Part of a student's difficulty with simplifying expressions is the blurry extension of real number operations beyond the dyadic operators of add and multiply in the field properties. For conciseness, an "agreed-to" order of operations must then include non-field operators (dyadic and monadic) and semi-logical operators. The field properties do not specify that multiplication comes before addition, and most certainly does not demand that exponents be done before multiplication. Indeed, radicals, division, and subtraction operations are usually defined at the elementary level in terms of conversion to their inverse operation: radicals to exponents, division to multiplication, and subtraction to addition.
Indeed, any algebraic expression with mixed operators requires punctuation in the form of grouping symbols to specify the priority of operations UNLESS the writer and reader agree to an order of operations for incompletely punctuated expressions. Even the simplest mixed expression such as 3+4X5 is ambiguous or not well-formed (3+X45) without punctuation and no agreed to order of operations. (3+4)X5 produces the same result as operating left to right, 35. 3+(4X5) produces the same result, 23, as operating right to left. Hence, 3+4X5 is ambiguous without punctuation or an agreed to order of operation.
Writing and deciphering an algebraic expression has its parallel in English. Consider the algebra of real numbers based on field properties and definitions, accompanied by grammar and stylistic considerations for writing and reading well-formed expressions. These expressions, through the use of algebraic models, may represent a real world situation or event in much the same way that a sentence in English may describe something from the real world. Hence, both an algebraic expression and an English phrase may contain information and have semantic content. At an elementary level, much of the algebra is concerned with rewriting a given expression into a different format (Simplified format, Factored format, Solved format, Visual/graphical format, Some canonical format) in the hopes that the re-expressed format will better reveal details of the real world situation modeled by the expression). Perhaps the most difficult exercise at the elementary level is the creation of an algebraic model for some real world situation, AKA word problems. From this point of view, elementary algebra is often the art of rewriting an algebraic expression in the hope that the revised format will reveal more information.
An adopted order of operations is usually the biggest influence on what if means to rewrite an expression into a "simplified" form. Hence the text book demand to SIMPLIFY an expression often has some standard format in mind.
The advent of personal computers and hand-held calculators has made at least one of these operation orders a standard usually described as algebraic logic that relies on left-to-right precedence within most given levels: GEMA -- Grouping symbols to over-rided the standard order, Exponents(Radicals), Multiplication(Division), Addition(Subtraction). However, for years, TI was known to tweak the order of operations on their calculators, sometimes by taking votes among math teachers about what "seems like" the most reasonable "next" operation in an incompletely punctuated expression. Note that the EMA order is arbitrary, yet math teachers do not usually vote for simple right-to-left, order-typed entry like APL (A Programming Language) logic nor even RPN (Reverse Polish Notation) logic although calculator chips often process expressions using stacks.
"Please Excuse My Dear Aunt Sally" is a mnemonic for the once-popular operational hierarchy Parenthesis, Exponents, Multiplication, Division, Addition, Subtraction. (Where division and subtraction were done left to right, and stacked exponents were ambiguous.) Yes. Look at it again. All of the multiplications were done before any divisions, and all the additions were completed before any subtractions. Unfortunately this can lead to inconsistencies if division is converted to multiplication using the monadic reciprocal operator and subtraction is converted to addition using the monadic negation operator. The advent of widespread use of algebraic logic on hand calculators and in programming languages effectively reduced this mnemonic to GEMA although some still quote the original and augment it by grouping multiplication and division with the left to right criteria, and similarly for addition and subtraction.
In the case of the order of operations, GEMA, a given hierarchical operation level in EMA has a distributive property across the NEXT lower level operation: I.E. Multiply distributes across Add. Exponents distribute across Multiply for reasons that may be apparent from the counting number definitions of these operations. Hence, the distributive property can be used to eliminate such grouping symbols. However, Addition is NOT the next lower level below Exponents. Thus, Exponents (and hence radicals) do not distribute across addition, so that (1+1)^2 <>1^2 + 1^2.
Because common algebraic expressions are often incompletely punctuated using grouping symbols, an agreed to order of operation is necessary. Parallel to this requirement, many common instructions such as "Simplify" are poorly defined, and any easy path to a canonical form for real algebraic expressions seems unlikely. Usually, SIMPLIFY means to remove grouping symbols (if not too difficult) and carry out as many operations in an algebraic expression as seems feasible to reduce the number of outstanding operations and produce a "clearer, more compact" expression: Very Vague. Seldom is an algorithm offered to students, perhaps for good reason, yet most of the "Simply" instructions in textbooks are for algebraic expressions that will yield to an algorithm. The advents of algebraic systems now offers the ability to have your calculator "simplify" expressions according to some internal algorithm.
Aside: A local newspaper used to drive me crazy with its daily "math" feature in which readers were encouraged to SOLVE not only 3x+5 = 11, but also to SOLVE 3+5*2 and to SOLVE 5x+3y +7x + 4. I pity the poor student attempting to absorb the concepts of SOLVE or SIMPLIFY after reading that feature.
A large part of beginning real number algebra consists of creating and interpreting algebraic MODELLING usually in the form of algebraic expressions or simple first or second degree equations in one variable. Associated with these models are the skills to SIMPLIFY, FACTOR, SOLVE, and GRAPH. Each of these skills allows re-expressing the model in the hope of extracting more useful insights and information about the situation modeled. Today, I've focused on SIMPLIFY, and I hope argued that simplification is directly related to the adopted order of operations. Hence, I conclude that the order of operations form an outline for simplification without rising to the level of an algorithm. In particular,
Given the instruction to SIMPLIFY an algebraic expression, Use the following outline to accomplish the simplification:
GEMA
G (Grouping symbols)
1. Use the distributive property for exponents across multiplication to remove grouping symbols by distributing exponents across grouped multiplications and divisions. Distribute the exponent to each factor in the parenthesis. Do not distribute exponents across terms (additions or subtractions). Leave expressions such as (a+b)^n intact when the counting number exponent n <>2 and n<>1 unless instructed to EXPAND (see binomial expansion). Also do not attempt to simplify radical expression unless the radicand is FACTORED, that is, do not distribute a radical sign (a fractional exponent) across a sum or difference. Exponents do not distribute across sums or differences.
2. Use the distributive property of multiplication across addition to remove parenthesis. Distribute the factor to each term in the parenthesis.
E (Exponents and radicals)(Goal: as few radicals as possible)
(Exponents) Rewrite every product/quotient using the rules of exponents so that no base appears in the product/quotient more than once. The rules for stacked exponents 3^4^2 are not universally agreed on. Consult your textbook. For clarity always use grouping symbols rather than stacking exponents.
(Radicals) The advent of calculators makes conversion of radicals to exponent form and then simplify an alternative approach.)
1. Factor the radicand and simplify the radicand according to the rules for exponents
2. For positive bases, insure that every exponent in the radicand is smaller than the index. Simple division accomplishes this for Nth root of b^P when P>=N. Divide N into P and obtain the quotient Q and the Remainder R. Then Nth root of b^P simplifies to b^Q * Nth root of b^R. Remember that b^0 =1 and 1 to any power =1 to finish the simplification. Repeat for every base b with an exponent larger than the index.
3. Consult your textbook to see whether a radical expression with a rationalized denominator or numerator is desirable.
4. Consult textbook to see whether reducing the index when possible is desirable.
M (Multiplication and Division)
Use reciprocals to convert all divisions to multiplications. This conversion can be done before the exponent step. Complete feasible multiplications. Note 3^2 = 9 is a good completion. 3^100 does not look better when completing the multiplication.
A (Addition and Subtraction)
Use negations to convert all Subtractions to Additions.
Combine like terms and complete feasible arithmetic. Basically this is limited factoring.
3x + 4x + 5y + sqrt(7) y = (3+4)x + (5+sqrt(7))y
= 7x +(5+sqrt(7))y
But that re-introduces parenthesis back into the expression.
We make no claim whether 5y + sqrt(7)y is actually "simpler" or easier to use than (5+sqrt(7))y, merely that it is an alternative form that may be more useful sometimes.
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