Problem 57 - Algebra for College Students

Question

Simplify each numerical expression. $$16-18+19-[14-22-(31-41)]$$

Step-by-Step Solution

Answer

The simplified expression is 15.

Step 1: Simplify Inner Parentheses

Start by simplifying the expression inside the innermost parentheses: \(31 - 41\). This simplifies to \(-10\).

Step 2: Simplify Middle Parentheses

Substitute \(-10\) back into the middle parentheses: \(14 - 22 - (-10)\). The expression \(-(-10)\) becomes \(+10\), so the expression now becomes \(14 - 22 + 10\).

Step 3: Simplify Middle Parentheses Further

Calculate \(14 - 22 + 10\) by first simplifying \(14 - 22 = -8\), then \(-8 + 10 = 2\). So, the expression in the brackets is \(2\).

Step 4: Replace Bracket with Simplified Value

Replace the entire bracket with its simplified value: \(19 - 2 = 17\). Now the expression is \(16 - 18 + 17\).

Step 5: Simplify Remaining Expression

Simplify the remaining expression: \(16 - 18 + 17\). First, calculate \(16 - 18 = -2\), then \(-2 + 17 = 15\).

Key Concepts

Order of Operations

When you're working with mathematical expressions, a consistent set of rules known as the Order of Operations guides you. This ensures you get the right answer every time. Mathematically, this is often remembered by the acronym PEMDAS:

• **P**arentheses
• **E**xponents
• **M**ultiplication and **D**ivision (from left to right)
• **A**ddition and **S**ubtraction (from left to right)

In our original problem, everything centers around following these steps in order. Start by solving the Parentheses first, which means diving into those nested brackets and the operations within them. Get each part done meticulously to avoid errors. Remember, Addition and Subtraction should be tackled from left to right at the very end, after handling all Parentheses, Exponents, and any Multiplication or Division. By following the correct order, the complexity of the expression becomes manageable and clear.

Parentheses in Mathematics

Parentheses are like a traffic signal for math problems. They tell you what needs your attention first. In mathematical problems, solving what's inside parentheses takes priority. Take our original expression as an example. Notice how solving inside the innermost set of parentheses \(31 - 41\) played a crucial role. You solve this piece first, making it \(-10\). This change allowed us to focus on the next set of operations. Following this, we plugged \(-10\) into the expression \(14 - 22 - (-10)\). Handling the negative sign is part of understanding Parentheses, also known as simplifying. This gradual peeling away of each layer ensures that when you reach the outermost operations, you have the simplest numbers to work with. Parentheses help maintain clarity in complex expressions and make sure no step is skipped.

Combining Like Terms

Combining like terms is the process of adding or subtracting terms that have the same variables and powers. In numeric expressions, this involves straightforward arithmetic operations where you combine numbers. In our expression, after simplifying within and between the parentheses, we had the expression \(16 - 18 + 17\). Here, 'like terms' means constant terms or numbers we can directly operate on.

• First, do the subtraction: Calculate \(16 - 18\) which equals \(-2\).
• Then, perform the addition: Combine \(-2\) with \(+17\) to get the final result of \(15\).

Each term is a part of the equation that directly interacts with the others. Simplifying requires the correct sequence of math operations as well as the cleaning up of unnecessary elements, ensuring that the solution is as simple as possible.