Problem 38 - Prealgebra

Question

Simplify each expression. $$(4 \cdot p) \cdot 6$$

Step-by-Step Solution

Answer

The simplified expression is \(24p\).

Step 1: Understand the expression

The expression given is \((4 \cdot p) \cdot 6\). It involves multiplying three terms together: the constant 4, the variable \(p\), and the constant 6.

Step 2: Apply the Associative Property of Multiplication

The Associative Property states that when three or more numbers are multiplied, the way they are grouped does not affect the product. Therefore, we can rewrite \((4 \cdot p) \cdot 6\) as \(4 \cdot (p \cdot 6)\).

Step 3: Multiply the Constants

We can reorder and multiply the constants together first: \(4 \cdot 6 = 24\). Now the expression simplifies to \(24 \cdot p\).

Step 4: Write the Final Simplified Expression

The simplified expression is \(24p\). Write this as the product of the simplified constant and the variable.

Key Concepts

Simplifying Expressions

Simplifying expressions is a crucial skill in algebra. It involves reducing expressions to their simplest form, making them easier to handle and understand. When simplifying, the goal is to create a new representation of an algebraic expression that retains the original value.

• Focus on reducing terms by combining like terms or simplifying operations within the expression.
• Use properties of arithmetic, such as the distributive, associative, and commutative properties, to rearrange and combine elements effectively.

Simplifying the expression \((4 \cdot p) \cdot 6\) involves using the associative property that allows us to group and multiply parts of the expression in a different order. This approach reduces the expression to a simpler and more manageable form.

Multiplication of Constants

When you encounter constants in an expression, multiplication is straightforward. Numbers with no variables involved are constants, and when multiplying them, it results in a single constant product.
In our example \((4 \cdot p) \cdot 6\), constants are 4 and 6. Here, we can multiply these constants together first, aligning with the associative property, to make the calculation easier.

• By multiplying \(4 \cdot 6\), we get 24.

After handling constants, we then focus on the remaining variable part of the expression. This approach provides a neat and tidy expression \(24p\) by the end of the process.

Algebraic Expressions

Algebraic expressions are combinations of variables, constants, and arithmetic operations. Variables represent unknown values, and along with constants, they form the fundamental parts of algebraic expressions.

• In our expression, \(p\) is a variable, and it interacts with constants 4 and 6 through multiplication.
• Algebraic expressions can be simplified by following specific arithmetic rules and properties.

Understanding how variables and constants coexist within an expression helps in identifying the appropriate operations necessary to simplify it. Mastering this makes dealing with more complex equations more intuitive and effective.