Problem 70 - Prealgebra

Question

Simplify the given expression. \(17.98-(10.07-(-10.1))\)

Step-by-Step Solution

Answer

The simplified expression is \(-2.19\).

Step 1: Evaluate the Inner Parentheses

The expression inside the parentheses is \(10.07 - (-10.1)\). When you subtract a negative number, it's the same as adding the absolute value of that number. Therefore, we have \(10.07 + 10.1\).

Step 2: Calculate Inside Parentheses

Calculate \(10.07 + 10.1\). Adding these gives us \(20.17\).

Step 3: Rewrite the Expression

With the inner parentheses calculated, the expression simplifies to \(17.98 - 20.17\).

Step 4: Subtract the Numbers

Now, subtract \(20.17\) from \(17.98\). Since \(20.17\) is greater than \(17.98\), the result will be negative: \(17.98 - 20.17 = -2.19\).

Key Concepts

Expression Simplification

Simplifying an expression means making it as simple as possible while retaining its value. To do so, we need to perform all possible arithmetic operations in a logical sequence. This entails adhering to the order of operations which is usually remembered by the acronym PEMDAS - Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction.

The goal is to break down complex expressions into easily manageable steps. In our original exercise, simplification involves handling both the parentheses and the arithmetic operations nested within. When each part of the expression is simplified progressively, it aids in clear understanding and avoids errors.

This approach not only leads to the correct answer but also enhances one's ability to tackle similar mathematical problems effectively in the future.

Parentheses Evaluation

Parentheses play a crucial role in determining the order in which calculations within an expression are performed. They signify which parts of an expression should be evaluated first. Let's look at our example: inside the parentheses, we have the operation \(10.07 - (-10.1)\).

When evaluating expressions with parentheses, follow these steps:

• Identify the most innermost parentheses and work outward.
• Within each set of parentheses, apply standard arithmetic operations adhering to the order of operations.

In this case, the operation inside the parentheses is "subtraction involving a negative number," which needs careful attention. Evaluating this correctly by converting subtraction into addition can significantly simplify the process.

Subtraction Involving Negative Numbers

Subtraction involving negative numbers can be confusing at first, but once understood, offers a powerful tool for simplifying expressions. When you see a subtraction sign combined with a negative, remember that it is equivalent to adding the absolute value of the negative number.

For instance, in our problem \(10.07 - (-10.1)\), we transform the expression by removing the double negative. This is based on the principle that subtracting a negative is the same as addition:

• The expression becomes \(10.07 + 10.1\).

This transformation makes the arithmetic much simpler and often reduces confusion, leaving only basic addition to handle.

By rewriting subtraction involving negative numbers as addition, you can navigate more easily through algebraic expressions and arrive at the solution without stumbling over potential areas of misunderstanding.