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In algebra, exponents are used to represent repeated multiplication of the same number. An exponent is written as a small number above and to the right of the base number, indicating how many times to multiply the base by itself. For example, the expression \((-7)^2\) contains an exponent of 2, which means we need to multiply -7 by itself once.

Understanding Positive Exponents

When the exponent is a positive integer, the operation is straightforward; just multiply the base, -7 in our case, the number of times indicated by the exponent, 2. Thus, \((-7)^2 = -7 \times -7\).

Dealing With Negative Bases

In our example, the base is a negative number (-7). It's vital to note that the negative sign is enclosed in parentheses, which means the exponent applies to the negative sign as well. If the negative sign were not in parentheses, only the number 7 would be squared, and the answer would be negative.

Exponents can greatly simplify expressions and are a powerful tool in algebra and higher mathematics.

Negative numbers are integers less than zero. They are the opposite of positive numbers and are represented with a minus sign (-) in front. Working with negative numbers is a fundamental part of algebra, and understanding their properties is important for simplifying algebraic expressions.

Multiplying Negative Numbers

When multiplying negative numbers, two key rules are useful: multiplying two negative numbers together yields a positive result, while multiplying a positive number by a negative number gives a negative result. For example, \(-7 \times -7 = 49\), because both numbers are negative. This rule is essential when simplifying expressions with negative bases raised to even exponents.

Remember that negative numbers may affect the outcome of an expression significantly, and correctly identifying and handling them is crucial for reaching the correct solution.

Arithmetic operations are the foundation of all mathematical calculations and include addition, subtraction, multiplication, and division. In the context of algebraic expressions, these operations help us to simplify and solve for unknowns.

Order of Operations

It's important to follow the correct order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This rule ensures that calculations are performed in the right sequence to obtain the correct answer.

For the given expression \((-7)^2\), we first address the exponent before we would perform any other operations if they were present. Understanding how to properly execute arithmetic operations, especially when combined with a solid grasp of how exponents and negative numbers work, allows for correct simplification of even the most complex algebraic expressions.