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Understanding numerical expressions is crucial for solving various mathematical problems. A numerical expression is essentially a mathematical phrase that includes numbers and operation symbols, but no equal sign. For instance, in the exercise, \( \frac{1.6 \times 10^{-19} \times 15}{36^2} \) is a numerical expression since it involves numbers and operations but no equals. To effectively interpret these expressions, one must carefully follow the order of operations: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right) — often remembered by the acronym PEMDAS. Applying this understanding step-by-step ensures accurate results.

Mathematical simplification is the process of reducing a complex expression into a simpler, more manageable form without changing its value. The goal is to make calculations easier. In the provided exercise, simplifying the expression was achieved through several steps: first by squaring the denominator (\( 36^2 = 1296 \)), then by multiplying the constants in the numerator ( \( 1.6 \times 15 = 24 \)), and lastly simplifying the fraction (\( \frac{24}{1296} = 0.0185 \)). Once simplified, converting the result into scientific notation yielded the final answer. Simplifying expressions helps in reducing errors and making complex calculations more approachable.

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. It is expressed as \( a^n \), where 'a' is the base and 'n' is the exponent, meaning 'a' is to be multiplied by itself 'n' times. In scientific notation, exponentiation plays a pivotal role. For example, in the exercise, terms like \( 10^{-19} \) are used to represent very small numbers. When multiplying numbers in scientific notation, the exponents are added, as seen in the final step: \( 0.0185 \times 10^{-19} = 1.85 \times 10^{-2} \times 10^{-19} = 1.85 \times 10^{-21} \). Mastery of exponentiation allows for efficient handling of large or small values and is a fundamental skill in mathematics.