Problem 41 - Prealgebra

Question

Simplify each expression. $$21+(k+16)$$

Step-by-Step Solution

Answer

The simplified expression is \(37 + k\).

Step 1: Identify the expression parts

The expression given is \(21+(k+16)\). This expression is composed of three parts: the number 21, the variable term \(k\), and the constant 16 within the parentheses.

Step 2: Eliminate the parentheses

Since there is no operation inside the parentheses that needs to be performed first (according to the order of operations), and since all terms within the parentheses are added, we can simply remove the parentheses. The expression then becomes \(21 + k + 16\).

Step 3: Combine like terms

To simplify the expression further, we combine like terms. The constants 21 and 16 are like terms because they are both real numbers without variables. Their sum is \(37\). Thus, the simplified expression is \(37 + k\).

Key Concepts

Simplifying Expressions

Simplifying expressions means rewriting them in a more concise and comprehensible form without changing their value. It involves performing operations and making the expression easier to work with. In our example, you'll notice we started with the expression \[21 + (k + 16\] Given the lack of operations inside the parentheses, the next step was straightforward. By eliminating the parentheses, we simplified it to \(21 + k + 16\), making it easier to identify and combine like terms. Aim to express the mathematical relationship as clearly as possible.

Order of Operations

The order of operations is a collection of rules that prioritize the steps to be taken when evaluating a mathematical expression. You can remember these rules using the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

• "Parentheses" indicate that operations enclosed within brackets should be performed first.
• "Exponents" mean you handle powers and roots second.
• "Multiplication and Division" are performed next, as they appear from left to right.
• Finally, "Addition and Subtraction" are handled in order from left to right.

In our problem, the expression was simple as \(21 + (k + 16)\), with the only operations being addition. The parentheses didn't alter the sequence as there were no nested operations within them. Thus, removing the parentheses was the first and simple step.

Combining Like Terms

Combining like terms is an essential aspect of simplifying expressions. It involves merging terms that have identical variables raised to the same power or constants, meaning terms that are fundamentally similar. In the expression \(21 + k + 16\), 21 and 16 are constants, also known as coefficients, without any variable attached to them. These can easily be combined to form a single term \(37\), thereby making the expression simpler: \(37 + k\).Always group together the constants separately from any terms with variables. This method ensures that the expression is reduced to its simplest form, making it far more manageable and easy to interpret.