Learning subtraction of square roots

A square of a number a is a number x. therefore x2=a .A number x whose square is a. Every positive real number a has a unique positive square root, called the principal square root. Square root denoted by a radical sign as sqrt of a. For positive ax, the principal square root can also be written in exponent notation, as a1/2. We can undo a exponent with a radical, and a radical can undo a power. The “`sqrt(a)` “symbol is called the "radical “symbol..The line across the top is called the vinculum.

Subtracting Square Root Terms

Subtracting square roots

Subtracting square roots is combining like terms when we need to do that with algebraic expressions. The induces (a square roots index is 2 `root(2)(a)` , a cube roots index is 3  `root(3)(a)` , a 4th roots index is 4 `root(4)(a)` ,a 5th roots index is 5 `root(5)(a)` etc.) or the radicands (enclosed by parentheses after SQRT or the expression under the root sign) are the same.

Just as with "regular" numbers, square roots can be subtracted together. But you might not be able to simplify the subtraction all the way down to one number. Just as "you can't subtract apples and oranges", so also you cannot combine "unlike" radicals. To subtract radical terms together, they have to have the same radical part.

Simplifying Square-Root Terms

Simplify a square root, we take out anything that is a perfect square; that is, we take out front anything that has two copies of the same factor.

We can raise numbers to powers other than just 2; we can cube things, raise them to the fourth power, raise them to the 100th power, and so forth.

(ab)^2=a^2b^2  and`sqrt(ab)`   = `sqrt(a)``sqrt(b)` but we can’t write this subtraction of square root  `sqrt(a-b)`  = `sqrt(a)` - `sqrt(b)`

Example Problems on Subtracting Square Root

Example Problems

1. (4 * `sqrt(2)` ) - (5 * `sqrt(2)` ) + (12 * `sqrt(2)` )
Solution: Combine like
= (4 - 5 + 12) * `sqrt(2)`
Answer is 11 * `sqrt(2)`

2. (53 * `sqrt(5)` ) - (5 * `sqrt(5)` )
Solution: Combine like terms
(53 - 5) * `sqrt(5)`
= 48 * `sqrt(5)`

3. (40 * `sqrt(5)` ) - (48* `sqrt(5)` )
Solution: Combine like terms by subtracting the numerical coefficients.
(40 - 48) * `sqrt(5)`
= -8 *`sqrt(5)`
`sqrt(3x+1)`-( -`sqrt(2x-1)` )= 1    subtracting 2 square roots with variables
`sqrt(3x+1)` = 1 - `sqrt(2x-1)`
Take square both sides
3x + 1 = 1 - 2 `sqrt(2x-1)` + 2x-1
3x + 1 - 1 -2x + 1 = -2 `sqrt(2x-1)`
x+1 = -2 `sqrt(2x-1)`
take Square both sides again
x^2 + 2x + 1 = 4(2x -1)
x^2 + 2x + 1 = 8x -4
x^2 -6x + 5 = 0
(x-5)(x-1) = 0
x1 = 5, x^2 = 1

Ogive – The cumulative line graph

In statistics, a frequency chart displays the given data, in which the frequency of each data item is found.  What does frequency mean? Frequency as we use in case of the frequency of the flight from one place to another means the number of times the particular flight travels from one place to another.  In statistics, frequency is used to display the number of times the data item occurs in a data set.  Tally marks or tallies are used to record and show the frequency of an item in a data. Now, let us learn about cumulative frequency. It is the total of the frequency and all the frequencies below it in a frequency distribution.  In simple words, it is the running total of frequencies. Given frequency of a set of data, the Ogive chart looks something similar to the chart given below:

Age Frequency      cumulative frequency
8      4 4
9      6 4+6 =10
10    15 10+15=25
11     9 25+9=34
12    18 34+18 = 52
13    10 52+10 =62

The Ogive Definition can be given as a distribution curve in which the frequencies are cumulative
Now that we have the cumulative frequencies, we shall now plot the graph. To plot the graph we take the ages on the x-axis and the cumulative frequencies on the y-axis as we plot a normal line graph. Once all the points are plotted, we now join the points. The curve we get is the cumulative frequency curve, also called the Ogive

We can define Ogive as a cumulative frequency graph which is a curve or graph showing the cumulative frequency for a given set of data. When the given data is an un-grouped data, to get Ogive, we find the cumulative frequency of the data and plot that on the y-axis and the given data to which cumulative frequency is calculated is taken on the x-axis. The graph we get is the Ogive of ungrouped data. When the data given is a grouped data, we divide the group into classes with upper and lower boundary which is taken on the x-axis and the cumulative frequency of the data on the y-axis. The graph we get here is the Ogive of a grouped data.

Ogive Example
For example, let us assume the amount of savings for the months of January and March as $200 and the savings of $125 for the months February, April and May.  For the given data, the ogive displays a running total of the savings with the amount saved in dollars on the y-axis and the months on the x-axis.

Trigonometric Identities | Theorems Based on Trigonometric Identities

Trigonometric Identies are some identies used in Trigonometry in order to make the calculations easier.
Trigonometry is a word consisting of three Greek words " Tri" means three, "Gon" means side, and "Metron" means measure. Thus, trigonometry is a study related to the measures of sides and angles of a triangle. Trigonometry is mainly used by captains of ships to find the direction and distance of islands and light houses from sea. Trigonometry is also used in astronomy, geography and engineering.
Trigonometric Ratios
In any right-angled triangle ABC,
let  angle B = 90 o  and angle C = T.                                                      

Line segment AC is the hypotenuse.
With reference to angle C, we can say that,
Line segment AB is the opposite side of Line segment BC is the adjacent side of Therefore, trigonometric ratios are given as,


Trigonometric Identities
Basic trigonometric identities are:
sin^2 T + cos^2 T = 1.
tan^2 T + 1 = sec^2 T
1 + cot 2T  = cosec^2 T
Theorems Based on Trigonometric Identities

Theorem 1: sin^2 T + cos^2 T =1
In right-angled triangle ABC, let angle< B = 90, angle< C = T.
Let AB = a, BC = b, and AC = c
By Pythagorean theorem we can say,
(hypotenuse)^2 = ( side)^2 + (side)^2
From figure we can say,
(AC)^2 =  (AB)^2 + (BC)^2
c^2 =  a^2 + b^2
divide throughout by c^2, we get,
(c^2 ) / c^2 =  ( a^2 + b^2 ) / c^2
1  =  a^2 / c^2 + b^2 / c^2
=  ( AB )^2 / ( AC)^2 + ( BC)^2 / (AC)^2
=   (AB / AC)^2 + ( BC / AC)^2
=   ( sin T )^2 +  ( cos T )^2
Therefore,
1 = sin^2 T + cos^2 T

Theorem 2: tan2 T + 1 = sec2 T
We have sin^2 T + cos^2 T = 1
Divide on both sides by cos^2 T,
( sin^2 T + cos^2 T ) / cos^2 T  =  1 / cos^2 T
(sin^2 T / cos^2 T) + (cos^2 T / cos^2 T)  =  1 / cos^2 T
By using trigonometric ratios,
sin T/ cos T  =  tan T
1 / cos T  =  sec T
substitute the values we get,
( sin T / cos T )^2 + 1  =  ( 1 / cos T)^2
(tan T)^2 + 1  =  ( sec T )^2
tan^2 T + 1  =  sec^2 T
Theorem 3: 1 + cot2 T  = cosec^2 T
We have sin^2 T + cos^2 T = 1
Divide on both sides by sin^2 T,
( sin^2 T + cos^2 T ) / sin^2 T  =  1 / sin^2 T
(sin^2 T / sin^2 T) + (cos^2 T / sin^2 T)  =  1 / sin^2 T
By using trigonometric ratios,
cos T/ sin T  =  cot T
1 / sin T  =  cosec T
substitute the values we get,
1 +  ( cos T / sin T )^2   =  ( 1 / sin T)^2
1 +  (cot T)^2  =  ( cosec T )^2
1 +  cot 2T  =  cosec^2 T

Standard Deviation of Mean in a nutshell

Standard deviation of Mean is the measure of the spread of the data about the mean value. If the standard deviation is low it shows that the values of the data are not spread out much and if the standard deviation is high it shows that the values of the data are spread out. At times we come across data which has the same mean but different range; to compare the sets of data standard deviation is very useful.  The average squared deviation from the mean is called the Variance. The square root of variance is the Standard Deviation of Mean. It is a statistical measure to know how the data is spread in the distribution, in simple words statistical measure of dispersion. Standard Deviation Means is also called the Mean of the Means.

In a population Variance is given by the formula: sigma^2 =summation[x – mu]^2/n
Where, x is each value in the data, mu is the mean of the data, n is the total number of values in the data.  Usually variance is estimated from a sample in a population. Variance calculated from a sample is given by the formula: sigma^2 = summation[x – x bar] ^2/ (n-1), here, x is each value from the sample, x bar is the mean of the values in the sample; n-1 is one less than the total number of values in the sample.  One Standard Deviation of the Mean is given by sigma= sqrt [summation[x – x bar] ^2/ (n-1)]

Standard Deviation of the Mean Equation
The equation or the formula to be used to calculate the standard deviation depends on whether the data is grouped or non-grouped. For example, given data, 42, 35, 48, 53, 47 is a non-grouped data.

In such a case, the standard deviation of the mean is calculated using the equation:
sigma = sqrt [summation (x- x bar) ^2/ (n-1)] where sigma is the standard deviation, (x-x bar) ^2 is the square of the deviations of the data values and n is the total number of values. Let us consider the data given below
Hours of components Frequency
300-400                   13
400-500                   25
500-600                   66
600-700                            58
700-800                   38
Understanding statistics problems is always challenging for me but thanks to all math help websites to help me out.
The data is a grouped data, here the standard deviation of the mean is estimated using the equation given by, sigma = sqrt [summation f(x-x bar) ^2/summation (f)] where sigma is the standard deviation, f is the frequency, x is each value of the data, x bar is the mean of the data values, summation is the sum of.

Introduction to indefinite integrals

In derivatives we learn about the differentiability of a function on some interval I and if it is differentiable, how to find its unique derivative f’ at each point of I. In application of derivatives we learn that using derivative we can find the slope of the tangent at any point on the curve, we can find the rate of change of one variable with respect to the other. Now let us look at an operation that is inverse to differentiation. For example we know that the derivative of x^5 with respect to x is 5x^4. Suppose the question is like this: derivative of which function is 5x^4∫ Then it may not be that easy to find the answer. It is a question of inverse operation to differentiation.

The answer to the question: " Whose derivative is a given function f ∫ " is provided by an operation called anti derivation. It is possible that we may not get an answer to this question or we may have more than one answer. For example, (d/dx) (x^4) = 4x^3, (d/dx)(x^4 + 3) = 4x^3 and in general (d/dx)(x^4+C) = 4x^3, where c is some constant.

Definition of integration (integrals): If we can find a function g defined on the interval I such that (d/dx)(g(x)) = f(x), for all x belonging to I, then g(x) is called a primitive or anti derivative or indefinite integral of f(x). It is denoted by ∫ f(x) dx and is called indefinite integral of f(x) with respect to x. The process (operation) of finding g(x), given f(x) is called indefinite integration.

Thus the question when can we find the integral of f cannot be easily answered. There are some sufficient conditions such as, continuous functions and monotonic functions have integrals. Sin x/x is continuous, hence ∫(sin x/x)dx is defined but cannot be expressed as any known elementary function. Similarly, ∫ v(x^3+1) dx and ∫v(csc x) are defined but canoe be expressed as known elementary functions. If anti derivative of f exists, then it is called integrable function.
(1) ∫ f(x) dx means, integral of f(x) with respect to x.
(2) In ) ∫ f(x) dx, f(x) is called the integrand.
(3) In ) ∫ f(x) dx, ∫ …. dx indicates the process of integration with respect to x.

For evaluating indefinite integrals we use the following standard table of indefinite integrals: