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When we talk about exponents, we're referring to the process of raising a number to a certain power, which tells us how many times to multiply that number by itself. The exponent is the small number placed slightly above and to the right of a base number. In the expression (5^{3}), 5 is the base and 3 is the exponent. To evaluate (5^{3}), you multiply 5 by itself 3 times: (5 * 5 * 5).

This equals to 125. Exponents are not just about multiplying a number by itself; they are a shorthand notation to express very large numbers like (2^{10}), which equals to 1024, or very small numbers like (5^{-3}), which is (1 / 125). Understanding exponents is crucial because it simplifies the writing and calculating of numbers that would otherwise be cumbersome to handle.

Arithmetical operations are the building blocks of mathematics. They include addition, subtraction, multiplication, and division. In the context of our exercise, we focus primarily on addition. After evaluating the exponent expression (5^{3}) and getting 125,we need to add this value to 12. This operation is simple: we just put the numbers together to get a sum. It's crucial for students to follow the proper order of operations, commonly remembered by the acronym PEMDAS—Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Order of operations is vital; it's the road map that tells us which path to take through the numbers so that everyone ends up at the correct destination. In our example, after powering we proceed with the addition, resulting in (125 + 12), which equals 137.

Algebraic expressions are combinations of numbers, variables (letters that represent unknown numbers), and arithmetic operations. Unlike our example which is a simple numeric expression, algebraic expressions can get a bit trickier with variables such as (x^{2} + 4y).

Evaluating algebraic expressions involves substituting numbers for variables and then following the order of operations to simplify. They are a cornerstone of algebra and essential for solving equations. Understanding how to work with algebraic expressions enables students to form equations based on real-world scenarios, and solve for unknowns. It opens the door to higher level math, encouraging logical thinking and problem-solving skills.