Problem 33 - College Algebra and Calculus: An Applied Approach

Question

Identify the rule(s) of algebra illustrated by the statement.\(57 \cdot 1=57\)

Step-by-Step Solution

Answer

The rule illustrated by the statement \(57 \cdot 1 = 57\) is the 'Multiplicative Identity' rule.

Step 1: Identify the operation

The operation in this statement is multiplication as indicated by the \(\cdot \) sign. The values being multiplied are 57 and 1.

Step 2: Recognize the special property of multiplication by 1

The number '1' has a special property in multiplication. According to this property, any number multiplied by '1' remains the same. Therefore, 1 is referred to as the Multiplicative Identity.

Step 3: Identify the rule from the property

Considering the special property of '1' in multiplication, the rule illustrated by the statement \(57 \cdot 1 = 57\) is the 'Multiplicative Identity' rule, because the product of any number and '1' is the number itself.

Key Concepts

Multiplication Properties

Multiplication, one of the four fundamental operations in arithmetic, adheres to several key properties that make it a unique and predictable mathematical process. Among the most fundamental of these is the Multiplicative Identity Property.

Understanding the properties of multiplication is crucial in algebra and other higher-level math courses. It forms the foundation upon which further rules and operations can be built. Specifically, we look at how multiplication interacts with numbers, particularly focusing on special numerals like the identity element '1'.

Another essential property closely linked to our discussion is the Associative Property of Multiplication, which states that the way in which numbers are grouped does not affect their product. For instance, \( (2 \cdot 3) \cdot 4 = 2 \cdot (3 \cdot 4) = 24 \). This concept is important when rearranging terms in algebraic expressions.

The Commutative Property of Multiplication also plays a pivotal role, emphasizing that changing the order of the numbers in a multiplication problem does not affect the outcome. An example being, \( 5 \cdot 3 = 3 \cdot 5 \). These properties form a reliable set of rules that can be used to simplify and solve all kinds of multiplication problems.

Algebraic Rules

Algebraic rules serve as the tools that mathematicians use to manipulate equations and expressions. Rules like the Distributive Property, which combines addition and multiplication in expressions such as \( a \cdot (b + c) = a \cdot b + a \cdot c \), are indispensable in solving algebraic problems.

Distributive Property

Applying this property helps to expand expressions and to simplify computations without changing the values. For instance, \( 3 \cdot (4 + 5) = 3 \cdot 4 + 3 \cdot 5 \).

The Power of Exponents

Regarding exponents, another set of algebraic rules come into play, such as \( (a^{m})^{n} = a^{m \cdot n} \) and \( a^{m} \cdot a^{n} = a^{m+n} \). These exponent rules are also critical in algebraic manipulation, enabling us to perform operations involving powers more efficiently.

Algebraic rules form the language of algebra; mastering their use is essential to perform a wide variety of problem-solving tasks within mathematics. They are the building blocks that allow for the translation of real-world problems into mathematical equations that can be systematically solved.

Identity Property of One

The Identity Property of One, also known as the Multiplicative Identity Property, is a key algebraic concept. It states that any number multiplied by 1 will result in that same number. Symbolically, this is expressed as \( a \cdot 1 = a \).

The beauty of this property is its simplicity and universal applicability across all real numbers. No matter how complex the expression, when a term is multiplied by 1, it maintains its identity unaffected.

For example, in the expression \(57 \cdot 1 = 57\), 57 maintains its value when multiplied by one. This property is essential when dealing with more complex mathematical operations, such as simplifying algebraic expressions or solving equations.

Recognizing the use of the Identity Property of One is often a step towards finding a simpler form of an expression or isolating a variable in an equation. It is one of the most straightforward algebraic rules, yet it is a powerful tool in a mathematician's toolkit.