Scientific Notation Multiplication

Introduction to scientific notation multiplication:
                     In scientific notation multiplication, a number is said to be written in Scientific notation if  it is expressed as m×10⁴ where b is a terminating decimal such that 1< m < 10, and n is an integer. Scientific notation multiplication is used to convey very large numbers. A number in scientific notation multiplication is printed as the product of a number either it is integer or decimal or it is a power of 10. The numbers have one digit to the left of the decimal point. The decimal of power that ten indicates how many places the decimal point was moved.

Examples for Scientific Notation Multiplication:

Scientific notation:

      To achieve the exponent scientific notation calculate the number of places from the decimal to the end of the number.

       The term 9, 99, 00,000,000,000 there are 11 places.

       Hence can we write the term 9, 99, 00,000,000,000 as?

                            999×10¹¹

   Example:   77700000  

1)      777×10⁵

Scientific notation multiplication: 

Make a note of (a^m.a^n) as of the note the example problem have similar base.

    Example 1: (7 x 10²) * (5 x 10³)

                           = 35 x 10⁵

            Move the decimal point over to the right until the coefficient lies between 1 and 10. For each place shift the decimal in excess of the exponent will be lowered 1 power of ten.

               35x10 ⁵= 3.5 x 10⁶in scientific notation multiplication.

     Example 2: (4x 10²) * (2 x 10²)

                            =8 x 10² x10²

                             =0.8 x10⁵        

         Move the decimal point over to the right until the coefficient lies between 1 and 10. For each place shift the decimal in excess of the exponent will be lowered 1 power of ten.

     Example 3:

         Write the following in the scientific notation 56.

     Solution:

       We can write 56 in scientific notation as follows,

       0.56 x 10² , 5.6 x 10¹ , 56 x 10⁰ etc..
 

Practice Problem for Scientific Notation Multiplication:

Problem 1: (8x 10⁶) `xx` (2 x 10³)

                            = 16 x 10⁹

Problem 2: (2 x 10 ⁸)`xx` (7 x 10³)

                           = 0.14 x 10¹³

 Problem 3: (64x 10⁶)`xx` (8 x 10³)

                           = 512 x 10⁹

 Problem 4: (16x 10⁶) `xx` (8 x 10³)

                           =128x10⁹

Problem 5: (1x 10⁶) `xx` (7 x 10³)

                            =0.7x10¹⁰

Square Root Property Calculator

Introduction for square root property calculator:       

     When calculating square root of any real numbers, we can use a square root property calculator to support your answer. The square root representation in mathematically shown by   √. The square root property calculator is used to calculate the real value that the square root is calculated from inside of the square root symbol. The radicand is a real number; it is inside the square root symbol. The square root property calculator used to manipulate the square root value of the form is `sqrt(x)` where x is radicand that any real numbers.

Square Root Properties:

1.   Product Property of Square Roots:

          Let us take any real numbers a and b, where a ≥ 0 and b ≥ 0, the square root of the product a and b is same as the product of each square root.

       `sqrt((a).(b))` = `sqrt(a)` .`sqrt(b)`   For  example   `sqrt((5)(7)(4))` = `sqrt(5)` .`sqrt(7)` .`sqrt(4)`

2.    Quotient Property of Square Roots:

        We consider any  real numbers a and b, where a ≥ 0 and b > 0, and then the square root of the quotient  `a / b` is equal to the quotient of each square root.

           `sqrt((a)/(b))`   = `sqrt(a)`  `-:` `sqrt(b)`

Square Root Property Calculator :

 The common procedure of using square root property calculator are,

       Step 1:   Evaluate `sqrt(4)`

                       In square root property calculator, KEYSTROKES :   [ √ ]4  2

       Step 2:  To find a root other than a square root, choose the ^x√   function from the   menu.

Example:

     Evaluate the following expressions by using the square root property calculator.

Solution:

           In square root property calculator, we perform the manipulation of the expression.

1. Evaluate: `root(2)(144)`

                        Keystrokes:    [ √ ] 144  12

2. Evaluate: `root(2)((8)(2))` 

                        Keystrokes:  [ √ ]  82   4

3. Evaluate:  `root(2)(625)` 

                        Keystrokes :    [ √ ] 625   25

4.Evaluate:  `root()(((16)(7^2)))` 

                         Keystrokes:   [ √ ] 16 x 7²  28

5. Evaluate:  `root()(1089)` 

                          Keystroke :  [ √ ] 1089  33

6. Evaluate:  `sqrt(676)`

                          Keystrokes :  [ √ ] 676   26

Tabulate the above expressions values by using square root property calculator.

Expression	Value
`root(2)(144)`	12
`root(2)((8)(2))`	4
`root()(625)`	25
`root()((16)(7^2))`	28
`root()(1089)`	33
`root()((676))`	26

Dealing with Decimals

Introduction:
                  Decimal: A dot in between a decimal number, as like present in 85.937, to point out the position where values alter from positive to negative orders or powers of 10. Dealing with decimals, related with the operations which are said to be as addition, subtraction, multiplication and division.

Important while Dealing with Decimals:

Addition and Subtraction - when dealing with decimal, we require keeping in mind the rules for adding or subtracting you need to be careful while placing the decimal point in the same column as it is placed above.

Comparison - In case of dealing with decimal numbers first we have to think the whole number section. If the number has further number of digits in the left part of the decimal then it is without a doubt that is the larger number. On the other hand the case of equal whole number, we have to evaluate the integer towards the right of the decimal.

Multiplication - In case of dealing decimals with multiplications, we should not include the decimal points in the account at the beginning. Simply continue with the multiplication like any other multiplication. All we have to do is to total up the decimal places which are present from the right and place the decimal point in appropriate places. The count starts from the right hand side to the left.

Division - If the decimal dealing with division then the dividend allows a decimal and the divisor does not, then our requirement to do is put the decimal place in the quotient accurately on top of the decimal point in the dividend.

Example Problems for Dealing with Decimals:

Dealing decimal with addition:

Add the decimals 4.5 and 4.4

     4.5
 +  4.4     8.9

Add the decimals 5.61 and 4.2

     5.61
+  4.20
    9.81 

Dealing decimal with subtraction:

Subtract the decimals 3.5 and 2.2 

3.5

-2.2

--------

1.3 

Subtract the decimals 5.81 and 5.7

  5.81

-5.70  

  .11

Dealing decimal with Multiplication:

1)    4.36

    x 0.8

    ------

      3488 answer before adding the decimal point.

     3.488 answers - it have 3 (three) decimal positions because from the given original number, the result is obtained.

 2)    0.25

   x 0.125

   -------

      3125 is the answer without decimal point.

    .03125 answers - it have 5 (five) decimal positions because from the given original number, the result is obtained.

Dealing decimal with division: 

A. 10/100 = ten divided by hundred.

              .10

            -----

    100) 10.0

            10 0

             ----

    So 10/100 = .10  

B. what about 3/9 or 3 divided by 9?

             .5

           -----

        6) 3.0

            3 0

            ----

    So 3/6 = .5 (five tenths)

Mix Word Problems Math

Introduction :

Two different quantities having different percentage of needed substances can be mixed together to get the desired amount. This is called mix word problem in math. For example: Let X and Y be two different solutions. Each are having different percentage of Salt in it. Those two X and Y can be mixed together to get the required percentage of the final solutions.

Now let us see few problems of this kind.

Example Problems on Mix Word Problems Math:

Ex 1:  Two salt solutions having 12% and 36% of salts respectively in them. They are added together to form 40% of salt solution. If 280 liters of 12% salt solution is added, how many liters of 36% has to be added?

Soln: Let x be the salt solution.

Given: `36/100`   x   +   `12/100` `xx` 280 = `40/100`    (X + 80)

`=>` 36x + 3360 = 40x +3200

`=>` 40x – 36x = 3360 – 3200

`=>` 4x = 160   `=>` x = 40 liters.

Ex 2: Two oils have to be mixed together. One of the oil has 15% of the required content and the other has 30% of the required content. How many liters of 30% of oil have to be added to 290 liters of 15% of the oil, so that we can have a mix of 42% of the oil content?

Soln: Let X be the amount of oil with 30% content in it.

Given: `30/100` (x) + `15/100` (290)   =   `42/100` (x + 90)

`=>` 30x + 15`xx` 290 = 42x  + 42`xx` 90

`=>` 12x = 4350 – 3780

`=>` x = `570/12` = 47.50 liters

More Example Problem on Mix Word Problems Math:

Ex 3: Two herbals have to be mixed together. One of them has 25% of the required vitamins and the other has 35% of the required vitamins. How many liters of 25% of herbal have to be added to 120 liters of 45% of herbal, so that we can have a mix of 40% of vitamins?

Soln: Let x be the amount of herbal with 25% vitamin in it.

Given: `25/100` (x) + `45/100` (120) = 40/100 (x+120)

`=>` 25x + 5400 = 40x + 4800

`=>` 15x = 5400 – 4800 = 600 `=>`x = `600/15`

x = 40liters