Problem 62 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The simplified expression is -25.

Step 1: Simplify Inside the Innermost Parentheses

Begin by simplifying the expression inside the innermost parentheses: \(-12 + 8 = -4\).Now the expression is: \(-19 - [15 - 13 - (-4)]\).

Step 2: Simplify Inside the Brackets

Next, simplify the expression inside the brackets: First, evaluate the subtraction: \(15 - 13 = 2\).Replace the expression inside the brackets: \(-19 - [2 - (-4)]\).

Step 3: Evaluate the Double Negative

Now simplify the expression by evaluating the double negative:\(2 - (-4) = 2 + 4 = 6\).So, the expression is: \(-19 - 6\).

Step 4: Final Simplification

Finally, subtract to get the numerical answer:\(-19 - 6 = -25\).

Key Concepts

order of operations

Understanding the order of operations is crucial for simplifying mathematical expressions correctly. This rule is commonly remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. Each of these operations should be handled in this specific order to avoid mistakes in calculations.

Here's how it works:

• Parentheses – Always perform computations inside parentheses or brackets first.
• Exponents – Solve any exponents or powers after dealing with parentheses.
• Multiplication and Division – These are prioritized equally and should be evaluated from left to right as they appear in the expression.
• Addition and Subtraction – Like multiplication and division, handle these operations from left to right.

In the example given, the expression was tackled by addressing the operations within the parentheses first. This ensures clarity and precision in deriving the final result.

parentheses and brackets in expressions

Working with parentheses and brackets can sometimes be confusing, but they play a crucial role in organizing operations within algebraic expressions. In mathematics, parentheses \(( )\) and brackets \([ ]\) are used to group numbers and operations to signify which part of the expression should be evaluated first.

Consider the original expression:\[-19 - [15 - 13 - (-12 + 8)]\]Here’s how to proceed:

• Innermost Parentheses: Evaluate \(-12 + 8\) because it's inside the parentheses first. This helps in reducing complexity.
• Brackets: After simplifying the inner parentheses, address the operations in the brackets \([ ]\). This keeps the operations in the correct order and ensures structured simplification.

Using brackets and parentheses effectively allows more complicated expressions to be manipulated with confidence, succeeding one simplification step at a time.

working with negative numbers

Negative numbers are often a tricky part of mathematics, but understanding how they function can be very rewarding. When working with negative numbers, addition, subtraction, and even multiplication or division rules need careful handling.

For subtraction and addition with negatives:

• When subtracting a negative, it’s like adding a positive: for instance, \(2 - (-4)\) becomes \(2 + 4\).
• When adding a negative, it’s like normal subtraction: \((-19) - 6 = -25\) because 19 is added in a decreasing manner.

In our expression, understanding the double negative rule \'\(-(-4) = +4\)\' was key in accurately simplifying \(2 - (-4)\) to find the correct answer. Mastery over these simple rules can make working with negative numbers less daunting and more intuitive.