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The substitution method is a fundamental aspect of evaluating expressions, especially when variables are present. Understanding this concept is vital because it allows you to replace variables with their given values to simplify the expression.

For instance, when you are given the expression \( \frac{10,000}{x^{120}} \) and the value of \( x \) as 1.075, the substitution method entails replacing every instance of \( x \) with 1.075. The new expression will look like \( \frac{10,000}{1.075^{120}} \). This method ensures that you are working with actual numbers, which makes the solving process much simpler.

Remember, accurate substitution is crucial. A slight mistake in substituting the values can lead to an entirely different answer. Always double-check your substitution to ensure accuracy before proceeding to calculate the expression.

Exponentiation is the process of raising a number to the power of another number. It is represented as \( a^n \) where \( a \) is the base and \( n \) is the exponent. The exponent denotes how many times the base is multiplied by itself.

An example from the given exercise is \( 1.075^{120} \). Here, 1.075 is the base and 120 is the exponent, which means we multiply 1.075 by itself 120 times.

Calculating Exponents with a Calculator

When dealing with large exponents like 120, it is practical to use a scientific calculator. Enter the base (1.075), then use the exponentiation function (often labeled as '^' or 'exp'), followed by the exponent (120), to get the result. Manual calculation can be prone to errors and incredibly time-consuming.

Exponents powerfully compact the mathematical expressions and understanding this concept is crucial when dealing with growth scenarios, compound interest, or in physics when calculating force or energy.

The order of operations is a set of rules that mathematicians agree upon to avoid confusion when interpreting mathematical expressions. The most common acronym to remember this sequence is PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

When evaluating \( \frac{10,000}{1.075^{120}} \), it's vital to follow the order of operations. Begin with the exponentiation \( 1.075^{120} \), and once you have that value, you proceed with the division: 10,000 divided by the result of the exponentiation.

Skipping steps or rearranging them can lead to incorrect answers. In a classroom or exam setting, understanding the order of operations can prove to be the difference between a correct or incorrect solution. In real-life applications, such as programming or financial calculations, this knowledge is equally crucial for achieving accurate results.