Problem 25 - Algebra 1: Concepts and Skills

Question

Use the distributive property to rewrite the expression without parentheses. $$3(4+6 a)$$

Step-by-Step Solution

Answer

The expression \(3(4+6a)\) distributed without parentheses is \(12 + 18a\).

Step 1: Identify the Terms to Distribute

The distributive property involves taking the number or variable that is outside the parentheses and multiplying it to each term inside the parentheses. In this case, that's the number 3 and it will be distributed to both 4 and \(6a\).

Step 2: Distribute to the First Term

First, distribute the 3 to the 4. To do this, multiply the 3 and 4 together. \(3*4 = 12\).

Step 3: Distribute to the Second Term

Next, distribute the 3 the \(6a\). To do this, multiply the 3 and \(6a\) together. \(3*6a = 18a\).

Step 4: Write the Final Answer

The new, distributed expression is found by putting together the results from steps 2 and 3. So, the expression \(3(4+6a)\) without parentheses using the distributive property is \(12 + 18a\).

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operational symbols. An algebraic expression does not have an equal sign, which means it's not an equation. For example, the expression \(3(4 + 6a)\) consists of the number 3 outside the parentheses and the terms 4 and \(6a\) inside the parentheses. Here, '4' is a constant, whereas \(6a\) is a term with a variable. The variable 'a' stands for an unknown value that can change.
To simplify algebraic expressions, you often need to apply various mathematical operations, such as addition, subtraction, and multiplication. The goal is to make the expression easier to work with, often by removing parentheses using properties like the distributive property.
It's crucial to understand these components first before performing operations on them. Mastery of this basic concept sets you up for more complex algebraic challenges.

Multiplication

Multiplication is a mathematical operation that combines groups of equal sizes. It is one of the four basic arithmetic operations and is foundational in manipulating algebraic expressions.
In the expression \(3(4+6a)\), multiplication plays a key role. The distributive property allows us to multiply the 3 across each term within the parentheses. This means multiplying the number 3 by both 4 and \(6a\):

• Multiply 3 by 4 to get 12.
• Multiply 3 by \(6a\) to get \(18a\).

Understanding multiplication also requires recognizing how it interacts with other operations and aspects of numbers, such as associative and commutative properties. These properties ensure that the order and grouping of numbers don't affect the final result of multiplication.
By mastering multiplication, you lay the foundation for more advanced algebraic manipulations.

Mathematical Operations

Mathematical operations include addition, subtraction, multiplication, and division. These operations are the building blocks of mathematics and are crucial for solving equations and simplifying expressions.
In the given exercise, the focus is on using multiplication within the distributive property. Let's highlight a few key parts of mathematical operations applied here:

• **Distributive Property:** This property allows us to multiply a single term by each term inside a set of parentheses. For \(3(4+6a)\), it means \(3\times 4\) and \(3\times 6a\).
• **Combining like terms:** After distribution, we add or incorporate these results to write the expression without parentheses or in a simpler form. For example, the distributed terms \(12\) and \(18a\) are combined to form \(12 + 18a\).

Understanding these operations helps in efficiently managing and simplifying algebraic expressions, leading to quicker and more accurate solutions. Getting comfortable with these basics is vital for advancing into higher-level mathematics.