Problem 34 - College Algebra and Calculus: An Applied Approach

Question

Identify the rule(s) of algebra illustrated by the statement.\(1 \cdot(1+x)=1+x\)

Step-by-Step Solution

Answer

The rule illustrated by the statement \(1 \cdot (1+x) = 1+x\) is the Multiplication Property of One.

Step 1: Identify the property

This expression represents the Multiplication Porperty of One. In mathematics, the multiplication property of one states that the product of any number and one is the number itself, it doesn't change the value of a number.

Step 2: Implement the property

If we implement the property in the expression: \(1 \cdot (1+x)\) we simply get \(1+x\) on the right. In the expression \(1 \cdot (1+x)\), we can simplify it by multiplying each term in the parentheses by 1. So, \(1 \cdot 1 = 1\) and \(1 \cdot x = x\). We then get \(1+x\) which matches the right side in our original equation.

Key Concepts

Multiplication Property of One

In algebra, the Multiplication Property of One is a fundamental concept that helps simplify expressions by multiplying a unique factor. The principle is straightforward: multiplying any number by one results in the original number itself. This aspect is particularly useful when you want to rewrite or manage expressions without altering their value. Key points of the Multiplication Property of One include:

• Multiplying by one does not change the value of a number.
• It is often used to verify equality in equations or simplify expressions.
• Such properties make equations less complex and more manageable.

Consider an expression like \(1 \cdot (1+x)\). Applying this property, each term inside the parentheses is multiplied by one, simplifying directly to \(1+x\). Hence, understanding and leveraging this property can facilitate solving more complex algebraic problems.

Simplifying Expressions

Simplifying expressions plays a vital role in algebra, as it allows us to present complex information in a more digestible form. When we simplify an expression, we reduce it to its most basic elements without changing its intrinsic value. This process often involves:

• Combining like terms.
• Applying algebraic properties like the Multiplication Property of One.
• Removing unnecessary parentheses.

For instance, in the statement \(1 \cdot (1 + x) = 1 + x\), we recognize that multiplication by one doesn't change the outcome. Therefore, when simplifying \(1 \cdot (1+x)\), each term gets multiplied by 1, resulting directly in \(1 + x\), removing the need for additional steps. By mastering these simplification techniques, you'll be able to approach algebraic problems with increased efficiency and accuracy.

Basic Algebra Rules

Algebra is built upon fundamental rules and properties that guide the manipulation of expressions and equations. These are crucial for making sense of more complicated mathematical concepts. By mastering these basic rules, you can solve algebraic problems systematically. Key rules include:

• Commutative Property: Order does not affect the sum or product (\(a+b = b+a\), \(ab = ba\)).
• Associative Property: Grouping does not affect the sum or product (\((a+b)+c = a+(b+c)\), \((ab)c = a(bc)\)).
• Distributive Property: Multiply across addition (\(a(b+c) = ab + ac\)).

Applying these rules, along with the Multiplication Property of One, allows you to simplify and solve equations more efficiently. For instance, recognizing the expressions and equations, understanding that simplifying terms like \(1\cdot x\) automatically reduces them, equates directly to \(x\). As you familiarize yourself with these fundamental properties and rules, they become intuitive tools for any algebraic challenge.