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2-6-3 Exponent Rules

By Alicia Mason,2014-05-07 14:58
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2-6-3 Exponent Rules

    2-7-1 Number Properties

Properties are the rules of the game. If the rules are changed a different kind of math is played.

    These properties are for everyday math.

    Commutative a + b = b + a 7 + 8 = 8 + 7 The commutative property says the order property of addition. (-3)+4=4+(-3) doesn’t matter for addition. Commutative property of a x b = b x a 7 x 8 = 8 x 7 The commutative property says the order

    multiplication. (-3)4=4(-3) doesn’t matter for multiplication.

Associative property (a+b)+c = a+(b+c) (7+8)+2=7+(8+2) The associative property says the

    of addition. 15+2 = 7 + 10 grouping doesn’t matter for

     addition. Associative property (axb)xc=ax(bxc) (7x8)x2=8x(7x2) The associative property says the

    of multiplication. 56x2=8x14 grouping doesn’t matter for

    multiplication.

An identity for a particular operation doesn’t change the identity of the number when the operation is

    done.

    7 + 0 =7 Additive Identity a+0=a Zero plus a number is that number.

    8x1=8 Multiplicative Identity 1xa=a One times a number is that number.

An inverse for a particular operation is the number that returns the identity.

    7 + (-7) =0 Additive inverse a + (-a) =0 The additive inverse is the negative of

    -7 is the the number. additive inverse

    of 7

    3 x 1/3 = 1 Multiplicative inverse The multiplicative inverse of a rational 1??a??1 ??number is it’s reciprocal. a??

     23x(4x)=3(4)xx=12x using both the commutative and associative properties of multiplication.

Distributive a(b+c)=ab+ac Multiplying a sum by some number ??53?4?5?3?5?4

    property is the same as multiplying each term by that same number. 2??3x2x?8?6x?24x

Practice: List the property illustrated by the following example.

    a) 5+0=5 R+6=6+R 3+(7+8)=(3+7)+8 8+(-8)=0

b) 4t(7t8t)= (4t7t)8t 0+t=t 2z+7=7+2z ??4?9?2?4?(?9)?4?2

c) xy=yx 3x=3x+0 1x=x 7(2x-8)=14x-56

    34123451????????9t?12?12t?16 d) m?m4??1 2?1?1?1??????44434565??????

     2e) -8d+8d=0 0-7y=-7y 15y+10=5(3y+2) -2e(3e-7)=-6e+14e

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    2-7-2 Distributive Property

To illustrate the distributive property look at the problem 3(2x-7).

    This problem says 3 times 2x-7 or 3 of (2x-7). Multiplying by 3 means add 2x-7 over and over.

    2x-7+2x-7+2x-7 Simplify this by using the commutative property. 2x+2x+2x-7-7-7=3(2x)+3(-7)

To use the distributive property, multiply each term inside the parenthesis by the multiplier. Study the

    following examples. Look where exponent rules are used. Watch the multiplication when negatives are

    involved. After the distribution, combine any like terms.

    Examples:

    2223 5381540xx???3762118xxxx????????4652420zzzz??????

     ?????5858ss42394618218xxxxx????????????

    2323325 3752115wywywywy???53653626???????xxx????

    There can be more than two terms inside the parenthesis.

    2222 ???????3468121824dddd5759352545xxxx?????????

    222332324 6523301218xyxyxyxyxyxyxy???????

    33233232 37563756876ghghghghghghghghghghgh????????????

    222216121612xxxx?16121612xxxx?22 ????43xx?????43xx

    444???444

    35435421525751525755wwwwwww?? ??????15w3333151515153wwww

    Practice:

    a) 325x??2523y??78x????????

    b) 538??x337??x10025??x??????

    c) ???42x???58x???42y??????

    d) 438x?????10515x???278y??????

    2e) 576yy?????876hhkk125????????3

    22321 ???45mm???zz610f) ???365jjj ????2??

    2 ???412l???u8g) ????958aa??????

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    h) 339xx???????9258hh975xx?????????

    53235283i) 272yzyz??mnmn85??vw??????41231215vwv?? ??5??

    72j) 859???x539???x1069???x??????38

    k) 8539???x????955x13739???xh??????

    222l) ????8265xx9368ww???????qq68??????

    2223m) 4104???yy1559xx????????5781uu??????5

    322322515pp?xx?12n) 2xyxyxyxy????? ??

    5?1

    222221612xx?aa?2o) uvuvuvuv?????? ? 12?25

    22322222642612ee?1010xx?p) stststst???3??335 ??5?1006

    33233232q) ghghghgh????xyxyxyxy????3????34

2222223232733r) 4105abghghgh????vwvwvwvw????????1288

    35835125302118vvv??152575hhh??s) ??5313v?5h

    35454715575uuu??35k?21k?77k ?

    3ut) 3?7k

    534573534573321ijijij??32112xyxyxy????u) 2352?3ij?9xy

    4132555334473321efefef??5wvwvwv??v) ?? 255?efwv

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    2-7-3 Factoring

Assume 12x+6 is the result of distribution. What did it look like before the distribution was done?

    There was something outside a set of parenthesis and two terms inside the parenthesis.

     ( + )

    6 ( 2x + 1 )

    Separating an expression so that the smallest possible pieces called factors multiply together to get the

    original expression is called factoring.

Look at each term and find the largest factor that is in all terms.

    2Example:726036xx?? Find the largest number that divides evenly into 72, 60 and 36.

    2This number goes in front of the parenthesis. 726036xx , , 121212Divide each term by the common factor.

     12 ( + + ) Fill in the positions in the parenthesis. Distribute 2to check the factoring. 12 ( 6x + 5x + 3 )

    2222 The largest factor in all terms is 8xy. When the division is done there 241656xyxyxy??

    shouldn’t be any negative exponents. Notice the negative in the third term.

    Divide each term by the common factor.

     8xy ( + - ) 2222 241656xyxyxy?, , 8xy ( 3xy + 2x - 7y ) 888xyxyxy

    The division can be done in your head, but some students need to write it down.

Practice: Factor the following.

    a) 15y-25=5(3y-5) 14z+56= 39u-13= 81p-18= 7t+21=

    b) 8+8t= 14-21w= 9v-81= 2x-10= 3w+9=

     2102c) -/y-/=-/(y+5) 4y81015r5r264333y?? = = ?? ??2525Note the negative. 55121233

     222222d) 6x-9x+12= 3(2x-3x+4) 14e+21e+56= 2x-4x-8= 8a+4a+4= 10b+5b-25

     22222e) 3+3w+3w= -5-5t-5t= -8+16e-24e= 12-24y+36y= 15r+15r-25

    f) ax+2a=a(x+2) bx-7x= 3s-3ts= 4ef+3f= 2w-wx=

     22222g) 4x-12x=4x(x-3) 14y+21y= 81z+18z= 10q-20q= 8a-8a=

     222222232h) 2ax+4ax-8a= 12xy+16xy+24x= ax+ax+a= 2ax-4a= e-5ex=

     32222322422325 i) 3xy+9xy-12xy=3xy(x+3-4y) 25ab+45ab-15ab=

    43

    75875443223ababab???5j) 24efefef???

     422322362544 k) 25ab + 75ab - 100ab 21xy+14xy-28xy

    2aax1521231? l) Hint: Factor out a ?. x??44884

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