Problem 11 - Prealgebra

Question

Simplify expression. \(8 e-4(2 f+5 e)\)

Step-by-Step Solution

Answer

The simplified expression is \[-12e - 8f\].

Step 1: Expand the Expression

Begin by distributing the -4 across the terms inside the parentheses in the expression: \[8e - 4(2f + 5e)\]This results in:\[8e - 4 imes 2f - 4 imes 5e\]Simplifying this results in:\[8e - 8f - 20e\]

Step 2: Combine Like Terms

Now, combine the like terms in the expression. The like terms here are the terms with 'e'. Thus, you combine \(8e\) and \(-20e\):\[(8e - 20e) - 8f\]This simplifies to:\[-12e - 8f\]

Key Concepts

Distributive Property

In mathematics, the distributive property is a useful tool that helps us to simplify expressions by removing parentheses. It involves multiplying a number or variable outside the parentheses by each term inside the parentheses. When you see an expression like \(a(b + c)\), the distributive property allows you to transform it into \(ab + ac\).
In our specific exercise, we apply the distributive property to \(-4(2f + 5e)\). This means that \(-4\) must be multiplied by both \(2f\) and \(5e\).

• Multiply \(-4\) by \(2f\) to get \(-8f\).
• Multiply \(-4\) by \(5e\) to get \(-20e\).

When you perform these operations, you end up with the expanded expression: \(8e - 8f - 20e\). The distributive property is particularly helpful in prealgebra for breaking down complex expressions into manageable parts. In essence, it distributes the operation evenly across the terms within the parentheses.

Combining Like Terms

Once you have applied the distributive property, the next step is to combine like terms in the expression. Like terms are terms within an expression that have the same variable raised to the same power. In simpler terms, these are terms that "look alike" or "belong" together because they share the same letter and exponent.
In the expression \(8e - 8f - 20e\), the like terms are \(8e\) and \(-20e\). They both contain the variable \(e\). Combining like terms involves adding or subtracting these coefficients. For \(8e - 20e\), you:

• Subtract 20 from 8, resulting in \(-12e\).
• The term \(-8f\) remains unchanged because there are no other terms with \(f\) to combine with.

By combining like terms, you effectively simplify the expression further, managing to reduce its complexity. This results in a cleaner and easier-to-read expression: \(-12e - 8f\). Combining like terms streamlines expressions and is a crucial skill in solving algebraic equations.

Simplification

Simplification in math refers to the process of reducing expressions to their most basic form. It involves using various algebraic techniques like the distributive property and combining like terms. Simplifying an expression makes it easier to interpret and work with, especially in solving equations. This process aims to create an equivalent expression that has fewer terms and is devoid of any extraneous components.
In our example expression, \(8e - 4(2f + 5e)\), simplification involved two major steps:

• First, applying the distributive property to eliminate the parentheses.
• Second, combining like terms \(8e\) and \(-20e\) to reduce the number of terms.

The result of these simplifications is the expression \(-12e - 8f\). By simplifying, we reduce the expression to its core components, making it much easier to understand and use in further calculations or applications.
Simplification is a fundamental skill in algebra that keeps equations orderly and manageable, allowing for a clearer analysis or resolution of mathematical problems.