Problem 89 - College Algebra

Question

In Exercises 85-94, factor and simplify each algebraic expression. $$(x+3)^{1 / 2}-(x+3)^{3 / 2}$$

Step-by-Step Solution

Answer

The simplified expression after factoring and simplifying is \((x+3)^{1 / 2} * [-x - 2]\).

Step 1: Identify the Common Factor

In the expression \((x+3)^{1 / 2}-(x+3)^{3 / 2}\), both terms share a common factor of \((x+3)^{1 / 2}\).

Step 2: Factor out the Common Factor

Take the common factor \((x+3)^{1 / 2}\) out of both terms, which gives \((x+3)^{1 / 2} * [1 - (x+3)]\)

Step 3: Simplify the Expression Inside the Brackets

Simplify [1 - (x+3)] to [-x - 2]. So our simplified expression becomes: \((x+3)^{1 / 2} * [-x - 2]\).

Key Concepts

Common Factor

A common factor is an element that is present in all terms of an algebraic expression and can be factored out or removed from each term it appears in, simplifying the overall expression. In the context of our exercise, the expression contains the terms \( (x+3)^{1 / 2} \) and \( (x+3)^{3 / 2} \).
Both these terms have \( (x+3)^{1 / 2} \) as a shared element that can be factored out.
By identifying and extracting the common factor, you are essentially grouping similar parts together, which allows simplification.
This is an important process because it reduces the expression to a more manageable form.

• Saves time and effort when calculating or further manipulating the expression.
• Provides insights into the structure of the expression.

Exponents

Exponents are symbols used to denote repeated multiplication of a number or term by itself. In \( b^n \), \( b \) is the base and \( n \) is the exponent, meaning \( b \) is multiplied by itself \( n \) times.
Understanding how to work with exponents is crucial when factoring and simplifying expressions involving powers. In our problem, the terms involve exponents of \( 1/2 \) and \( 3/2 \.\)
These represent square root and cube root operations.Working with exponents involves basic rules such as:

• Adding exponents when multiplying terms with the same base.
• Subtracting exponents when dividing terms with the same base.
• Distributing the exponent over multiplication inside parentheses.

Understanding these rules allows for seamless manipulation and simplification of expressions with exponential terms.

Simplification

Simplification is the process of making an algebraic expression easier to understand or work with, generally by reducing it to a simpler form without changing its meaning.
In the original expression \( (x+3)^{1 / 2}-(x+3)^{3 / 2} \,\) simplification starts by factoring out the common factor and then operating within the bracket.
The operation \([1 - (x+3)]\) unfolds as \([1 - x - 3]\), which simplifies directly to \(-x - 2\).Furthermore, simplification helps:

• Clarify complex expressions.
• Eliminate redundant calculations.
• Make expressions more accessible for further mathematical operations or problem-solving.

Simplification is key in mathematical problem-solving and helps pave the way for more advanced reasoning.

Algebraic Expressions

Algebraic expressions form the cornerstone of algebra and consist of variables, numbers, and operational symbols. They represent quantities that can change and are often used to model real-world situations.
An expression like \( (x+3)^{1 / 2}-(x+3)^{3 / 2} \) is an algebraic expression, where \( x \) could be any number depending on the context.
Each part of an algebraic expression plays a significant role in its simplification and manipulation.Aspects of algebraic expressions include:

• Variables: Symbols (like \( x\)) that stand for unknown values.
• Constants: Regular numbers present within expressions.
• Operations: Addition, subtraction, multiplication, and division applied to the numbers and variables.

Understanding the structure of algebraic expressions allows you to manipulate and factor them, leading to simplified solutions or forming equations for solving specific problems.