Problem 46 - Prealgebra

Question

Simplify each expression. $$(t+4) 3$$

Step-by-Step Solution

Answer

The simplified expression is \(3t + 12\).

Step 1: Distribute the Constant

To simplify the expression \[(t + 4) 3\],you need to apply the distributive property, which states that for any numbers \(a\), \(b\), and \(c\), \((a + b) c = ac + bc\). Apply this to the expression as follows:Multiply \(t\) by 3 and then multiply 4 by 3.

Step 2: Multiply Each Term

First, multiply \(t\) by 3.\[3 imes t = 3t\]Next, multiply 4 by 3.\[3 imes 4 = 12\]

Step 3: Combine the Results

Now combine the results from Step 2:\[3t + 12\]This is your simplified expression.

Key Concepts

Simplification

Simplification is a fundamental concept in algebra where the goal is to make an expression easier to understand or work with. By breaking down or reducing the terms in an equation, you can achieve a more concise and often more manageable form.

It involves considering all parts of the expression carefully and finding ways to condense it using mathematical rules or properties. When simplifying, you often use one or more of the basic operations—such as addition, subtraction, multiplication, or division. This process can involve combining like terms or applying mathematical properties like the distributive property to break down expressions with parentheses.

• Helps in solving equations faster by dealing with simpler expressions.
• Keeps expressions neat and optimized for easier understanding.
• Reduces the possibility of making arithmetic errors, as fewer steps mean fewer chances for mistakes.

By paying attention to simplification, you build a stronger foundation to understand more complex algebraic equations down the road.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. Unlike equations, expressions do not have an equals sign. They can often be viewed as phrases within mathematical language, helping to convey specific ideas or actions.

• Variables are symbols such as \(t\) that represent unknown quantities.
• Constants, like the number 4 in the given expression, are fixed values.
• Operators denote the operations being performed, for example, addition or multiplication.

The beauty of algebraic expressions lies in their versatility; they can represent real-world situations, model problems across various fields, and solve unknowns through manipulation of their terms. Expressions can be simplified or manipulated using different rules such as distributing multiplication across addition, which is the case for our expression \((t + 4) 3\). This example demonstrates how to express the same relationship in a simpler manner, without altering the context or value.

Prealgebra Concepts

Prealgebra serves as the building block for more advanced algebra concepts. It encompasses various essential arithmetic and basic algebra topics, like understanding numbers, arithmetic operations, and properties of operations such as the distributive property.

The main aim of prealgebra is to transition students smoothly from arithmetic into the more abstract concepts found in algebra. It includes guiding principles related to variables and expressions.

• Variables act as placeholders and help in formulating real-world or theoretical problems mathematically.
• Simplifying expressions by applying arithmetic properties sets the stage for deeper algebraic learning.
• Understanding orders of operations ensures consistency and correctness when handling complex expressions.

Prealgebra concepts play a crucial role in ensuring a strong mathematical foundation, fostering confidence in dealing with more complex algebraic problems in the future. By mastering prealgebra, students develop critical logical reasoning and problem-solving skills necessary for success in subsequent math courses.