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When we talk about 'exponential values', we're referring to an expression where a number, known as the base, is raised to the power of an exponent. The exponent tells us how many times the base is multiplied by itself. For instance, in the expression \(2^{2}\), 2 is the base and the exponent is 2, meaning that 2 is being multiplied by itself once: \(2 \times 2 = 4\).
Exponential growth can be incredibly fast because each multiplication could substantially increase the number. This sort of rapid increase is seen everywhere—from finance in compound interest, to science in population growth. A good grasp of exponential values is crucial not only in algebra but also in higher-level math and many scientific fields.
In our exercise, the exponential expressions \(2^{2}\) and \(3^{3}\) must first be calculated independently before they are used in further arithmetic operations. The power of an exponential expression like this lies in its simplicity for representing complex or large count operations.

Substitution is a foundational technique in algebra where we replace variables with their respective numerical values. In the context of our exercise, this means taking the algebraic expression \(x^{x}-y^{y}\) and replacing \(x\) and \(y\) with the given numbers.
To correctly evaluate the expression for \(x=2\) and \(y=3\), we 'substitute' the variable \(x\) with the number 2, and \(y\) with the number 3, resulting in a new expression \(2^{2}-3^{3}\). The substitution here is straightforward because each variable matches its own exponent, but it gets more complex in other equations where this isn't the case.
Understanding how and when to substitute is key to solving algebraic expressions, and practice with different equations can help solidify this essential skill. It's worth noting that substitution doesn't change the original expression's meaning or value—a principle that's crucial in preserving the integrity of equations.

Arithmetic operations are the bread and butter of mathematical computation, encompassing addition, subtraction, multiplication, division, and more. These operations are the building blocks for more advanced math concepts.
After evaluating the exponential values \(2^{2}\) and \(3^{3}\), our next step is to perform the arithmetic operation of subtraction to find the difference between these two values. We take the result of \(2^{2}\), which is 4, and subtract the result of \(3^{3}\), which is 27, yielding \(4 - 27 = -23\).

Solving The Subtraction

In the context of the given exercise, subtraction is used to combine the results of the two exponential calculations. Since the value of \(3^{3}\) is greater than \(2^{2}\), this results in a negative number. It's critical to keep track of the sign when performing such operations, as it affects the final outcome, and a mistake here can lead to an incorrect solution.
Arithmetic operations follow a particular order called the Order of Operations, or PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), which must be observed to correctly solve expressions like the one in our exercise. However, in our case, we are only dealing with exponents and subtraction, so it's a little more straightforward.