Factor and simplify each algebraic expression. [12x(-1/2)+6x^(-3/2)]

Verified step by step guidance

1. Identify the common factor in the expression \(12x^{-1/2} + 6x^{-3/2}\).
2. Notice that both terms have a common factor of \(6x^{-3/2}\).
3. Factor out \(6x^{-3/2}\) from the expression: \(6x^{-3/2}(2x^{1} + 1)\).
4. Simplify the expression inside the parentheses: \(2x + 1\).
5. The factored and simplified expression is \(6x^{-3/2}(2x + 1)\).

Key Concepts

Factoring

Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. In algebra, this often involves identifying common terms or using techniques such as grouping or applying special formulas like the difference of squares. Understanding how to factor is essential for simplifying expressions and solving equations.

Exponents

Exponents represent repeated multiplication of a number by itself and are a fundamental concept in algebra. They can indicate how many times a base is multiplied, and understanding the rules of exponents, such as the product of powers and power of a power, is crucial for simplifying expressions that involve variables raised to powers. This knowledge is particularly important when dealing with negative or fractional exponents.

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form while maintaining equivalence. This can include combining like terms, factoring, and applying the distributive property. The goal is to make the expression easier to work with, which is essential for solving equations or performing further operations in algebra.