Problem 58 - Algebra for College Students

Question

Simplify each numerical expression. $$-19-[15-13-(-12+8)]$$

Step-by-Step Solution

Answer

The simplified expression is -25.

Step 1: Simplify the Innermost Parentheses

Look at the expression inside the innermost parentheses: \(-19 - [15 - 13 - (-12 + 8)]\).Simplify \(-12 + 8\) to get \(-4\).Substitute back into the expression:\(-19 - [15 - 13 - (-4)]\).

Step 2: Simplify the Brackets

Now simplify the expression inside the brackets:\(15 - 13 - (-4)\).First, simplify \(-(-4)\) to \(+4\), making the expression \(15 - 13 + 4\).

Step 3: Perform Addition/Subtraction Inside the Brackets

Simplify the expression \(15 - 13 + 4\):First, perform \(15 - 13 = 2\), then add \(4\) to get \(2 + 4 = 6\).So the expression inside the brackets is \(6\).

Step 4: Final Simplification

Now substitute back into the full expression:\(-19 - [6]\).Simplify this to get:\(-19 - 6 = -25\).

Key Concepts

Simplification

Simplification is the process of making an algebraic or numerical expression easier to work with. It often involves reducing the number of operations by combining like terms and following the rules of arithmetic. In the context of the given exercise, simplification involves breaking down the expression to find its simplest form.

• First, identify parts of the expression that can be simplified, such as terms inside parentheses.
• Next, reduce these components step by step, solving one piece at a time.
• Finally, continue simplifying until you reach the simplest possible version of the expression.

Taking these steps not only helps in finding the solution more quickly but also ensures accuracy in complex calculations.

Numerical Expressions

A numerical expression is a mathematical phrase involving numbers and operational symbols (such as addition, subtraction, multiplication, and division). Numerical expressions do not include any variable and are solved to find a single number result.

• The expression from the exercise is a good example:

    -19 - [15 - 13 - (-12 + 8)]
  

  Every part of it consists of numbers and basic operations.
• They require calculating each part in sequence, respecting the hierarchy of operations.
• The goal is to perform the necessary arithmetic operations to simplify the expression to a single number.

Working with numerical expressions demands attention to detail, especially when dealing with multiple terms and operations.

Order of Operations

The order of operations is crucial in simplifying and evaluating mathematical expressions correctly. In mathematics, there is a universally accepted sequence of operations that must be followed, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

• Start with calculations inside parentheses or brackets.
• Next, handle any exponents (though not present in this exercise).
• Perform multiplication and division as they appear from left to right.
• Finally, execute addition and subtraction from left to right.

In the given exercise, the steps followed this exact order. First, simplifying within the innermost parentheses:

-12 + 8 = -4.

Then handle the subtraction and addition within the brackets. Finally, simplify the entire expression by performing the subtraction.
Following the order of operations ensures you simplify expressions correctly and arrive at the correct solution.