Solving Root of a Number

 If any number is expressed as x × x, then x is the product of two same numbers. We know that 5^2 = 5× 5 =25.Here 25 is called the square of 5 and 5 is called the square root of 25. And (2/3)^2 = (2/3)× (2/3) = 2 ×2/3 ×3 = 4/9. 4/9 is called as square of 2/3 and is also called as square root of 4 / 9.

Examples of Solving Root of a Number

1. Simplify 7^2 = 7 ×7 = 49

Here 49 is called the square of 7 and 7 is called the square root of 49

2. (0.4)^2 = (0.4)× (0.4) = 0.16

Here 0.16 is called the square of 0.4 and 0.4 is called the square root of 0.16

3. Simplify 9^2 = 9 ×9 = 81

Here 81 is called the square of 9 and 9 is called the square root of 81

4. Simplify 121^2 = 121 ×121 = 14641

Here 14641 is called the square of 7 and 7 is called the square root of 14641

5. Simplify 81^2 = 81 ×81= 49

Here 6561 is called the square of 81 and 81 is called the square root of 6561

Multi Step Square Root Examples:

(1). Find the square root of 144

Solution:

Split the number into the product of prime factors.

                              144 = 3 × 3 × 2 × 2 × 2 × 2

                            √144 = √3^2 × 2^2 × 2^2

                                    =3 × 2 × 2

             Therefore √144 = 12

(2). Find the square root of 5^3 × 5^5

Solution:

                         5^3 × 5^5 = (5 × 5 × 5) × (5 × 5 × 5 × 5 × 5)

                                       =5^2 × 5^2 × 5^2 × 5^2

       Therefore √5^3 × 5^5 = √5^2 × 5^2 × 5^2 × 5^2

                                       = 5 × 5 × 5 × 5 × 5

                                      = 625

      Therefore √5^3 × 5^5 = 625

Discuss of Solving Root of Number

(1)   Find the square root of 36
                
(2)   Find the square root of 6^2×7^2

(3)   Find the square root of 8100


Answers:
1. 6    2. 42   3.  90

Learn Online Limits

In the mathematical expression the main concept of limit is used to express a value that a sequence or function approaches as the input or key approaches of some value. The limit is typically reduced as lim as in Lim(xn) = x or represent by the right arrow (→) as in an → a. Let us consider this function f(x) = x2. Examine that as x take values very close to 0, the value of f(x) also move towards 0. We say limits  f(x) = 0  x →0

Rules For how to solve limits


Rule1: In learning online limits, given limits function put x=a .If f(a) is a definite value then

limits  f(x) = f(a)
         x →a

Rule2: In learning online limits, If  proving limits  f(x) is a rational function then factorize the numerator and the denominator.Cancel out the  common factors and then put x=a

Rule3: If the given learning online limits function contains a surd then simplify it by using conjugate surd's.After simplification,put x =a

Rule4: If the given  proving learning online  limits  function contains a series which is capable of being expanded then after making proper expansion and simplifying,cancel the common factors in the numerator and denominator,if any Then, put x =a


Limits Examples


1) Evaluate  proving limits lim     (xm -am ) / (xn -an)
                                             x →a

Solution for proving  limits:  lim    (xm -am /xn -an)  =   lim    {xm -am /x-a) ÷ (xn -an /x-a)}
                                                   x →a                                   x →a
Limits =   lim     (xm -am /x -a)   ÷   lim(xn - an /x -a)
                  x →a                                 x →a
Limits =     (ma n-1) ÷ (nan-1)

Limits =   ma m-1 / na n-1   = (m) /(n a m-n)


2) Evaluate proving  limits lim (x+2)3/3 -  (a +2)3/2 / x-a

                                              x→a

Solution proving  limits:   lim (x+2)3/3 - (a +2)3/2 /  x-a
                                            x→a
                        (x +2)3/2 - (a+2)3/2
=      lim     ------------------------------------------
                (x+2)→(a+2)            (x +2) - (a +2)
          ------------------------------------------------------
=   3/2. (a+2)(3/2 -1) =     3/2(a +2)1/2                      [ lim   (xn -an /x -a)   =  nan-1]
                                                                                       x→a

3) Find Limit (x →2) {3x2-5x+7}

Solution:- Given Limit ( x →2)    {3x2-5x+7}

= 3(2)2-5(2)+7   = 12-10+7 = 9


4) Show that Limit (x →3)  (x2+2x-5)  /  (2x2-5x-1) = 5/2

Solution:-Limit (x →3) (x2+2x-5) / (2x2-5x-1)

= Limit (x →3) (x2+2x-5) /  Limit ( x →3)  (2x2-5x-1)

=[ (3)2+2(3)-5)]  / [ 2(3)2-5(3)+1]  =  (9+ 9 - 5) /  (18-15+1)    = 10/ 4 = 5/2.

We can be solved these practice problems on limits  by learning these limits problems.

Fraction Decimal Percent Table

FRACTION DECIMAL PERCENT TABLE

Decimals, Fractions and Percentages are just different ways of showing the same value:

A Half can be written...

As a fraction:       1/2

As a decimal:        0.5

As a percentage:   50%

A Quarter can be written...

As a fraction:       1/4

As a decimal:        0.25

As a percentage:   25%

Example Values

Here is a table of commonly occurring values shown in Percent, Decimal and Fraction form:

Percent       Decimal      Fraction

1%                0.01                      1/100

5%                0.05                       1/20

10%              0.1                1/10

12½%           0.125            1/8

20%               0.2                1/5

25%               0.25                        1/4

331/3%          0.333...        1/3

50%               0.5               1/2

75%            0.75           3 /4

80%            0.8             4/5

90%            0.9           9/10

99%           0.99        99/100

100%          1               1

125%          1.25        5/4

150%            1.5         3/2

200%             2           2

Converting Between Percentage and Decimal

Percentage means "per 100", so 50% means 50 per 100,

or simply 50/100.If we divide 50 by 100 you get 0.5 (a decimal number).

So, to convert from percentage to decimal: divide by 100 (and remove the "%" sign).The easiest way to divide by 100 is to move the decimal point 2 places to the left.

Example: Convert 8.5% to decimal

Move the decimal point two places: 8.5 -> 0.85 -> 0.085

Answer 8.5% = 0.085

Converting From Decimal to Percentage

To convert from decimal to percentage, just multiply the decimal by 100, but remember to put the "%" sign so people know it is per 100.The easiest way to multiply by 100 is to move the decimal point 2 places to the right.

Example: Convert 0.65 to percent

Move the decimal point two places: 0.65 -> 6.5 -> 65.

Answer 0.65 = 65%

To change a Decimal into a Fraction

Take the decimal, drop the decimal point, and place the result into the numerator  of a fraction.

To determine the denominator , write a 1, followed by zeros --- as many zeroes as it takes to match the original length of the decimal.

Examples:

0.75 becomes 75/100

0.034 becomes 34/1000

2.5 becomes 25/10

Roman Numerals Learning

Roman numeral is a symbol, roman numerical learning is used to represent a number. (Our digits 0-9 are often called as Arabic numerals.) In learning of roman numerals are written as the combinations of the seven letters.

Those seven letters are,

I =1             L=50

V = 5           C=100      M=1000

X=10           D=500


Note:

If a lesser numbers follow larger numbers, then numbers are added.

If a lesser number precedes bigger number, then the smaller number is subtracted from the larger.

How to write roman numerals and rules for subtracting letters -roman numerals learning:


Here, how 1100 will be written as Roman Numerals Learning, you would state M for 1000 and then put a C after it used for 100; Otherwise 1,100 = MC in Roman Numerals Number.

Some examples:

• VIII = 5+3 = 8

• IX = 10-1 = 9

• XL = 50-10 = 40

• XC = 100-10 = 90

• MCMLXXXIV = 1000 + (1000 -100) + 50 + 30 + (5 - 1) = 1984

Rules for subtract letters- Roman numerals learning:


•   Subtract powers of ten, such as I, X, or C. Writing VL for 45 be not suitable: write XLV as a replacement
•   Subtract only a distinct letter from a single digit. Write VIII for 8, not IIX; 19 is XIX, not IXX.
•   Don't subtract letter from a unlike letter more than ten times larger. This means you can just subtract the I from V or X, and X from L or C, so MIM is against the law.


Let’s found with an addition problem: 13 + 58. In Roman numerals learning, that's XIII + LVIII. We'll begin by writing.
these  two numbers subsequently to each other:Next, we  are rearrange the letters so that the numerals are in descending order: LXVIIIIII. Now we have six be, so we'll rewrite them as VI: LXXVI. The two Vs are the same as an X, so we simplify again and get LXXI, or 71, this is our final answer.

Learn discrete random variables

If the Random variable X assumes only finite or countably infinte set of values it is known as discrete random variable.

Probability density function of Discrete Random variable:-

Suppose X is a Random variable which can take at the most a countable number of values X1, X2, X3, ..................... Xn with each value of  " X ". We associate a number

pi = P ( X = Xi ) ; i = 1,2,..............n

which is known as the probability of Xi and satisfies the following conditions:

pi = P ( X = Xi ) `>=` 0  ( i = 1,2,..............n )    i.e., pi 's are all non- negative and
`sum` pi = p1 + p2 +................... + pn = 1
i.e., the total probability is one.

The function pi = P ( X = Xi ) ; i = 1,2,..............n is called the probability function or more precisely probability mass function of the random variable X


Cumulative distribution function of F(x) of discrete random variable


Cumulative distribution function F( x ) of a discrete random variable X is denoted as F( X = xi ) and defined as               F ( X = xi ) = P ( X = xi )

F ( X = xi ) = P ( X = x1 ) + P ( X = x2 ) + ............................ + P ( X = xi )

F ( X = xi ) =    `sum_(n=1)^i` P ( X = xn )


Example of learn discrete random variables


Let x denote the minimum of two numbers that appear when a two dice is thrown once. find the discrete probability distribution?

Solution:-  The sample space S = { ( 1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) }

n ( S ) = 62 = 36

Given that X = min ( a, b )

P( 1 ) = P ( X = 1 ) = { (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (3,1) (4,1) (5,1) (6,1) }

= 11 / 36

P( 2 ) = P ( X = 2 ) = { (2,2) (2,3) (2,4) (2,5) (2,6) (3,2) (4,2) (5,2) (6,2) }

= 9 / 36

P( 3 ) = P ( X = 3 ) = { (3,3) (3,4) (3,5) (3,6) (4,3) (5,3) (6,3) }

= 7 / 36

P( 4 ) = P ( X = 4 ) = { (4,4) (4,5) (4,6) (5,4) ( 6,4) }

= 5 / 36

P( 5 ) = P ( X = 5 ) = { (5,5) (5,6) (6,5) }

= 3 / 36

P( 6 ) = P ( X = 6 ) = { (6,6) }

= 1 / 36

X    1    2    3    4    5    6
P ( X = x)    11/36    9/36    7/36    5/36    3/36    1/36

What is variance in statistics

Definition:

Variance in statistics of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically are .It Measures the variability in the data from the mean value.Variance is defined as

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This definition of variance in statistics can be used for both discrete and continuous random variables.Variance will never be negative, provided it is defined, because the squares are positive or zero . The unit of variance is the square of the unit of observation . Example: The variance of a set of heights measured in centimeters will be given in square centimeters. This is an inconvenient result, and so the standard deviation is generally used . The standard deviation is the square root of the variance .



The variance of random variable X is typically designated as Var(X), , or simply σ2 (pronounced “sigma squared”). If a distribution does not have an expected value, as is the case for the Cauchy distribution, it does not have a variance either.


The formula is:


The unbiased formula is (for a sample):



with

x = the mean.

N = the population size.

n = the sample size.



Standard Deviation


The standard deviation formula is very simple: it is the square root of the variance in statistics. It is the most commonly used measure of spread.

An important attribute of the standard deviation as a measure of spread is that if the mean and standard deviation of a normal distribution are known, it is possible to compute the percentile rank associated with any given score. In a normal distribution, about 68% of the scores are within one standard deviation of the mean and about 95% of the scores are within two standard deviations of the mean.



Variance and Standard Deviation: Step by Step

1. Calculate the mean, x.
2. Write a table that subtracts the mean from each observed value.
3. Square each of the differences.
4. Add this column.
5. Divide by n -1 where n is the number of items in the sample  This is the variance in statistics.
6. To get the standard deviation we take the square root of the variance.

Example

The owner of the Ches Tahoe restaurant is interested in how much people spend at the restaurant.  He examines 10 randomly selected receipts for parties of four and writes down the following data.

44,   50,   38,   96,   42,   47,   40,   39,   46,   50

He calculated the mean by adding and dividing by 10 to get

x  =  49.2

Below is the table for getting the standard deviation:

x	x - 49.2	(x - 49.2 )2
44	-5.2	27.04
50	0.8	0.64
38	11.2	125.44
96	46.8	2190.24
42	-7.2	51.84
47	-2.2	4.84
40	-9.2	84.64
39	-10.2	104.04
46	-3.2	10.24
50	0.8	0.64
Total		2600.4

Now

2600.4
------- =  288.7
10 - 1

Hence the variance in statistics is 289 and the standard deviation is the square root of  289 = 17.

Solving Calculus Derivative Problems

Two mathematicians, Namely Gottfried Leibniz and Isaac Newton, developed calculus. Calculus problems can be dividing into two branches: Differential Calculus problems and Integral Calculus problems. Differential calculus is use to measure the rate of change of a given quantity whereas the integral calculus is use to measure the quantity when the rate of change is known.

The output of a function will change when we change the input value of a function.The measure of the change in the function is called as Derivative. The solving of  derivative of a function can be calculated by differentiating the function.Let us see how to solving the derivative problems.


Calculus derivative example problems:


The following solving problems  are based on the  derivatives.

Solving problem 1:

Determine the derivative dy/dx of the inverse of function f defined by

f(x) = (1/8) x - 2

Solution:

The first is used to find the inverse of f and differentiate it. To find the inverse of f we first write it as an equation

y = (1/8) x - 2

Solve for x.

x = 8y + 16.

Change y to x and x to y.

y = 8x + 16.

The above gives the inverse function of f. Let us find the derivative

dy / dx = 8


Solving problem 2:

Determine the critical number(s) of the polynomial function f given by

f(x) = x 4 - 108x + 100

Solution:

The domain of f is the set of all real numbers. The first derivative f ' is given by

f '(x) = 4 x 3 - 108

f '(x) is defined for all real numbers. Let us now solve f '(x) = 0

4 x 3 - 108 = 0

Add 108 on both sides,

4x 3– 108 108=108

4x 3= 108

x 3 = 27

x = 3 or x = -3

Since x = 3 and x = -3 are in the domain of f they are both critical numbers.

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Calculus derivative Practice Problems:


1) Determine the derivative dy/dx of the inverse of function f defined by

f(x) = x/2+ 3x/2 - 2

2) Determine the critical number(s) of the polynomial function f given by

f(x) = x 3 - 48x + 10

Answer Key:

1). dy / dx = 2

2).X = 4 or X= -4