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BIDMAS stands for Brackets, Indices, Division and Multiplication, Addition and Subtraction. It's a helpful acronym used in the UK to remember the correct sequence to approach solving mathematical equations. Just like a recipe in cooking, steps in BIDMAS must be followed in order to get the intended result.

When you look at an algebraic expression, start by solving any calculations inside brackets. Next, take care of any exponents which are indicated by indices. Afterwards, perform any division or multiplication as they appear from left to right. Finally, address any addition or subtraction in the expression, again moving from left to right.

PEMDAS is the North American counterpart to BIDMAS, indicating the order of operations: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction. Important to note that multiplication does not necessarily come before division, nor does addition always precede subtraction—instead, multiplication and division should be addressed as they appear from left to right, as should addition and subtraction.

Understanding PEMDAS helps to avoid common calculation errors and ensures that everyone solves the equations consistently. For instance, in our expression \(\frac{3}{4} \cdot 8 - 6\), we first handle the multiplication before tackling the subtraction, resulting in the correct answer of 0.

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They represent values that can change; the numerals and the operations tell us exactly how those values relate to each other.

In the given problem, \(\frac{3}{4} \cdot 8 - 6\), we're dealing with a numerical expression, where all parts are numbers and defined operations, rather than variables. By skillfully evaluating expressions—like knowing when to multiply and when to subtract—you can simplify the expression to find a single numerical answer.

The order of operations is a fundamental principle in mathematics that determines the sequence in which operations should be performed. This ordering helps avoid ambiguity in mathematical expressions.

Without a universally agreed-upon order, different people might interpret an equation in various ways, leading to multiple answers. The order, as prescribed by BIDMAS or PEMDAS, brings consistency and clarity to mathematical computations. As applied to our textbook exercise, respecting this order allowed us to effectively evaluate \(\frac{3}{4} \cdot 8 - 6\) and reach the conclusion that the expression is equivalent to 0.