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Exponentiation is a mathematical operation, involving numbers called the base and the exponent. The exponent tells you how many times to multiply the base by itself. For example, in the expression \(3^{2}\), 3 is the base, and 2 is the exponent. It means 3 multiplied by itself: \(3 \times 3 = 9\).

When dealing with negative bases, always be careful. A negative base raised to an even power will result in a positive number, while a negative base raised to an odd power will result in a negative number. For example, \(-2^{2}\) means \(-2 \times -2 = 4\). Notice that we get a positive result because the exponent is 2 (even number). Always perform the exponentiation first before any other arithmetic operations!

Evaluating mathematical expressions correctly requires following a specific order called the order of operations. This ensures that everyone gets the same result. The common acronym to remember this is PEMDAS:

• P - Parentheses first
• E - Exponents (powers and roots)
• MD - Multiplication and Division (left to right)
• AS - Addition and Subtraction (left to right)

In our example \( -2^{2} + 5(3)^{2} \), we start by addressing the exponentiation: \(-2^{2}\) and \(3^{2}\). Next, perform multiplication: \(5 \times 9 = 45\). Finally, we add the results together: \(4 + 45 = 49\). Breaking it into steps makes it easier to follow and reduces the chance of errors.

Arithmetics is the branch of mathematics dealing with basic operations like addition, subtraction, multiplication, and division. Let's take a look at how these operations apply to our problem:

• Addition: The operation of finding the total, or sum, by combining two or more numbers. Example: \(4 + 45 = 49\).
• Subtraction: The operation of finding the difference between two numbers. Not needed in our example.
• Multiplication: The operation of calculating the total of one number added several times. Here, \(5 \times 9 = 45\).
• Division: The operation of finding how many times one number is contained within another. Not needed in our example.

Understanding these fundamental operations is crucial for tackling more complicated mathematical problems with confidence.