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Exponentiation is a mathematical operation that involves raising a number, called the base, to a power, which is the exponent. When we write something like \(4^3\), it means \(4\) is multiplied by itself \(3\) times: \(4 \times 4 \times 4\). This results in \(64\).
When handling numbers with an exponent, you must multiply the base by itself according to the number of times equal to the exponent. Exponents are a way to express large numbers in a compact form.
In the given problem, there are two exponentiations: \(-4^3\) and \(3^2\). It's important to note the negative sign in \(-4^3\), which leads us to consider rules for negative numbers as well.

Negative numbers are numbers less than zero. They are often depicted with a minus sign, such as \(-3\) or \(-64\). When working with negative numbers and exponentiation, a common point of confusion is how to treat the negative sign.
For example, in \(-4^3\), we interpret this as the negative of the result of \(4^3\), as the exponent only affects the base \(4\) and not the negative sign. Thus, \(4^3 = 64\) and \(-4^3 = -64\).

• Always remember, unless the negative sign is in parentheses, it remains outside of the exponentiation.
• Negative signs in front of exponents simply require attaching the sign to the final result after calculation.

The order of operations is a rule to follow to solve mathematical expressions correctly. A common way to remember the order is PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). This ensures that we perform operations in the correct sequence.
In the expression \(-4^{3} - 3^{2}\), following the order of operations tells us to handle the exponentiation first, then the subtraction, which is exactly what the solution steps outline. By calculating \(-4^3\) as \(-64\) and \(3^2\) as \(9\) first, we ensure accuracy in later subtraction: \(-64 - 9\).

• Always begin with computations inside parentheses; then handle exponents.
• Follow with multiplication or division as they appear from left to right, ending with addition or subtraction from left to right.

By adhering to these rules, we eliminate confusion and errors in calculations.