Problem 38 - College Algebra and Calculus: An Applied Approach

Question

Identify the rule(s) of algebra illustrated by the statement.\(\frac{1}{7}(7 \cdot 12)=\left(\frac{1}{7} \cdot 7\right) 12=1 \cdot 12=12\)

Step-by-Step Solution

Answer

The statement illustrates the Distributive Rule, the Associative Property of Multiplication, and the Identity Property of Multiplication.

Step 1: Identify the Distributive Rule

The distribution rule states that \(a(b + c) = ab + ac\). By extension, in the given equation \(\frac{1}{7}(7\cdot 12)\), multiplication is distributed over the terms within the bracket. This rule demonstrates that these operations can be carried out in any order without changing the end result.

Step 2: Identify the Associative Property

The associative rule states that the way numbers are grouped does not change their result. In the expression \((\frac{1}{7} \cdot 7) 12\), the product of a fraction and a whole number (\(\frac{1}{7}\) and \(7\)) is computed before multiplication with \(12\). This shows that products can be found no matter how the terms are grouped.

Step 3: Identify the Identity Property of Multiplication

The identity property of multiplication states that any number multiplied by 1 remains the same. In the equation \(1 \cdot 12=12\), it demonstrates that \(12\) multiplied by identity \(1\) gives the same result, \(12\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that involves distributing multiplication over addition or subtraction. In simple terms, it tells us how to deal with expressions like \(a(b + c)\). Instead of first adding \(b\) and \(c\) and then multiplying by \(a\), you can distribute \(a\) across the parentheses: \(a(b + c) = ab + ac\). This principle is vital for simplifying complex expressions and solving equations.

• It works with both addition and subtraction, e.g., \(a(b - c) = ab - ac\)
• Helps in expanding expressions to make them easier to work with

In the given exercise, the expression \(\frac{1}{7}(7 \cdot 12)\) indicates distributing multiplication over the terms inside the bracket. Though there are no additional terms, it showcases that multiplication across terms can be rearranged without affecting the outcome. This flexibility is one reason why the distributive property is so powerful in algebra.

Associative Property

The associative property helps simplify expressions by allowing numbers to be regrouped without changing the final outcome. This property applies to both addition and multiplication. It is encapsulated in the expressions \((a + b) + c = a + (b + c)\) for addition and \((ab)c = a(bc)\) for multiplication.

• Important for reordering terms when simplifying expressions
• Enables more efficient calculations by grouping numbers in strategic ways

In the step-by-step solution, the expression \((\frac{1}{7} \cdot 7) 12\) uses the associative property to group \(\frac{1}{7}\) and \(7\) together before multiplying with \(12\). This rearrangement shows that the result is the same, no matter how the numbers are grouped, crucial for tackling algebraic problems with more than two factors.

Identity Property of Multiplication

The identity property of multiplication is like a magic trick that keeps numbers unchanged when multiplied by one. Simply put, multiplying any number by 1 leaves the number exactly as it is: \(a \cdot 1 = a\). This is often used in algebra to simplify expressions without altering their value.

• An essential tool in maintaining numerical value while transforming expressions
• Commonly used to 'clean up' equations by eliminating the multiplication sign

In our exercise, we see this property in action with \(1 \cdot 12 = 12\). This shows that whenever you multiply a number by one, it remains the same. Recognizing this property helps you see through complex algebraic manipulations and ensures your results stay accurate throughout various transformations.