Problem 40 - Prealgebra

Question

Simplify each expression. $$(17+p)+9$$

Step-by-Step Solution

Answer

The simplified expression is \(p + 26\).

Step 1: Identify the Expression

The given expression is \((17 + p) + 9\). Our task is to simplify it by combining like terms if possible.

Step 2: Apply Associative Property

The associative property of addition states that the way in which numbers are grouped does not affect the sum. Therefore, we can rewrite the expression as \(17 + (p + 9)\).

Step 3: Simplify by Combining Like Terms

In the expression \(17 + (p + 9)\), the number terms can be combined. Add 17 and 9 to get 26, yielding the simplified expression \(p + 26\).

Key Concepts

Associative Property of Addition

The Associative Property of Addition is an essential concept in prealgebra. It tells us that the grouping of numbers in an addition operation does not impact the final result. This means we can change the grouping of terms and still end up with the same sum. For example:

• When we have \[(a + b) + c\] it's the same as \[a + (b + c)\].
• Even if the numbers and variables are different, the property still holds true, for example:\[(17 + p) + 9 = 17 + (p + 9)\]. The parentheses can shift around without changing the sum.

Understanding how this property works helps in rearranging the numbers in a way that makes the calculation easier. For our example, by grouping 17 and 9 closer together, we can effortlessly add them first, simplifying our expression.

Simplifying Expressions

Simplifying expressions involves making them easier to read or solve without changing their value. It means reducing the complexity of an expression while keeping its equivalence. When simplifying expressions, we aim to:

• Clear any unnecessary parentheses.
• Combine terms where possible to reduce the number of separate operations.

In the expression \((17 + p) + 9\), by using associative property, we shifted the parenthesis to obtain \(17 + (p + 9)\). By simplifying further, unnecessary parentheses can be removed, leaving us with
\(p + 26\), a neatly simplified expression.
This process ensures that our final expression is as clear and concise as possible for further calculations or understandings.

Combining Like Terms

Combining like terms is a foundational skill in simplifying algebraic expressions. A 'like term' shares the same variable and exponent. For instance, \(3x\) and \(4x\) are like terms because they both contain 'x'. They can be added or subtracted effectively.

• In our expression analysis, we seek terms without variables to pair and simplify them.
• With our expression \(17 + (p + 9)\), the numbers 17 and 9 are like terms. They are constants, which means they can be combined to form a new numeric value.

Adding these, we obtain 26, thus producing a refined expression \(p + 26\).This step significantly simplifies the expression, making it easier to handle or solve in algebraic problems.