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Evaluating expressions is the process of finding the value of an expression with given numbers. In the exercise we have, it's essential to follow the order of operations, often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This order ensures that calculations are performed correctly.

Let's break this down:

• Parentheses: Always the first step, solve what's within parentheses.
• Exponents: Next, address any exponents in the expression.
• Multiplication/Division and Addition/Subtraction: These follow based on what comes first from left to right.

In our example, the expression was simplified by first evaluating the expression inside the parentheses, then solving the exponent, and finally performing the multiplication. This systematic approach ensures accuracy in the final result.

Exponents represent repeated multiplication of a base number. The expression \((4-3)^5\) is a perfect example of using exponents. To begin with, the base is calculated inside the parentheses before further steps are taken. So, \(4 - 3 = 1\), which means the base is 1.

Once you have the base, the exponent tells you how many times to multiply this base by itself. In mathematical terms, \(1^5\) means multiplying 1 by itself five times: \((1 \times 1 \times 1 \times 1 \times 1)\).

• Any number raised to the power of one equals itself.
• Conversely, any number (except zero) raised to the power of zero equals one.

In this particular example, \(1^5 = 1\), because multiplying 1 by itself any number of times still gives 1. Understanding exponents is critical for dealing with more complex expressions and mathematical problems.

Simplifying an expression means reducing it to its simplest form, where no further algebraic operations can simplify it further. In the given expression \(9(4-3)^5\), simplification occurs at several steps:

Starting with the simplest operations, like resolving parentheses, ensures that the expression is as easy to handle as possible at each subsequent step. This process involves:

• First simplifying inside the parentheses: \(4 - 3\), which simplifies to 1.
• Next, addressing any exponents: \(1^5 = 1\).
• Finally, simplifying the multiplication: \(9 \times 1\) simplifies to 9.

By following these steps, you streamline the expression to its simplest form, making it much easier to evaluate correctly. This exercise shows how critical simplification is when dealing with expressions, allowing for quick and accurate calculations.