+3)^3/2

Verified step by step guidance

1. Identify the common factor in the terms of the expression. In this case, the terms are \((x+3)^{1/2}\) and \((x+3)^{3/2}\). The common factor is \((x+3)^{1/2}\).
2. Factor out \((x+3)^{1/2}\) from the expression. This will leave you with \((x+3)^{1/2} \cdot (1 - (x+3))\).
3. Simplify the expression inside the parentheses. Subtract \((x+3)\) from \(1\), resulting in \(1 - x - 3\).
4. Combine like terms inside the parentheses. Simplify \(1 - x - 3\) to \(-x - 2\).
5. Rewrite the factored expression as \((x+3)^{1/2} \cdot (-x - 2)\). This is the simplified and factored form of the given expression.

Key Concepts

Factoring Algebraic Expressions

Factoring involves rewriting an expression as a product of its factors. In the context of algebraic expressions, this often means identifying common terms or using special formulas, such as the difference of squares or the sum/difference of cubes. Understanding how to factor is essential for simplifying expressions and solving equations.

Exponents and Radicals

Exponents represent repeated multiplication of a base, while radicals are the inverse operation, indicating the root of a number. In the given expression, the terms (x+3) raised to fractional exponents can be rewritten in radical form, which aids in simplification. Mastery of these concepts is crucial for manipulating and simplifying expressions involving powers and roots.

Simplifying Expressions

Simplifying an expression involves reducing it to its most basic form, often by combining like terms and eliminating unnecessary components. This process may include factoring, reducing fractions, and applying algebraic identities. A clear understanding of simplification techniques is vital for effectively solving algebraic problems and preparing for more complex mathematical concepts.