Problem 110 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. $$ 70 a \cdot 10 $$

Step-by-Step Solution

Answer

The simplified expression is \( 700a \).

Step 1: Identify the Multiplicands

The expression given is \( 70a \cdot 10 \). Here, we have two multiplicands: \( 70a \) and \( 10 \).

Step 2: Rearrange the Expression

We can rearrange the expression using the associative property of multiplication: \( 70a \cdot 10 = 70 \cdot a \cdot 10 \).

Step 3: Simplify the Numerical Multiplication

First, multiply the numerical values, \( 70 \) and \( 10 \): \[ 70 \cdot 10 = 700 \]

Step 4: Combine the Results

Substitute the result of the numerical multiplication back into the expression to give:\( 700 \cdot a \).

Step 5: Write the Simplified Expression

The fully simplified expression is \( 700a \).

Key Concepts

Associative Property

The associative property is a fundamental rule in mathematics that makes computations easier and more flexible. It allows us to group numbers and variables in a multiplication expression without affecting the result.
This property is defined for both addition and multiplication. For multiplication, it implies that when three or more numbers are multiplied, the grouping of numbers can be changed without changing the product.
For example, in the expression \( (a \cdot b) \cdot c = a \cdot (b \cdot c) \), the product remains the same regardless of how the multiplicands are grouped.
Here's how the associative property works in our original problem:

• We started with the expression \( 70a \cdot 10 \).
• Using the associative property, we rearranged it to \( 70 \cdot a \cdot 10 \).
• This helped in focusing on numerical multiplication first, making simplification straightforward.

Understanding this property helps us not only to simplify expressions by rearranging them for easier calculations but also to structure complex problems more effectively.

Multiplicands

Multiplicands are the quantities being multiplied together in a multiplication problem. Identifying them correctly is crucial for solving any multiplication expression.
In algebra, these multiplicands could be numbers, variables, or a combination of both, like \( 70a \) and \( 10 \) in our example.
When working with expressions:

• First, identify all parts involved in the multiplication.
• Group numbers and variables as separate multiplicands if possible.
• This makes subsequent steps, like applying the associative property or simplifying, easier.

In our original exercise, realizing that \( 70a \) and \( 10 \) are multiplicands helped us navigate through the steps, using properties effectively to find a simplified solution.

Numerical Multiplication

Numerical multiplication involves multiplying the numbers within a given expression. This is a key step for simplification, especially when variables are involved.
In our original exercise, the numerical values \( 70 \) and \( 10 \) are recognized and multiplied first.
Here's how it simplifies the process:

• Isolate numerical values, as we did by rearranging to \( 70 \cdot 10 \cdot a \).
• Simplify by multiplying the numbers: \( 70 \cdot 10 = 700 \).
• Reinsert back into the expression with the variable, leading to \( 700a \).

By focusing on numerical values first, we ensure that each part of the expression is simplified and manageable.
Numerical multiplication often determines the first major step in algebraic simplification, making it a crucial skill.