Problem 44 - Prealgebra

Question

Apply the distributive property to expression, and then simplify. \(8(6-3 y)\)

Step-by-Step Solution

Answer

The simplified expression is \(48 - 24y\).

Step 1: Apply the Distributive Property

The distributive property allows us to multiply a single term by each term inside the parentheses. Apply it to the expression \(8(6-3y)\) by multiplying 8 by each term inside the parentheses. This gives: \(8 \times 6 - 8 \times 3y\).

Step 2: Perform the Multiplication

Carry out the multiplication for each term separately. Multiply 8 by 6 to get 48, and 8 by 3y to get 24y. The result of this step is: \(48 - 24y\).

Step 3: Simplify the Expression

Re-write the expression in a simplified form, which in this case, is just the expression without any like terms to combine. The expression \(48 - 24y\) is already simplified because there are no like terms.

Key Concepts

Simplifying Expressions

Simplifying expressions is a key aspect of algebra that helps to make equations more manageable and easier to understand. When we simplify an expression, we combine like terms and perform operations to reduce the expression to its shortest form. This process does not change the value of the expression but presents it in a way that is easier to work with.

In the original exercise, the expression given is \(8(6 - 3y)\). Once we apply the distributive property and multiplication, we arrive at \(48 - 24y\). In this expression, there are no like terms to combine because 48 is a constant and \(-24y\) is a variable term. Thus, the expression is in its simplest form.

To simplify effectively:

• Identify like terms: These are terms that have the same variables raised to the same power.
• Combine like terms: Add or subtract the coefficients of these terms.
• Perform operations: Simplify by carrying out any multiplication or division.

Simplifying makes it easier to evaluate or solve equations in further steps.

Multiplication

Multiplication is a fundamental operation in mathematics that combines groups of numbers. In the context of algebra, multiplication can also involve distributing a number over terms in parentheses.

For the expression \(8(6 - 3y)\), we use multiplication to apply the distributive property. We multiply 8 by each term inside the parentheses:

* Multiply 8 by 6: \(8 \times 6 = 48\)
* Multiply 8 by \(-3y\): \(8 \times -3y = -24y\)

This results in the expression \(48 - 24y\). Understanding how to perform multiplication, especially with variables and constants, is essential for more advanced math topics. It allows us to simplify expressions and solve equations as well.

Prealgebra Concepts

Prealgebra lays the groundwork for algebra by introducing essential concepts such as variables, operations, and properties. One of the cornerstones of prealgebra is understanding how to manipulate and simplify expressions using properties such as the distributive property.

In the original exercise, we start with an expression \(8(6 - 3y)\). Key prealgebra concepts here involve:

• Using the distributive property: This property states that \(a(b + c) = ab + ac\), allowing us to simplify expressions.
• Understanding variables: Recognizing that 'y' is a placeholder for numbers and operates under the same mathematical principles as numbers.
• Simplifying expressions: This includes combining like terms and performing arithmetic operations.

Grasping these prealgebra concepts ensures a smoother transition into algebra, where these principles are applied in more complex situations.