If 4ab 2 equals 4b 2a

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Simplify terms

Terms are arithmetic expressions that consist of numbers, variables (i.e. letters that stand for unknown numbers of a certain "type") and arithmetic symbols. An equation is not a term, but a statement that two terms have the same value. A term therefore does not contain an equal sign!

If a term consists only of numbers, it can easily be calculated. You have to follow the usual rules:
* Parentheses come first
* Powers before point calculation
* Dot before line calculation
* calculate from left to right

Example:

3 · (4 - 3,5)² - ((5 + 4) - 3 · 11) = 3 · 0,5² - ( 9 - 33 ) = 3 · 0,25 - (-24) = 0,75 + 24 = 24,75

If you do not understand arithmetic with letters at all, you should consider that our numerical signs, such as the 1, are also only symbols. The fact that you can also write 5 · 1 (or equal to 5) for 1 + 1 + 1 + 1 + 1 will probably not lure a mouse out from behind the stove. Calculating with other symbols is also not more complicated, but rather simpler, because you have to calculate less. You simply group similar symbols together by "counting" them.

The five (identical) hearts ♥ + ♥ + ♥ + ♥ + ♥ are together as many as five times a such a heart, so 5 · ♥ or simply 5 ♥.
For the sum ♣ + ♦ + ♠ + ♠ + ♠ + ♥ + ♦ + ♣ + ♦ + ♥ + ♣ + ♠ you can write 3 · ♣ + 4 · ♠ + 2 · ♥ + 3 Painting points): 3 ♣ + 4 ♠ + 2 ♥ + 3 ♦.

It works with letters in the same way as with the card symbols: You combine the same letters and, so to speak, only take one letter of each type times with the total number available. As a reminder: the summands of a sum can be rearranged (commutative law). Hence, e.g. a + c + b + a + b + a is the same as a + a + a + b + b + c or 3a + 2b + c. If only a Letter of a certain "variety" is included, the 1 is left out, as in the last example with c.

Of course, the number can also be reduced by subtraction: a + a + a - a corresponds to 3a - a, so "three a minus one a", and that of course results in "two a", so: a + a + a - a = 2a.

Examples

x + x + x = 3x x + x - x + x = 2x a + 2a + 3b + 2b = 3a + 5b x + y + x = 2x + y 3a + 4a = 7a 4g - g = 3g h - 7h = -6h 3m + 1 - 5m = 1 - 2m a - b + a = 2a - b 2x + 4 - x + 5 = x + 9 10x - 2.1x = 7.9x 3a + 6b = 3a + 6b a + 2b + c - a = 2b + c 2e - 4f - 3e + 4f = -e

You can't always simplify everything. In the third to last example, for example, you can't do anything at all, since a and b only appear in one summand and the two Not can be "offset" with each other. (Also the factors 3 and 6 not, because they only show the respective numbers of the a and the b !!!) In the penultimate example the 2b and the c simply remained for the same reason.

If a variable is left with zero, as in the penultimate example of a or in the last example of f, then one does not write 0a or 0f, but simply omits the variable entirely.

Instead of -1e you just write -e.

Individual numbers, as in the eighth and tenth example, are also summarized if possible and, of course, they are calculated immediately (see tenth example: 4 + 5 = 9).
Individual numbers and variables from other summands may not be "offset".

Multiplications

The expression 3a means written out a + a + a, as we have just seen, not a · a · a. Therefore there is another abbreviated notation, namely the number of equal factors in a product noted at the top right of the letter: a · a · a = a³. The superscript 3 is also called the exponent or exponent. a³ reads "a to the power of three". In addition to "a to the power of two", a² is also said to be "a squared" or simply "a square".

Is it just about a Product, the variables can be summarized by exponents in a similar way to addition:

a · a · a · a = a4 a * a * b * b * b = a2· B3 x y y x x y z = x2· Y3· Z

In the last example the variables are "mixed up". You can still summarize them, because the commutative law applies, i.e. you can first rearrange the factors. With this intermediate step, the last example would be

x y y x y z = x x x y y y y z = x2· Y3· Z

If the product contains numbers, these are extracted, calculated and written to the front. (The commutative law also applies to this "type of multiplication", i.e. the factors may be rearranged even if numbers and variables are mixed.)

4a * 5a = 4 * 5 * a * a = 20a2 x³ x x² = x x x x x x x x = x5 x³ x 4y x x x 5z² x 7 = 4 x 5 x 7 x x3X x y z2 = 140x4·Y Z2

In such products, the painting dots are usually left out, i.e. instead of 140x4·Y Z2 you write 140x4Y Z2.

Caution: A very "popular" source of error is confusing factor and exponent. Memorize the difference very well! The number in front of the letter is a number and actually means that the variable so often added becomes. In contrast, the exponent indicates how often the variable multiplied becomes.

As a rule x is7 not the same as 7x, because 7x = x + x + x + x + x + x + x, but x7 = x x x x x x x x x x x. You just have to think of some number for x and calculate it with it to see the difference. For example, take the number 2 for x, then 7x is 14, and x7 = 27 = 2·2·2·2·2·2·2 = 128.

x x x x x Not equals 3x (even if there are three x). In this case, get used to talking and thinking about "x times x times x" or rather "x to the power of three" instead of "three x"!

In general, one can check its simplifications by inserting the same numbers for all the same variables and calculating both the given term and the simplified term. If both results agree, the simplification should be correct in most cases (although an example is not a proof !!!!), but if they do not agree, the simplification was certainly wrong.

Products from different variables cannot be simplified any further. Instead, you can at most, or write.

Summing up mixed summands

In expressions such as, which have different "combinations" (better products) of variables and exponents in their summands, only those may be summarized that exactly match in all variables and associated exponents.

xy + 3x²y - 5xy + 7xy² + 3xz = -4xy + 3x²y + 7xy² + 3xz from + bc + ac + abc = from + bc + ac + abc (no simplification possible!) a²x + 2a³x² - ax + 2ax + 7a²x = 8a²x + 2a³x² + ax
Brackets

In principle, parentheses with variables are resolved in the same way as parentheses containing only pure numbers. The main difference is that most of the time you can't really "calculate" the content of the brackets, but here too you start with the simplifications in the innermost brackets.

Number or variable or product times brackets
Each summand in brackets is multiplied by the number (or the variable / product) before (after) the bracket (distributive law). Note the sign rules!

4 · (a + 5b - 2c²) = 4a + 20b - 8c² x · (a + 5b - 2c²) = ax + 5bx - 2c²x (alphabetical order!) -3a · (a + 5b - 2c²) = -3a² - 15ab + 6ac² (x + 3y - z²) x 2 = 2x + 6y - 2z² (x + 3y - z²) x 2yz = 2xyz + 6y²z - 2yz³

Plus bracket
The brackets can simply be left out if there is a plus directly in front of the bracket (without a number!):

3 + (a + 5b - 2c²) = 3 + a + 5b - 2c² 3a + (a + 5b - 2c²) = 3a + a + 5b - 2c² = 4a + 5b - 2c²

Minus bracket
All signs in brackets are reversed, the brackets and the minus in front of the bracket are omitted:

3 - (a + 5b - 2c²) = 3 - a - 5b + 2c² 2b - (a + 5b - 2c²) = 2b - a - 5b + 2c² = -a - 3b + 2c² 4a - (-a + 5a² - 7c³ ) = 4a + a - 5a² + 7c³ = 5a - 5a² + 7c³ (Caution: 5a and -5a² do not cancel out because of the different exponents!)

Bracket times bracket
Each addend in the first bracket is multiplied by each addend in the second bracket (those with a negative sign are also considered as addends). Attention: Note the sign rules!

(3a + 4) · (x - 7y) = 3ax - 21ay + 4x - 28y (2a - 3b + c²) · (5x³ - 7y) = 10ax³ - 14ay - 15bx³ + 21by + 5c²x³ - 7c²y (2a - b) · (3a + 5b) = 6a² + 10ab - 3ab - 5b² = 6a² + 7ab - 5b² (4x - 5y) * (5y + 4x) = 20xy + 16x² - 25y² - 20xy = 16x² - 25y²

If you have three brackets, you first combine a pair, which you leave in brackets, and then multiply with the third bracket.

(x + 2) * (3a - b) * (2a - x) = (3ax - bx + 6a - 2b) * (2a - x) = 6a²x - 3ax² - 2abx + bx² + 12a² - 6ax - 4ab + 2bx

An exponent in brackets means the same as with numbers: The brackets must be taken as often as the exponent indicates.

(x - y) ² = (x - y) x (x - y) = x² - xy - xy + y² = x² - 2xy + y² (2a + 3b) ² = (2a + 3b) x (2a + 3b) = 4a² + 6ab + 6ab + 9b² = 4a² + 12ab + 9b² (5x - 8y) ² = (5x - 8y) · (5x - 8y) = 25x² - 40xy - 40xy + 64y² = 25x² - 80xy + 64y² (Such expressions can can be solved directly with the binomial formulas) (a - b) ³ = (a - b) · (a - b) · (a - b) = (a² - ab - ab + b²) · (a - b) = ( a² - 2ab + b²) * (a - b) = a³ - a²b - 2a²b + 2ab² + ab² - b³ = a³ - 3a²b + 3ab² - b³

Further examples

3x² - 3x - 4x · (3 - x) = 3x² - 3x - 12x + 4x² = 7x² - 15x -a · (2a + 3b) ² - (a - b) ³ = -a · (4a² + 12ab + 9b²) - (a³ - 3a²b + 3ab² - b³) = -4a³ - 12a²b - 9ab² - a³ + 3a²b - 3ab² + b³ = -5a³ - 9a²b - 12ab² + b³ 3 · (x - y) ² - ((5 + y²) - x² · 11) (compare very first example) Calculation not possible. 4 = (x - y) ² * (x - y) ² = (x² - 2xy + y²) * (x² - 2xy + y²) (see above) = x4 - 2x³y + x²y² - 2x³y + 4x²y² - 2xy³ + x²y² - 2xy³ + y4 = x4 - 4x³y + 6x²y² - 4xy³ + y4

© Arndt Brünner, September 29, 2003
Version: December 28, 2004