Factor out Square Roots

The square root is the radical form of the method that are used in the calculation mathematical problems. The symbol for the square root is meant by root √. This could be in the form to describe their nature of working with the square roots. There are numerous methods are available in the rooting, they are square (second) root `sqrt(x)` , cube (Third) root `root(3)(x)` up to nth root `root(n)(x)` .  Here we are going to see about how to factor out square roots and solved example problems based on the factor out square roots.

Some Examples for Square Root Operations:

`sqrt(x) xx sqrt(y)` =`sqrt(xy)`   [By the first property]
`xsqrty` = `sqrt(x^2y)`              [By the second property]
`sqrt(x/y)` = `sqrtx/sqrty`          [By the third property]


Steps to factor out square roots:

Get the values to find the factor out square root values.
Now check for the factors in the given square root. The factor should be in the squared form for factoring the value out of the root symbol.
The squared values are factor out from the square root symbol and
Write the calculated value in the correct form.

Factor out Square Roots - Example Problems:

Factor out square roots -  Problem 1:

Solve for factor out square root of 448

Solution:

Factoring out the square root of 448

`sqrt 448`

Finding factors by L Division method

2  |448

2  |224

2  |112

2  |56

2  |28

2  |14

7

The factor for the `sqrt(448)` is `sqrt(2^6xx7)`

=  ` sqrt(2^6 xx 7)`

= `8 sqrt 7`

The factor out value for the square root value of 448 is `8sqrt7`

Factor out square roots -  Problem 2:

Solve for factor out square root of 648

Solution:

Factoring out the square root of 648

`sqrt 648`

Finding factors by L Division method

2 |648

2 |324

2 |162

3 |81

3 |27

3 |9

3

The factor for the `sqrt(648)` is `sqrt(2^3xx 3^4)`

=  ` 2xx9 sqrt(2 )`

= `18 sqrt 2`
The factor out value for the square root value of 448 is `18sqrt2`


Factor out square roots -  Problem 2:

Solve for factor out square root of 528

Solution:

Factoring out the square root of 528

`sqrt 528`

Finding factors by L Division method

2 |528

2 |264

2 |132

2 |66

3 |33

11

The factor for the `sqrt(528)` is `sqrt(2^4xx 3 xx 11)`

=  ` 2sqrt(33)`
The factor out value for the square root value of 528 is ` 2sqrt33`

Integers Prime Numbers

In math, the whole numbers are called as integers. Likewise, the prime numbers are always integers that are whole numbers. It does not contain any sign in front of the number. It is also called as natural numbers. Now we are going to see about prime numbers are integers.
Explanation for Integers Prime Numbers

Some notes about prime numbers in math:

A number which has only two distinct divisors is called as prime numbers that is indicated as integers. The property primality is included in prime numbers. In two distinct divisors, 1 is constant divisor of all prime numbers. The prime number’s factor series is n = p1, p2,……pt.

Determine the prime numbers in math:

Step 1: The multiple factors of number are listed.

Step 2: State the given number is prime number when it has two divisors. If there are more than two divisors, we can call that number is composite number.

If we take the number as greater than 100 means, how we find whether that number is prime or not. Do the following procedures.

Procedure 1: Find out the square root of number.

Procedure 2: List the prime numbers of rounded value.

Procedure 3: Find out whether the listed prime numbers are divide the rounded value or not.

Procedure 4: If the listed prime numbers are not divide the given number means that is called as prime number.

More about Integers Prime Numbers

Example problems for integers prime numbers:

Problem 1: State the given integer is prime or not.

134

Answer:

Given integer are 134.

Square root value of 134 is 11.57.

The square root value is rounded as 12.

List the below 12 prime numbers are 2, 3, 5, 7, 11.

In above prime numbers, the integer 2 is dividing the 134.

Finally, the given integer 134 is not a prime number.

Problem 2: Find out which is not a prime number from the below list.

31, 43, 53, 29, 23, 60

Answer:

The number is 60 is not a prime number. Because it has factors more than two.

Exercise problems for integers prime numbers:

1. Find out the prime numbers from the list.

73, 60, 25, 80, 30

Answer: The prime number is 73.

2. Find out the prime number from the list

120, 101, 140, 90

Answer: The prime number is 101.

Metric Conversion Equations

Let us see about metric conversion equations. The conversion is mainly done between the measurements. Generally, there are two types of measurements like standard measurement and metric measurements. There are more measurements in metric. The metric measurements used to measure the area, distance, energy, force, mass, power, pressure, velocity or speed, volume or capacity and volume of liquid.

Conversion Equations of Metric:

The metric conversion equations are following below. The equations are forming by the table format. The basic metric units are meter, gram and liter. Some prefixes are,

1000  = kilo.

100 = Hecto.

10 = deca.

0.1= deci.

0.01 = centi.

0.001 = milli.

Conversion of length:

The metric conversion of length equation is following below.

Length conversion
1 kilometer	0.62137 miles
1 meter	3.2808 feet
1 centimeter	0.3937 inch
1 mile	1.6093 kilometer
1 foot	0.3048 meter
1 inch	2.54 centimeter

Conversion of weight:

The metric conversion of weight equation is following below.

Weight conversion
1 kilogram	2.2046 ibs.
1 gram	0.0353 ounce
1 short ton	907.1847 kilos
1 pound	0.4536 kilos
1 ounces	28.3495 grams.

Conversion of volume:

The metric conversion equation of volume is described below.

Volume conversion
1 hectoliter	100 liter
1 liter	100 centiliter
1 gallon	3.7853 liters
1 cubic inch	16.3871 cubic cm
1 hectoliter	0.1 cubic meter
1 liter	1 cubic diameter
1.0567 quarts	1 cubic decimeter

Conversion of Area:

The metric conversion of area equation is described below.

	Area conversion
	1 square kilometer	1 million square meters	0.3861 square miles
	1 hectare	10,000 square meters	2.471 acres
	1 square meter	10.7639 square feet	1550.003 sq inch
	1 square mile	640 acres	258.9988 hectares
	1 square foot	144 square inches	929.0304 sq centimeters

These are the conversion equation for the standard to metric.

Examples of Conversion Equations:

Let us see some examples of metric conversions.

Example 1:

Convert the 2 millimeter scale into inches.

Solution:

1 millimeter = 0.03937 inches.

So, 2 millimeter = 2 * 0.03937.

= 0.07874.

Example 2:

Convert the 2 inch thread into centimeter.

Solution:

1 inch = 2.54 centimeter.

2 inch =  2.54 * 2.

= 5.08 centimeter.

Example 3:

Convert the 2 square foot into square inches.

Solution:

1 square foot = 144 square inches.

2 square foot = 144 * 2.

= 288 square inches.

Exponents

Introduction to exponents:

           The exponent is usually shown as a superscript to the right of the base. The exponentiation an can be read as: a raised to the n-th power, a raised to the power [of] n, or possibly a raised to the exponent [of] n, or more briefly as a to the n. Some exponents have their own pronunciation: for example, a2 is usually read as a squared and a3 as a cubed.

(Source – Wikipedia.)

Properties of Exponents:

           The following four properties are generally used in exponents. These properties are very helpful to crack problems in exponents.

• The first and foremost significant uniqueness fulfilled by integer exponentiation is

           p^x+y= p^x . p^y

• This distinctiveness has the result

           p^x-y = p^x /p^y

• for p ≠ 0, and

           (p^x)^y=p^x.y

• Another basic identity is (p. q)ⁿ = pⁿ . qⁿ

Examples Word Problems for Exponents:

Word problems exponents- Example 1:

         Solve: (s⁴) (s³)

Solution:

         Conditions of what those exponents indicate. "To the 4th" demonstrates multiplying four reproductions and "to the 3rd" demonstrates multiplying three reproductions. By the generalization method the factors are then multiplied.

          (d⁴)(d³) = (d d d d) (d d d) [multiply the terms]
                     = d d d d d d d [simplify the terms]
                     = d⁷

          Answer is: d⁷

Word problems exponents- Example 2:

          Solve:  (n³) (n⁶)

Solution:

         Conditions of what those exponents indicate. "To the 3rd" demonstrates multiplying three reproductions and "to the 6th" demonstrates multiplying six reproductions. By the generalization method the factors are then multiplied.

          (n⁴)(n⁵) = (n n n n) (n n n n n) [multiply the terms]

                        = n n n n n n n n n [simplify the terms]

                        = n⁹

         Answer is: n⁹

Word problems exponents- Example 3:

         Solve: (st)⁴/s³

Solution:

         Conditions of what those exponents indicate. "To the 4th" demonstrates multiplying four reproductions and "to the 3rd" demonstrates multiplying three reproductions. By the generalization method the factors are then multiplied.

          (st)⁴ / s³= t⁴ s^4-3 [simplify the terms]

                         =t⁴s¹

          Answer is: t⁴s¹

Word problems exponents- Example 4:

        Shorten the following Zero exponents using the quotient rule:

          7³/ 7³

Solution:

        Given 7³/ 7³

           = 7⁴×7^-4  [multiply the terms]

           = 7⁰ [add the exponents]

           = 1

           Answer is: 1

Practice word Problems for exponents:

• Solve : (s⁴) ( st³)            

             Answer: s⁵ t³

• Solve :(a³b³c⁴) (ab) (a² b c³) 

             Answer: a⁶ b⁵ c⁷         

• Simplify: (t³)⁴                      

             Answer: t⁷

• Solve: (a . b)⁶                     

             Answer: a⁶.b⁶

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