Problem 11 - Prealgebra

Question

Apply the associative property to expression, and then simplify the result. \(6(-10 y)\)

Step-by-Step Solution

Answer

The expression simplifies to \(-60y\) using the associative property.

Step 1: Understand the Expression

The given expression is \(6(-10y)\). The expression consists of multiplying 6 by \(-10y\).

Step 2: Apply Associative Property

The associative property of multiplication states that the way in which numbers are grouped in a multiplication problem does not change the product. For example, \((a \cdot b) \cdot c = a \cdot (b \cdot c)\). In the expression \(6(-10y)\), you can group numbers and variables as: \((6 \cdot -10) \cdot y\).

Step 3: Perform the Multiplication

Now calculate \(6 \cdot -10\):\[6 \cdot -10 = -60\]After applying the associative property, the expression becomes \(-60y\).

Step 4: Simplify the Expression

The result from Step 3, \(-60y\), is already in its simplest form.

Key Concepts

Multiplication

Multiplication is one of the fundamental operations in mathematics. When we talk about multiplication, we mean adding a number to itself a specific number of times. For example, if you multiply 3 by 4, this means you have four 3's added together: 3 + 3 + 3 + 3 = 12.
In algebra, multiplication involves numbers, variables, or both. Let's consider the expression \(6(-10y)\). Here, 6 is multiplied by \(-10y\), which means 6 is to be multiplied by both \(-10\) and \(y\) at the same time.

• The process remains similar: grouping numbers for simpler calculations.
• Remember that negative numbers cause changes in sign; thus, 6 times \(-10\) results in \(-60\).

By understanding how multiplication works in expressions, especially how negative signs operate, you can simplify complex algebraic terms easily.

Simplifying Expressions

Simplifying expressions is a crucial skill in learning algebra. It involves reducing a mathematical expression to its simplest form. This means performing all possible operations until you can no longer simplify the expression further.
For the expression \(6(-10y)\), you perform multiplication first. As mentioned earlier, multiplying 6 by \(-10\) results in \(-60\), then you multiply \(-60\) by \(y\), which doesn’t change the expression further, resulting in \(-60y\).

• The expression \(-60y\) is the simplest form as there are no more operations to conduct and no like terms to combine.
• Always ensure no common factors remain, other than 1.

Simplifying ensures that expressions are easier to read, interpret, and solve, making complex problems more manageable.

Prealgebra Concepts

Prealgebra serves as the foundational stage for algebra, focusing on basic mathematical concepts. These concepts are essential stepping stones for understanding more complex mathematics.
One critical concept in prealgebra is the associative property of multiplication, which this exercise highlights. This property tells us that the grouping of numbers in multiplication doesn’t matter — the product remains the same.

• Consider the expression \((a \cdot b) \cdot c = a \cdot (b \cdot c)\). No matter how numbers or variables are grouped, the product will always be the same.
• This property allows us to rearrange terms for easier computation and simplification. It’s a handy tool when faced with complex expressions that need simplifying or when calculating without a calculator.

Mastering these elementary concepts helps build confidence and skill in tackling more advanced algebraic problems as students progress in their mathematical journey.