Problem 42 - Prealgebra

Question

Simplify each expression. $$6 \cdot(y \cdot 2)$$

Step-by-Step Solution

Answer

The simplified expression is \( 12y \).

Step 1: Identify the Expression to Simplify

Examine the expression given: \( 6 imes (y imes 2) \). Notice that it involves multiplication and contains both numerical and variable components.

Step 2: Apply Associative Property of Multiplication

The associative property of multiplication states that the way in which factors are grouped in a multiplication problem does not change the product. For this problem, we can regroup \( (y imes 2) \) as \( (6 imes 2) \) to make multiplication easier. This changes the expression to: \( y imes (6 imes 2) \).

Step 3: Calculate Simple Multiplication

Now, compute the multiplication for the two constant numbers: \( 6 \times 2 = 12 \). Replace this in the expression, transforming it to \( y imes 12 \).

Step 4: Rearrange the Expression

Finally, place the constant (adjust by common convention to write the constant first) in front of the variable to finalize the expression as \( 12y \).

Key Concepts

Associative Property

The associative property is a fundamental concept in mathematics, especially useful in simplifying expressions. It's a rule that allows you to rearrange the grouping of numbers or variables in any multiplication operation without affecting the overall product. This means if you have three numbers a, b, and c, you can group them in any order:

• \((a \cdot b) \cdot c = a \cdot (b \cdot c)\)

For example, in the expression \(6 \cdot (y \cdot 2)\), by using the associative property, we regroup the expression to \(y \cdot (6 \cdot 2)\). This simplification doesn't change the product but makes the multiplication easier to handle. The associative property is particularly handy when working with both numbers and variables, ensuring that calculations are simplified step by step.

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form while maintaining the equality in mathematical operations. This process often includes:

• Applying arithmetic rules or properties (like associative property).
• Combining like terms.
• Rewriting expressions for clarity.

In the given expression, \(6 \cdot (y \cdot 2)\), we simplify by first applying the associative property to regroup the terms as \(y \cdot (6 \cdot 2)\). Next, we perform the arithmetic operation \(6 \times 2\) to get \(12\). The final simplified expression becomes \(12y\). Simplification helps in resolving complex expressions to a form that is easier to understand and work with.

Multiplication

Multiplication is one of the basic arithmetic operations that represents repeated addition. When simplifying expressions, multiplication is a key step. Often, it involves working with both numerical coefficients and variables. In our example, we encountered the expression \(6 \cdot (y \cdot 2)\). Through multiplication, we firstly isolated and calculated the numerical part: \(6 \times 2 = 12\). Then, by multiplying the result with the variable part \(y\), we derived the final form \(12y\). When multiplying, it is important to:

• Follow the order of operations to avoid errors.
• Apply properties like associative property to simplify.
• Express numeric results clearly in combination with any variables.

Remember, thorough understanding and careful calculation are crucial to mastering multiplication in algebraic expressions.