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Exponentiation is the process of raising a number to the power of another number. This is denoted by the superscript notation where a base number is followed by an exponent. For example, the expression
\( a^n \)
means to multiply the base number, \( a \) by itself \( n \) times. It's important to note when dealing with exponentiation, the order of operations, also known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), must be followed.

In the given exercise, the base number is negative three \( (-3) \) and it is raised to the 4th power. Raising a negative number to an even exponent will result in a positive number because the negative signs cancel out in pairs. Here, \( (-3)^{4} \) would be \( 3 \times 3 \times 3 \times 3 \), which equals \( 81 \) and not \( -81 \) because of the even number of negative factors.

Substitution is a fundamental concept in algebra that involves replacing a variable in an expression with a specific value. This technique allows for simplifying expressions and solving equations.

When faced with substituting a value into an algebraic expression, you should identify the variable and replace it with the given number. It’s crucial to substitute the value accurately while maintaining the structure of the original expression. For instance, if an expression contains \( -x \) and \( x \) is given as \( -3 \) like in our example, then \( -x \) should be substituted with \( -(-3) \) to properly reflect the change. Once the substitution is done, the resulting numerical expression can be evaluated using the order of operations.

Negative numbers are used to represent values less than zero and are symbolized by a minus sign \( - \) in front of a number. They play a crucial role in the world of mathematics, showing up in various contexts including debts, temperatures below zero, and elevations below sea level.

When evaluating expressions with negative numbers, it's essential to understand the rules pertaining to the operations with negative values. For example, multiplying or dividing two negative numbers yields a positive result, whereas the product or quotient of a positive number and a negative number is negative. In the context of our exercise, the exponentiation of a negative number raised to an even power, such as \( (-3)^{4} \), results in a positive number, while an odd power would result in a negative number. Additionally, applying a negative sign in front of a negative number, as in \( -(-81) \) flips the sign, turning it into a positive \( 81 \).