Problem 98 - Beginning and Intermediate Algebra

Question

Perform each indicated operation. $$ -7.62-[(-3.99+1.427)-(-2.8)] $$

Step-by-Step Solution

Answer

-7.857

Step 1: Simplify the Innermost Parentheses

Start by simplifying the innermost expression inside the parentheses: (-3.99 + 1.427). Calculate -3.99 + 1.427 = -2.563

Step 2: Simplify the Outer Parentheses

Next, substitute the simplified value into the expression: o [-2.563 - (-2.8)] Calculate this expression inside the brackets: -2.563 - (-2.8) = -2.563 + 2.8 = 0.237

Step 3: Subtract from -7.62

Now substitute the simplified value back into the entire expression: -7.62 - (0.237),and calculate: -7.62 - 0.237 = -7.857

Key Concepts

Order of Operations

Mathematics often involves complex expressions needing a clear set of rules for evaluation. This is where the 'order of operations' comes into play. The standard order is often remembered using the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (left to right), and Addition and Subtraction (left to right).
Following this order ensures that everyone evaluates the expression in the same way.
In our example, the expression is as follows: $$ -7.62 - [(-3.99 + 1.427) - (-2.8)] $$
We start by solving the innermost parentheses before dealing with other operations. This structured approach prevents mistakes and guarantees a correct result.

Parentheses

Parentheses are crucial for determining the order in which mathematical operations should be performed. They allow us to group parts of an expression, overriding the default order of operations if necessary. In our exercise, parentheses guide us through a step-by-step simplification process.
The expression involves multiple layers of parentheses: $$ -7.62 - [(-3.99 + 1.427) - (-2.8)] $$
We address the innermost parentheses first: $$ (-3.99 + 1.427) $$
Solving this gives us: $$ -2.563 $$
Then, the next outer parentheses: $$ [-2.563 - (-2.8)] $$
Correctly handling parentheses ensures clarity and accuracy in complex calculations.

Negative Numbers

Negative numbers can be tricky, especially when combined with other operations. Understanding how they work is key to accurate calculations. A negative number is simply a number less than zero. When you subtract a negative number, it becomes an addition.
In the problem, we encounter negative numbers multiple times: $$ -3.99, -2.8, -7.62 $$
For instance: $$ [-2.563 - (-2.8)] $$
The double negative here means we actually add the numbers: $$ -2.563 + 2.8 = 0.237 $$
Mastering negative numbers allows for more confident and precise handling of mathematical expressions.

Subtraction

Subtraction is one of the four basic arithmetic operations. It represents the operation of removing objects from a collection. Subtraction can get a bit more complicated when dealing with negative numbers.
The initial expression involved a couple of subtractions: $$ -7.62 - [(-3.99 + 1.427) - (-2.8)] $$
Breaking it down, after simplifying the innermost parentheses, we reached: $$ -7.62 - 0.237 $$
Continuing with the subtraction: $$ -7.62 - 0.237 = -7.857 $$
Properly managing subtraction, especially with negative numbers, ensures accurate results in arithmetic operations.