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To begin evaluating an expression, one of the initial steps is to substitute variables with their given values. This makes abstract formulas concrete and ready for computation.

Consider the expression \(\sqrt{m n}-5 n^{2}\) and the values \(m=27\) and \(n=3\). Substitution means replacing each \(m\) and \(n\) with 27 and 3, respectively. This turns our expression into \(\sqrt{27 \times 3}-5 \times 3^{2}\), which is a numerical expression free of variables.

The importance of accurate substitution cannot be overstated. A single error in this foundational step can lead to the wrong result at the end of your calculations. So take your time to substitute correctly!

A critical operation in algebra is calculating the square root of a number. The square root of a number is a value that, when multiplied by itself, gives the original number. When faced with an expression such as \(\sqrt{81}\), we're looking for a number that times itself equals 81.

In this case, \(9 \times 9 = 81\), so \(\sqrt{81} = 9\). It's really important to grasp that a square root can be a 'perfect square'—like 81—where the root is an integer, or it can be an irrational number that doesn't resolve neatly. Calculators are valuable tools when dealing with non-perfect squares. Still, understanding the concept helps you recognize when the roots are straight forward to calculate by hand.

Following the order of operations is akin to following a recipe—missing a step can lead to an unexpected result! In mathematics, we often refer to the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) to remember this sequence.

In the given expression, after we substitute, we first deal with the square root (which is an exponent-type operation), and this is how we obtain 9. Next, we execute the multiplication in \(5 \times 3^{2}\), which simplifies to \(5 \times 9\). Finally, we perform the subtraction: \(9 - 45\) resulting in -36. It's vital to stick to this order to correctly evaluate expressions. Hop, skip, or jump out of line, and just like in baking, we might end up with something inedible—mathematically speaking, of course.