Problem 37 - Prealgebra

Question

Simplify each expression. $$16+(7+d)$$

Step-by-Step Solution

Answer

The simplified expression is \\(23 + d\\).

Step 1: Understand Grouping

Identify the grouping present in the expression. Here, the parentheses group the expression \((7+d)\), indicating that operations inside should be considered together. However, since there are no operations here other than addition, we can remove the parentheses.

Step 2: Use the Associative Property of Addition

The associative property allows us to change the grouping of numbers being added without changing the result. We can rewrite \(16 + (7 + d)\) as \(16 + 7 + d\).

Step 3: Simplify

Perform the operation between the numbers: \(16 + 7 = 23\). Combine this with the remaining part of the expression: \(23 + d\).

Key Concepts

Associative Property of Addition

The associative property of addition is a fundamental concept in mathematics. It states that when you add numbers, no matter how you group them, the result will be the same. This property can be especially useful when simplifying expressions.
For example, let's consider the expression \(16 + (7 + d)\). When using the associative property, we can rearrange it to look like \((16 + 7) + d\). Here, we see that the grouping changes from \((7 + d)\) to \((16 + 7)\), but the result remains unchanged.

• The associative property does not involve the numbers themselves changing, only the grouping.
• It simplifies arithmetic expressions by allowing computations to be carried out in different orders.
• This property only applies to addition and multiplication, not to subtraction or division.

Understanding this property is crucial for simplifying mathematical expressions, especially when combined with other properties.

Grouping Symbols

Grouping symbols such as parentheses \(()\), brackets \([]\), and braces \({}\) play a significant role in mathematics. They specify the order in which operations are performed and help clarify the structure of expressions.
In the expression \(16 + (7 + d)\), the parentheses around \((7 + d)\) indicate that these terms should be considered together. Since addition is taking place inside the parentheses, the associative property allows us to rearrange without changing the outcome, meaning the parentheses can be removed.

• Grouping symbols help manage the operations inside them first, following the order of operations.
• They enhance understanding of complex expressions by clearly showing which components are related.
• Removing or changing grouping symbols must be done carefully to maintain the integrity of the expression.

Understanding how to work with grouping symbols and when they can be removed is a key skill in simplifying expressions.

Prealgebra

Prealgebra is a foundational level of mathematics that prepares students for algebra. It encompasses a range of basic math concepts including addition, subtraction, multiplication, division, fractions, and more. One of the essential parts of prealgebra is the simplification of expressions.
This involves understanding properties like the associative property of addition and how to correctly use grouping symbols. Simplifying expressions, as in the exercise \(16 + (7 + d)\), is a common prealgebra task that builds necessary skills for more advanced math.

• Prealgebra focuses on strengthening arithmetic skills and introducing basic algebraic concepts.
• It helps students develop problem-solving strategies and mathematical reasoning.
• Mastering prealgebra concepts is critical for success in algebra and higher-level mathematics.

By practicing with prealgebra exercises, students gain confidence in dealing with numbers and variables, laying the groundwork for future learning in mathematics.