91Ó°ÊÓ

Factoring involves breaking down an algebraic expression into simpler terms, or factors, that multiplied together give the original expression. This concept is key in algebra, especially when simplifying complex expressions. When we talk about factoring in the context of algebraic expressions like \((x+5)^{-\frac{1}{2}} - (x+5)^{-\frac{3}{2}}\), we aim to identify a common element or base.

In this example, the expression \((x+5)\) is a common base. Recognizing this helps in factoring, as we can take out this common base first, simplifying the expression as a whole. The process begins by determining the lowest power or exponent of the common base among the terms involved. In our exercise, the exponent \(-\frac{1}{2}\) is the smaller compared to \(-\frac{3}{2}\), hence factored out as \((x+5)^{-\frac{1}{2}}\). This brings simplicity and order into the expression, making further manipulation much more manageable.

Exponents, sometimes called powers, are a representation of the number of times a number, known as the base, is multiplied by itself. They are crucial in expressing large numbers or repeated multiplications concisely. In the expression \((x+5)^{-\frac{1}{2}} - (x+5)^{-\frac{3}{2}}\), the exponents are negative fractions: \(-\frac{1}{2}\) and \(-\frac{3}{2}\).

A negative exponent indicates that the base is on the denominator instead of the numerator, according to the rule \(a^{-n} = \frac{1}{a^n}\). The fractional part of the exponent denotes a root; hence, an exponent of \(-\frac{1}{2}\) signifies a reciprocal square root. Understanding this can greatly help in transforming and simplifying the expression by moving the elements using the properties of exponents. When working through such problems, keeping the base same but simplifying the exponents decouples some complexity from algebraic manipulations.

Simplification is the process of reducing an expression to its simplest form, maintaining the same value but becoming easier to work with. In algebra, simplification involves combining like terms, reducing fractions, or removing extraneous factors. After factoring and identifying exponents in an expression, the next logical step is to simplify.

In the expression we’re tackling, we begin with the factored form: \((x+5)^{-\frac{1}{2}}[1-(x+5)]\). Simplifying the part inside the brackets:

• First transform \(1 - (x+5)\) into \(-x - 4\)
• Once simplified internally, multiply/operate the expression outside the brackets, if necessary.

Simplification clears up algebraic expressions by reducing their complexity, making them easier to solve or evaluate.