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Understanding like terms in algebra is crucial for simplifying algebraic expressions. Like terms are terms that have exactly the same variables raised to the same power. For instance, in our expression \(18 x^{5} y^{2} - 11 x^{5} y^{2}\), both terms are considered 'like' because the variables \(x\) and \(y\) and their corresponding exponents (5 and 2, respectively) match.

When simplifying expressions, you only combine like terms. This is because algebra follows specific rules that do not allow the combination of terms with different variables or powers, which would be akin to trying to add apples to oranges—they are simply not the same thing. So, remembering to identify like terms accurately is a stepping stone to mastering algebraic simplification.

Subtraction in algebra, especially when it involves like terms, works by subtracting the numerical coefficients and keeping the variable part intact. This notion leans on the properties of subtraction in algebra, which dictate that only the numbers in front of the variables (coefficients) are subject to arithmetic operations, not the variable part.

Example: In the expression\(18 x^{5} y^{2} - 11 x^{5} y^{2}\), only the coefficients 18 and 11 are subtracted. The subtraction property lets us combine these like terms to simplify the expression while \(x^{5} y^{2}\) remains as the common variable factor. It’s essential to apply this property accurately; incorrect application could lead to errors in solving equations or simplifying expressions.

Simplifying numerical coefficients is a basic yet potent part of working with algebraic expressions. A numerical coefficient is a number placed in front of a variable or variables in an algebraic term. To simplify an expression, you combine these coefficients based on the operations given (addition, subtraction, multiplication or division).

For instance: \(18 x^{5} y^{2} - 11 x^{5} y^{2}\) simplifies to \(7x^{5}y^{2}\) after subtracting the coefficients (18 - 11 = 7). It's a straightforward process of arithmetic, yet it's at the heart of balancing equations and simplifying more complex expressions in algebra. Always ensure that your coefficients are accurately combined to maintain the integrity of the equation.