Problem 33 - Fundamentals of Mathematics

Question

Rewrite each expression in simpler form. $$ -(-15) $$

Step-by-Step Solution

Answer

The simpler form is 15.

Step 1: Identify the Double Negative

The expression \(-(-15)\) contains a double negative. This can be seen as the negative of a negative number.

Step 2: Simplify the Double Negative

The rule for double negatives in mathematics is that they cancel each other out. Therefore, the expression \(-(-15)\) simplifies to just 15, because negating a negative number results in a positive.

Step 3: Final Result

After canceling the double negative, we find that the simplest form of the expression is 15.

Key Concepts

Understanding Double Negatives

Double negatives in mathematics may initially appear tricky, but they follow a straightforward rule. A double negative occurs when two negative signs appear in succession in an expression. For example, in the expression \(-(-15)\), the two negatives sequentially affect the same number.

The concept is based on the property that multiplying or dealing with two negatives results in a positive. In simpler terms, the two negatives "cancel out." Therefore, applying this rule to our example, \(-(-15)\) simplifies to just 15.

This cancellation is like the subtraction of two negatives in arithmetic, which results in a positive value. Understanding this concept helps tremendously with simplifying expressions.

Key Mathematical Rules for Simplification

Simplifying expressions requires the application of several key mathematical rules. One essential rule is that a negative times a negative equals a positive. This principle underlies the simplification of double negatives.

Another important rule to remember is the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). While simplifying expressions, it's crucial to follow this sequence to ensure accuracy.

In the case of double negatives, parentheses help clarify the order in which operations should occur. Always handle what's inside the parentheses first, and apply the rule that two negatives make a positive to transition from \(-(-15)\) to 15.

Expression Simplification Techniques

Simplifying mathematical expressions is such an essential skill. It involves making an expression more compact without changing its value. Let's explore some strategies to master this!

• Identify Double Negatives: Recognize when double negatives are present and apply the rule that two negatives make a positive.
• Use Parentheses: Always pay attention to expressions inside parentheses and simplify them first. This helps prevent mistakes and guides the application of mathematical rules.
• Order of Operations: Follow PEMDAS to keep your simplifications accurate. This ensures that operations are performed in the correct sequence.

By internalizing these techniques, you'll be well-equipped to simplify expressions quickly and correctly, making solving more complex problems easier. Practice regularly to strengthen these skills!