Smart Math exponential number and Operations powers of integer arithmetic

Smart Math in exponential number.
quadratic terms of a number. Squared or the square of a number is multiplying a number by the number itself.Furthermore, the powers of a number means multiplying repeated with the same number. 
Note the following powers of prime number 2. 

2 ² = 2 x 2 (2 ² read 2 squares or 2 to the power 2)= 42 ³ = 2 x 2 x 2 (2 ³ read 2 to the power 3)= 8 

Smart Math Solution:
a. 9 ² = 9 x 9 = 81b. (-6) ³ = (-6) x (-6) x (-6) = 36 x (-6) = -216c. -5 = - (5 x 5x 5 x 5) = -625d. (-10) 4 = (-10) x (-10) x (-10) x (-10) = 10,000 Properties exponential number.

a. Multiplicative nature of the exponential number Note the following rank of integer multiplication.3 ² x 3 ³ = [3x3] x [3x3 x3]Means [3x3] = 2 factors and [3x3x3] = 3 factors So there are 5 factors add up, so be 3x3x3x3x3 

b.Nature of the distribution of exponential number Note the following integer division rank.
5 ³: 5 ² = [5x5x5]: [5x 5] the meaning [5x5x5] = 3 factors and [5x 5] = 2 factors So there is 1 factor is reduced, so that a 5
Smart Math Operation Count Mixed Integer 
In completing integer arithmetic operations, there are two things to note, that1. sign arithmetic operations;2. brackets.If in a mixed integer arithmetic operations are bracketed, workmanship are in brackets should be done first. 

If in an integer arithmetic operation there is no parentheses, the process is based on the properties of the following arithmetic operations.1. Operations of addition (+) and subtraction (-) as strong, meaning that operations are located on the left side done first.2. Multiplication (x) and division (:) as strong, meaning that operations are located on the left side done first.3.  

Multiplication (x) and division (:) is stronger than the addition operation (+) and subtraction (-), multiplication means (x) and division (:) done first rather than the addition operation (+) and subtraction (-).
Example: 
a. 24 + 56 x 42-384 : 12= 24 + (56 x 42) - (384: 12)= 24 + 2352-32=       2376-32= 2,344 

b. 28 x (364 + 2,875): (9756-9742)= 28 x 3,239: 14= 90 692: 14= 6,478
c. 80: ((11-7) x (-4))= 80: (4 x (-4))= 80: (-16)= -5

d. (-8 + 5) x (36: (6-9))= -3 X (36 (-3))= -3 X (-12)= 36 

Thus example Smart Math exponential number and Operations of integer arithmetic, may be useful for the reader.

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SMART MATH Calculate The Numbers Round Operation

SMART MATH Calculate The Numbers Round Operation 
a. Summation with tools.

In calculating the sum of two integers, can used by using the number line. Numbers are summed illustrated with arrows corresponding to the direction with the number.If the positive numbers, arrows pointing the right direction. Conversely, if the negative number, the arrows pointing in the direction of left. 
1. 6 + (-8) 
2. (-3) + (-4) 

b. Summation without tools 
The sum of the numbers of little value can be done with the help of a number line. However, for the numbers of great value, it can not be done. Therefore,we should be able to add up the integers without tools.1) Both numbers marked with the same.

If both marked with the same numbers (both numbers positive or both negative numbers), totalizing both numbers. The result is the same mark with the mark the two numbers. 

SMART MATH Calculate The Numbers Round Operation

Example: 

a) 125 + 234 = 359

b) -58 + (-72) = - (58 + 72) = -1302) 

The two numbers opposite sign.

If the two numbers opposite sign (positive numbers and negative numbers), reduce the number of greater value with a smaller number of value without notice sign. The result, according tick numbers are worth greater.
 

SMART MATH Calculate The Numbers Round Operation

Example: 

a) 75 + (-90) = - (90-75) = -15
b) (-63) + 125 = 125-63 = 62
a. 6 + 5 = 5 + 6 = 11
b. (-7) + 4 = 4 + (-7) = -3
c. 8 + (-12) = (-12) + 8 = -4
d. (-9) + (-11) = (-11) + (-9) = -20

c.Has the identity element.

Number 0 (zero) is the identity element in summation. That is, for any integer when added to 0 (zero), the result is the number itself. It can be written as follows. For any integers a, always applya + 0 = 0 + a = a.

 d.Associative properties.

Associative properties of nature also called grouping. This trait can be written as follows.For any integers a, b, and c, apply(A + b) + c = a + (b + c).  

Properties Integer Additiona.Closed nature,On the sum of integers, always produces integers as well. It can be written as follows. For any integers a and b, apply a + b = cwith c is also an integer. 

a. -16 + 25 = 9-16 And 25 are integers.9 is also an integer. 

b. 24 + (-8) = 1624 and -8 are integers.16 is also an integer. 

b.Commutative propertiesCommutative properties called exchange properties. Addition two integers always obtained the same results although both numbers are exchanged place. It can be written as follows.For any integers a and b, always apply 

a + b = b + a.
 a. 6 + 5 = 5 + 6 = 11 
b. (-7) + 4 = 4 + (-7) = -3 
c. 8 + (-12) = (-12) + 8 = -4 
d. (-9) + (-11) = (-11) + (-9) = -20c. 

Has the identity element Number 0 (zero) is the identity element in summation. That is, for any integer when added to 0 (zero), the result is the number itself.It can be written as follows.For any integers a, always apply a + 0 = 0 + a = a.d.Associative properties Associative properties of nature also called grouping. This trait can be written as follows.For any integers a, b, and c, apply(A + b) + c = a + (b + c).

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Smart Math in Multiplication Algebra

Smart Math in Multiplication  Algebra ShapeNote back in shape algebra distributive properties. Distributive propertiesis the basic concept of multiplication in the algebra. For more details,learned the following description.The multiplication rate with Two PartsIn order for you to understand the multiplication rate of the two-rate form of algebra,learn about the following example. 

Use the distributive law to complete the following multiplication.a. 2 (x + 3)
b. -5 (9 - y) 

c. 3x (y + 5)

d. -9p (5p - 2q)Answer:a. 2 (x + 3) = 2x + 6
b. -5 (9 - y) + 5y = -45
c. 3x (y + 5) = 3xy + 15x

d. -9p (5p - 2q) =-45p2 + 18pq

Smart Math in factoring with Distributive Properties In elementary school, you would have learned how factoring a numbers. Remember you regarding the matter? Basically,factoring a number means a number expressed inform of multiplication factors. In this section, will be studied in ways factoring an algebraic form by using the distributive properties.With these properties, the form ax + ay algebra can be factored into a (x + y),where a is a factor of ax and ay fellowship. Therefore, study 

Example Problem 

Factor the following algebraic forms.a. 5ab 10b + c. +-15p2q2 10pqb. 2x - 8x2yAnswer:a. 10b + 5abFor factoring 5ab + 10b, specify fellowship factor of 5 and10, then from ab and b. Fellowship factor of 5 and 10 is 5.Factors fellowship of ab and b is b.So, 5ab factored into 5b 10b + (a + 2).b. 2x - 8x2y  

Fellowship factor of 2 and -8 is 2.Factor of x and x2y fellowship is x.Thus, 2x - 8x2y = 2x (1 - 4xy).c. +-15p2q2 10pqFactors fellowship of -15 and 10 is 5.Factors fellowship of p2q2 and pq is pq.So-15p2q2 10pq + = 5pq (-3pq + 2). 

Smart Math in Difference Two Squares Notice the multiplication (a + b) (a - b). This form can be written(a + b) (a - b) = a² - ab + ab - b ²= a² - b ²So, the form a2 - b2 can be expressed in terms of multiplication (a + b) (a - b).a² - b² = (a + b) (a - b)Forms a² - b ² is called the difference of two squares.Factor the following forms.a. p² - 4c. 16 m² - 9N ²b. 25x² - y ²d. 20p² - 5q²Answer:a. p2 - 4 = (p + 2) (p - 2)b. 25x² - y² = (5x + y) (5x - y)c. 16m2 - 9n2 = (4m + 3n) (4m - 3n)d. 20p ² - 5q ² = 5 (4p ² - q ²) = 5 (2p + q) (2p - q) 

Smart Math in factoring quadratic form ax ² + bx + c with a = 1Note the multiplication of the following two parts.(X + p) (x + q) = x² + px + qx + pq= X² + (p + q) x + pq So, the form x² + (p + q) x + pq can be factored into (x + p) (x + q).Supose, x² + (p + q) x + pq = ax ² + bx + c so a = 1, b = p + q,and c = pq.Of the example, it can be seen that p and q are factors of c. If p and q are summed, the result is b. Thus for factoring the form ax ² + bx + c with a = 1, specify two numbers a factor of c and if the two numbers are added together,the result is the same as b.So that you may better understand the material, study the following example problems:

Fraction following forms.

• a.) x² + 5x + 6 
• b.) ax ² + 2x - 8

Answer:a. x ² + 5x + 6 = (x + ...) (x + ...)Suppose, x ² + 5x + 6 = ax2 + bx + c, obtained a = 1, b = 5, and c = 6.To fill in the blank, specify two numbers is a factor of 6and if the two numbers are added together, the result is equal to 5.Factor of 6 is 6 and 1 or 2 and 3, which is 2 and a qualified So, x ² + 5x + 6 = (x + 2) (x + 3) 

b. x2 + 2x - 8 = (x + ...) (x + ...)By way as in (a), obtained by a = 1, b = 2, and c = -8.Factors of 8 are 1, 2, 4, and 8. Therefore c = -8, one of the two numbers is sought must be negative. Thus, the two Eligible numbers are -2 and 4, because -2 × 4 = -8 and-2 + 4 = 2.So, x ² + 2x - 8 = (x + (-2)) (x + 4) = (x - 2) (x + 4)Factoring Form ax ² + bx + c with a ≠ 1Previously, you had factored form ax2 + bx + c with a = 1.Now you will learn how to factor in the form ax2 + bx + c with a ≠ 1.Note the multiplication of the following two parts.(X + 3) (2x + 1) = 2x + x ² + 6x + 3= 2x ² + 7x + 3In other words, the form 2x2 + 7x + 3 be factored into (x + 3) (2x + 1).As for how factoring 2x ² + 7x + 3 is the reverse phase binomial multiplication above.2x ² + 7x + 3 = 2x ² + (x + 6 x) +3= (2x ² + x) + (6x + 3)= X (2x + 1) + 3 (2x + 1)= (X + 3) (2x +1)

From the description you can know how to factor in the forma x ² + bx + c with a ≠ 1 as follows.1) Describe bx be the sum of the two tribes that if the two tribes    The same result multiplied by (ax ²) (c).2) Factor the form obtained using distributive properties

This Mathematics in Smart Math in Multiplication  Algebra

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Smart Math Algebra factorization in Mathematics School

Smart Math Algebra factorization in Mathematics School 
To remind again about Algebra? In Class grade 8, you have familiar form of algebra and arithmetic operations have also been studied in The algebraic form. Now, you will add to your knowledge of the algebra, in particular regarding the factorization algebra.Why do you need to learn algebra? Possible You do not realize that the concept of algebra is often used in everyday life.Each day, Nita save X amount of dollars. How big savings the child after one week? How much greater the savings after one month? After 10 days, saving money that bought two books that cost y dollars, what is the rest of the money savings Mark?When you are looking for a settlement of the case, then you're using algebraic concepts. Therefore, study the chapter The well Calculate Operation Algebra Shape In class grade 8, you have studied the terms of shape algebra, coefficient,variables, constants, tribal, and similar parts. To remind you again,learned the following examples. 
• 2PQ          • 5x + 4           • 2x + 3y -5           • x2 + 3x -2           • 9x2 - 3xy + 8 
Form of algebraic number (1) is called a single tribe or a tribal one because only consists of one syllable, is 2PQ. On the algebraic form, called 2coefficient, while p and q are called variable because the value of p and q can change able. The form of algebraic numbers (2) is called the binomial as The algebraic form has two parts, as follows.a. Parts containing variable X, the coefficient is 5.b. Tribes that do not contain the variable X, namely 4, called constants. Constants is a tribe whose value does not change.Now, in the form of algebraic numbers (3), (4), and (5), let you specify Which is the coefficient, variables, constants, and tribes?
Smart Solution Math Algebra factorization in Mathematics School Addition and reduction Algebra Shape In this section, you will learn how to add up and subtract similar tribes in the algebra. Basically, the properties of summation and reductions in force at the real numbers, applies also for addition and subtraction on algebraic forms, as following.a. Commutative properties + b = b + a, with a and b real numbers b. Associative properties(A + b) + c = a + (b + c), with a, b, and c real numbers c. Distributive properties (b + c) = ab + ac, with a, b, and c real numbers So that you may better understand the properties that apply to the algebra,Example: Question [Q] and Answers [A]Simplify the following algebraic forms Q. 6mn + 3mnA. 6mn + 3mn = 9mnQ 16x + 3x + 3 + 4A. 16x + 3 + 4 + 3x + 3x = 16x + 3 + 4
     = 19x + 7Q. - X - y + x - 3A. - X - y + x - 3 =-x + x - y - 3
     = - Y - 3Q. 2p - 2q + 3P2 - 3p + 5q2A. 2p - 3P2 + 2q - 5q2 + 3p = 2p + 3p - 3P2 + 2q - 5q2
     = 5p - 3P2 + 2q - 5q2
     =-3P2 + 5p - 5q2 + 2qA. 6m + 3 (m2 - n2) - 2m2 + 3N2Q. 6m + 3 (m2 - n2) - 2m2 + 6m + =
     3n2 3n2 - 3n2 - 3n2 = 2m2 +
     6m + 3m2 - 2m2 - 3n2 + 3n2 =
      m2 + 6m
This  Smart Math Algebra factorization in Mathematics School

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Smart Math Methods Comparison of Temperature

Smart Math Methods Comparison of Temperature
Size Comparison Using Temperature in Celsius, Fahrenheit and Reamur
Those required things considered in comparison of the size of the Celsius temperature, Reamur, Fahrenheit:

1. Denoted by C centigrade thermometer, thermometer Reamur symbolized by R, denoted by F.    Fahrenheit thermometer
2. Comparison of C: R: F [+32] = 5: 4: 9
3. When calculating the temperature in Fahrenheit should be added 32 °
4. If it comes to calculate the temperature in Fahrenheit to Celsius and 32 ° Reamur be reduced first.

example:

• 60 ° C = ...... ° R = ...... ° F 

Smart solution :

R = 4/5 X 60 ° = 48 °
F = 9/5 X 60 ° + 32 ° = 140 °
So 60 ° C = 48 ° R = 140 ° F

• 50 ° F = ..... ° C = ...... ° R

Smart solution :
C = 5/9 x [50 ° - 32 °] = 5/9 X 18 ° = 10 °
R = 4/9 x [50 ° - 32 °] = 4/9 X 18 ° = 8 °
So 50 ° F = 10 ° C = 8 ° R

• 25 ° C = ..... ° R = ..... ° F

Smart solution :
25 ° C = [4/5 X 25 °] R
4 X 5 = 20 ° R
25 ° C = [9/5 X 25 ° + 32 °] F
[9 X 5] = + 32 ° 77 ° F

• thermometer shows a temperature of 45 ° Celsius. What is the temperature indicated by a thermometer Reamur and Fahrenheit?

Smart solution :
C = 45 °
C: R: F = 5: 4: 9
R = 4/5 X 45 ° = 36 °
F = 9/5 x 45 ° + 32 ° = 113 °
So 45 ° C = 10 ° R = 113 ° F

Use Easy Methods Comparison of Temperature Although social math still using way as before.

• Temperature american city yesterday 18°C. Today it dropped 3°C. Temperature American city today If Reamur measured using a thermometer and Fahrenheit, is ....

Smart solution :

Temperature american city  = 18°C

Decrease in temperature  = 3°C

Temperatures today  =  18°C - 3°C = 15°C

Measurement with a thermometer Reamur

15°C = 4/5 X 15°R = 12°R

15°C =  [ 9/4  X 15 + 32 ] F = [ 27 + 32 ] F = 59°R

You can do Test below:

• Temperature of a meat storage room that is not being used is 25°C.After cooling the machine is turned on for one hour, the room temperature drops by 29°C. so that the temperature reached -16°C, how many degrees the temperature of the room again be lowered ?Answer is 12°C (you prove)
• CelsiusThermometer shows the temperature of 285°. how high the temperature in the show by Reamur and Fahrenheit thermometer ?
• Reamur thermometer shows a temperature of 260°. What is the temperature indicated by a thermometer celcius and fahrenheit?

• daytime temperatures in city A 31°C, at night dropped 12°C. when the morning temperature rises 7°C.What is the temperature in the city in the morning? (try it yourself in your Smart Easy Methods Comparison)

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