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The order of operations is a critical concept in math that helps us know which operations to perform first in an expression. To remember the correct order, we use the acronym PEMDAS. PEMDAS stands for:

• P: Parentheses
• E: Exponents
• M: Multiplication
• D: Division
• A: Addition
• S: Subtraction

Always start with calculations inside parentheses, followed by exponents. Then proceed with multiplication and division from left to right. Finally, perform addition and subtraction from left to right. By using PEMDAS, we ensure our calculations are accurate and consistent.

Parentheses are used in mathematical expressions to indicate which operations should be performed first. They help to group terms and manage operations that should take precedence. For example, in the expression \(7-5[7-5(7-5)]\), the innermost parentheses are \(7-5\). It is important to simplify these inner operations first, before moving on to the next outer set of parentheses. Parentheses may also be nested, meaning one set inside another. In these cases, always start from the innermost set and work your way outward. If there are brackets or braces, they serve the same grouping function, assisting in organizing complex expressions.

Simplifying expressions means breaking them down into simpler or more manageable parts. This involves performing the operations in the correct order and combining like terms. Let's simplify the example expression step by step:
1. Start with the innermost parentheses: \(7 - 5\), which simplifies to \(2\).
2. Substitute back into the expression: \(7 - 5 [7 - 5(2)]\)
3. Next, simplify: \(5(2)\), which equals \(10\).
4. Substitute: \(7 - 5 [7 - 10]\)
5. Simplify the expression in brackets: \(7 - 10\), which is \(-3\).
6. Substitute: \(7 - 5(-3)\)
7. Multiply: \(5(-3) = -15\)
8. Finally, subtract: \(7 - (-15)\), which is \(7 + 15 = 22\).
Simplifying expressions makes solving complex problems much more manageable.

Negative numbers can be tricky but are important in mathematics. They represent values less than zero and have unique properties, especially when involved in operations. When subtracting a negative number, it's the same as adding the positive equivalent. For example, \(7 - (-15)\) is equal to \(7 + 15\). Similarly, multiplying two negative numbers results in a positive number, while multiplying a positive number by a negative number results in a negative number. For instance:

• \(5 \times -3 = -15\)
• \(-5 \times -3 = 15\)

Understanding how negative numbers work is essential for accurate calculations. Practice with various problems to become comfortable with operations involving negative numbers.