Problem 65 - Elementary Algebra

Question

Simplify the following problems using the commutative property of multiplication. You need not use the distributive property. $$ 16 a b 2 c $$

Step-by-Step Solution

Answer

Answer: $32abc$

Step 1: Identify the commutative property of multiplication

The commutative property of multiplication states that for any two numbers, say A and B, we have the following relationship: $$ A*B = B*A $$ In other words, the order of the factors doesn't affect the result of the multiplication.

Step 2: Apply the commutative property of multiplication to the given expression

Given the expression: $$ 16ab2c $$ We can rearrange the factors using the commutative property of multiplication: $$ 16 \cdot a \cdot b \cdot 2 \cdot c $$

Step 3: Simplify the expression

Now, rearrange the factors to group the constants (numbers) together and the variables together: $$ 16 \cdot 2 \cdot a \cdot b \cdot c $$ Finally, multiply the constants and rewrite the expression: $$ 32abc $$ The simplified expression using the commutative property of multiplication is: $$ 32abc $$

Key Concepts

Simplifying Expressions

Simplifying expressions in algebra involves reducing them to their simplest form. This means combining like terms and using mathematical properties to make expressions easier to work with. In the given exercise, we started with the expression \(16ab2c\). The goal was to simplify this into a more manageable form using properties of multiplication. By rearranging and grouping the numbers and variables together, we made it a single compact expression. - Simplifying makes the expression clearer and helps when you need to evaluate or further manipulate it. - It reduces complexity and makes it easier to compare with other expressions.By understanding how to identify like terms and using properties like the commutative property, expressions can become straightforward and less intimidating.

Algebraic Terms

Algebraic terms are the building blocks of algebraic expressions. They consist of numbers, variables, and coefficients. In our exercise, the terms included in the given expression are constants (like 16 and 2) and variables (like \(a, b, \) and \(c\)). - **Variables** represent unknown values and are typically denoted by letters such as \(a, b, c, x, y,\) etc.- **Coefficients** are numbers that multiply the variables. In the expression \(16ab2c\), the coefficient 16 multiplies the variables \(a, b,\) and \(c\).- **Constants** are fixed values that are not multiplied by variables. These are sometimes seen adding or subtracting from terms in expressions.Understanding algebraic terms and their roles allow us to manipulate expressions correctly. By knowing terms and their properties, simplifying becomes a logical and simple process.

Multiplication Properties

Multiplication properties in mathematics simplify working with expressions and numbers. A key property is the commutative property. This states that you can change the order of factors without changing the product. For example, in our task, we rearranged \(16ab2c\) as \(16 \cdot 2 \cdot a \cdot b \cdot c\).**Key Multiplication Properties:**- **Commutative Property:** \(a \times b = b \times a\). The order of multiplication doesn't affect the result. This property makes rearranging terms straightforward.- **Associative Property:** \((a \times b) \times c = a \times (b \times c)\). The way numbers are grouped in multiplication does not change the outcome.- **Identity Property:** There is a multiplicative identity, which is 1; multiplying any number by 1 leaves it unchanged. Using these properties simplifies expressions and calculations. Understanding these foundational properties enhances mathematical insight and problem-solving efficiency.