Laws of Exponents

Let us understand the term exponents first, by directly jumping on the example as 3 rises to the power of 2. It can also be written in mathematics like 32. Let us understand the base and the laws of exponents as, this bigger number 3, we call this a base. And the smaller number that is 2 up here we call that the exponent. Just remember, that the bigger number is the base and the smaller number that is at the right hand corner is the expnt. So that is exactly do we use expnt. for? It tells us the total number of factors the base has. It is basically a short cut for repeated multiplication of the same number. In this case, it will be 3 times 3 as the expnt. is 2 so two times the base will get multiplied.

Let us understand how do we read this 24, “ 2 to the 4th power” here the base is 2  and the expnt. Is 4 though here the number 4 is bigger than the number 2 but the base is 2 thus 2 will be multiplied 4 times. That is 2times 2 times 2 times 2 equals to 16. Hence 16 is the solution.

Let us understand the laws exponents, when we multiply numbers with expnts. that have same base, just add the expnts. and keep the same base given. Laws of exponents problems are as follows, example 32 times 33 equals 3 (2+3) equals to 35. Multiplication law in exponents is quite simple just memorizing the thumb rule of adding the expnts.

When we divide numbers with expnts. That has the same base, just subtract the expnts. While keeping the same base. Example 26 divided by 22 equals 2(6-2) equals to 24.

When in expnt of an expnt, we multiply the expnts. for example, (42)3 equals to 4 (2 times 3)  that is nothing but 46. Is this topic coordinate planes hard for you? Watch out for my coming posts.

When we notice any negative expnt. Put 1 over that number to make a positive expnt. For example 7 -2 equals 1/72 equals to 1/ 14.

Laws are so simple to remember, look here we look back again to all we just studied. When we multiply the same base, add the expnts. When we divide the same base, subtract the expnts. for an expnt of an expnt. We do multiplication and for the negative expnt. Take 1 over the number to make it positive which becomes simpler and easier for solving tough equations. Laws of exponents lesson are for to remember the rules while solving the equations which will solve the problems quicker.

Right Angles Geometry

Introduction to right angles geometry:

In mathematics, the right angles are right angles triangle in the topic geometry. In geometry, when one of the angles is right angles in any of the three angles then it is called as right angles triangles. In geometry, the right angles have three sides and the two angles with one right angle. In geometry, the right angles have some special cases. The right angles with the special types and the formulas are given below.

Right Angles Geometry – Types:

In geometry, the right angles are classified into two different types. In geometry, the right angles are classified based on only by angles. The classifications of the right angles are given below.

The diagram for the right angles are given as,



Where,

x - Adjacent

y - Opposite

z - Hypotenuse. And

X & Y are angles.

In the right angles, the Pythagoras theorem is the theorem used to give about the right angles.

This theorem is explained by their sides and also the angles in the right angles.

The Pythagoras theorem for sides is given as,

x2 + y2 = z2

The Pythagoras theorem for angles is given as,

Sin A = x / z

Cos A = y / z

Tan A = x / y

The classification of the right angles is given below, Having problem with hard math problems for 12th graders keep reading my upcoming posts, i will try to help you.

Isosceles right angles

Scalene right angles

Isosceles Right Angles:

The structure for the isosceles right angles is given as,



In geometry, isosceles right angles are the right angles where this has two angles be 45 degree and one of an angles is 90 degree in geometry. The ratio of the sides for these types of is (1, 1, v (2)).

Scalene right angles:

The structure for the isosceles right angles is given as,



In geometry, scalene right angles are the right angles where this has one angles be 60 degree and another one is 30 and one 90 degree angles in geometry. The ratio of the sides for these types of is (1, v (3), 2).

Simple Introduction

Introduction to simple introduction of values

The simple introduction position value table by affecting a numeral single place from right to left, its place- value becomes ten times. In the similar method by poignant a digit singles position from left to right, its position-worth becomes one-tenth. So remote we have been discussion concerning seats from right to left beginning from ones place in the simple introduction. Extension of this table from left to right beyond ones place gives rise to the idea of decimals simple. I like to share this Simple Event with you all through my article.

Simple Introduction-basics

The simple introduction concept of decimals is also connected with the fractions denominators are 10, 100 or 1000 etc.

The place value represents the value of the numbers in the actual representations.

The face value represents the starting value of the numbers.

For example:

If we have 45 then the place value for 5 is 5 ones and the face value of 45 is 4

In this situation the educator may help kids to appreciate that a decimal can be spoken as a fraction and vice-versa.

The simple way to represent that "A decimal may surround a entire number module and a decimal module".

In a decimal, if whole number fraction is not specified, we designate it by inscription a zero on the left of the decimal position

For example, 803/100= 0.803;

The thought of simple percent is in use as an integral element of this section and require to be agreed a bit more significance as its utilize in our everyday life is extremely ordinary. Please express your views of this topic how to divide decimals with decimals by commenting on blog.

Basically simple introduction, it is not a original theme but its connection with the learn of fractions and decimals has a particular import. To complicated the notion of percent, we speak that percent earnings for each hundred or and indicate it by the symbol ‘ %’.

Simple Introduction-examples

Permit us think the place value of 4 in the numbers given below:

4365 The place value of 4 is 4000.

3465 The place value of 4 is 400.

3645 The place value of 4 is 40.

and 3674 The place value of 4 is 4.

Slope As A Derivative

In this page we are goimng to discuss about slope as a derivative concept . The trigonometrically tangent of the angle that a line makes with the positive direction of x – axis in anticlockwise direction is called the slope of the line.

The slope of a line in general is denoted by m. Thus m = tan Ø, where Ø is the angle which the line makes with the positive x-axis in anticlockwise direction.

The application of derivatives to plane geometry is related to slopes. Let us consider a curve whose equation is y = f(x).

On this curve we take a point P(x1, y1). Let the tangent at this point not be parallel to the co-ordinate axes, we can get the equation of tangent line at P.

The derivative of the function f(x) is the slope of the tangent.

dy /dx at P(x1,y1) is tan α = slope of the tangent at P, where α is the angle which the tangent at P(x1,y1) makes with the positive direction of x-axis.

The equation of straight line with slope m and passing through the point (x1,y1)is given as

y-y1 = m(x –x1).

For the curve y = f(x) we have  dy /dx =f’(x)

For the tangent line, slope = m = f’(x1)= dy / dx at (x1,y1).

The equation line of the tangent line will be of the form

y-y1 = f’(x1)(x –x1).

If the tangent at P is parallel to the x axis then α =0, `rArr`  tan α =0, `rArr`  slope =dy /dx =0.

If the tangent at P is perpendicular to the x –axis, or parallel to the y-axis then,

α = `\pi/{2}`, `rArr`  tan α =1,  `rArr` slope =dy / dx =1.

I like to share this Derivative of a Square Root Function with you all through my article.

Slope of the tangent  at P(x1,y1) on the curve y = f(x)  is the value of dy/dx at (x1,y1)
Slope of the Normal at P  is perpendicular  to the tangent  and is negative 1/ slope of the tangent.

Slope as a Derivative Examples

Below are the examples on slope as a derivative -

Example 1:
Find the slopes of tangent and normal to the curve y = x4 - 3x2 at (1, -2)
Solution:
Step 1 : Let us write the function y = x4 - 3x2
Step 2 : Then slope is dy/dx = 4x3 - 6x
Step 3 : Slope at (1,-2 ) is 4(1)3 - 6(1) = 4 - 6 = -2  Hence slope of the tangent is m= -2

Step 4 : Slope of the normal = -1/ slope of the tangent = - (-1/2) = 1/2. Hence  slope of the normal = 1/2
Let us find the slope and normal to a function which involves the time element also.

Example 2 :
Find the slope and normal to the curve x = ct and  y = c/t at t
Solution:-
Step 1 : Let us write the equation x = ct

Step 2 : Then the slope  of the function is dx/dt = c
Step 3 :  Let us find dy/dt which is equal to -c/t2
Step 4 :  We have to find dy/dx. Hence here we use chain rule . That is dy/dx = dy/dt ÷ dx/dt = -c/t2 ÷ c = -c/t2 * 1/c
= -1 /t2
Step 5 : Hence slope of the tangent to the curve x =ct , y = c/t is -1/t2
Step 6 : Slope of the normal  at 't'is the reciprocal of the tangent with sign change =  t2
Hence  slope of the normal at 't' is t2
Answer ; Slope of the tangent  at 't' is  -1/t2 and slope pf the normal at 't' = t2

Integrals is the opposite of differential equations.  Hence if slope of an equation is given , then we can find the equation using integrals.
Example 3: Find the equation of the curve whose slope at the point (x,y) is 3x2 + 2, if the curve is passing through the point (1,-1)

Solution:-
Step 1 : We have the slope dy /dx = 3x2 + 2
Step 2 : That gives us dy = (3x2 + 2) dx
Step 3 : Let us integrate both sides ∫dy  = ∫ (3x2+ 2) dx
y  =   3 x3 + 2x + c  = > x3 + 2x + c
3
Step 4 : This passes through (1,-1) Hence we get -1 = (1)3 + 2(1) + c
-1  =  1 +2+c
-1  =   3 + c
That is           c = -4
Step 5 : Let us now write the equation as y = x3 + 2x - 4

Answer :- Equation of the curve whose slope at a point (x,y) is 3x2 + 2 and the point is (1,-1) is y = x3 + 2x - 4
We have used  differential equation  to  find slope of the tangent and slope of the normal to the curve and  we have used integrals  to find the equation of the curve when slope is  given.

Basic Information on Algebra

Introduction for basic information on algebra:

Algebra is the division of mathematics connecting to the study of the regulations of operations and associations, and the organization and ideas occurrence from them, jointly with terms, polynomials, equations and algebraic structures. Collectively with geometry, examination, topology, combinatory, and number theory, algebra is one of the mainly significant divisions of pure mathematics.

Some Basic Information of Algebra:

The following information is very useful for basic algebra.

1. Make a revision of the basic formula with deference to the signs:

(plus +)    +  (plus +)    =   (plus+ )
(minus -)  +  (minus- )  =   (minus -)
(plus +)    *   (plus +)    =   (plus+ )
(minus -)  *   (minus- )  =   (plus +)
(minus -)  *   (plus + )   =   (minus -)
(plus +)    *   (minus- )  =   (minus -)


2. Practicing with the basic formulas:

(x + y) 2 =x2 + y2 + 2xy.
(x - y) 2 =x2 + y2 – 2xy.
x 2 – y 2 = (x + y) (x - y).


3. Attempt the quadratic equation with regard to x and y and solve correspondingly

Example Problems for Basic Information on Algebra:

Example 1 for basic information on algebra:

1) (s + 4) (s – 5)

Solution:

= (s + 4) (s – 5)

= s2 -5s + 4s– 20

= s2 - 1s- 20

Example 2 for basic information on algebra:

2) Solve this algebra expression: 5(a -3) + 4b - 5(a -b -3) + 12

Solution:-

Using Multiply Factors

= 5a - 15 + 4b -5a + 5b + 15 + 12

Groups like terms

= 9b + 12

Example 3 for basic information on algebra:

3) Using algebra equation solve: 3(-5x - 2) - (x - 2) = -2(8x + 4) + 4

Solution:-

Multiplying factors:

-15x – 6 - x + 2 = -16 x - 8 + 4

Group like terms:

-16x – 4 = -16x – 4

Add 16x + 4  to both sides and using the equations as fallows

0 = 0

Example 4 for basic information on algebra:

4) Calculate the value of the following algebra problem: `103^2`

Solution:

Given 1032

= (100+3)2

= 1002+ 2(100) (3) + 32

= 10000 + 600 + 9

= 10609.

Solving Basic Statistics Problems

Introduction to solving basic statistics problems:

Everyday we come across a wide variety of information’s in the form of facts, numerical figures, tables, graphs, etc. These figures are the numerical data collected with a definite purpose, are called data. Data is a word in a plural form of the Latin word datum. Every part of our lives utilizes data in one form or the other. This extraction of significant information is studied in a branch of mathematics called Statistics.

Solving Basic Statistics Problems-statistics Definition

Basically the word ‘statistics’ appears to have been taken from the Latin word ‘status’ meaning ‘a state (related to a political)’. In its origin, statistics is simply the collection of data on different aspects of the life of people, useful to the State. Basic Statistics deals with collection, organization, analysis and Interpretation of data. The word ‘statistics’ has held different meanings in different contexts, so that the calculation process is based on the meaning and data’s.Please express your views of this topic Definite Integrals by commenting on blog.

Solving Basic Statistics Problems
Example for basic stastistics problems:

Consider the marks obtained by 10 pupils in a mathematics test as given below:

55 36 95 73 60 42 25 78 75 62

The data in this form is called raw data.

By looking in this way, can you find the highest and the lowest marks?

Did it take you some time to search for the maximum and minimum marks? Wouldn’t it be a lesser amount of time consuming if these scores were arranged in ascending or descending order? So let us arrange the marks in ascending order as

25 36 42 55 60 62 73 75 78 95

Now, we can clearly watch that the lowest marks are 25 and the highest marks are 95.

The difference of the highest and the lowest marks in the basic stastistics data is called the range of the data. So, the range in this case is 95 – 25 = 70.

Arrangement of data in ascending or descending order can be quite time consuming, particularly when the number of observations in an experiment is large.

Define Rational Function

Introduction to define rational function:

Rational numbers is define as an integer that holds the value which is not equal to 0 and the value form of `x/y` , where x, y is two integers and` y != 0` . Get-together of each and every rational integer(numbers) is denoted by y. A rational integer(numbers) form is said to be proper form, when x and y having no similar factor other than 1. All integers(numbers) are said to be rational numbers. Rational function is referred as f(x). Let us discuss about the rational function in breif.

Define Rational Function:
Rational function is defined as a form that is,

`f(x) = ((p(x))/(q(x)))`

A rational function is generally a division of two polynomial functions mentioned as p(x) and q(x). That is, it is a polynomial divided by an another polynomial. Thus the rational function is defined with the definition and the form of the function is shown. Now let we have some example problems for clear understanding.

Example - Define Rational Function:

Example - Define rational function 1:

Determine the domain for the given rational function f(x) = `(x - 5)/(x^2 + 4x - 12)`

Solution:

Step 1: Given rational function is,

f(x) = `(x - 5)/(x^2 + 4x - 12)`

Step 2: Set the denominator of the given function f(x) = `(x - 5)/(x^2 + 4x - 12)` to zero and solve for x

x2 + 4x - 12 = 0

By rewriting the above equation,

x2 + 6x - 2x - 12 = 0

By separating common terms, we get

x(x + 6) - 2(x + 6) = 0

(x - 2)(x + 6) =0

By factoring each term, we get

x - 2 = 0

x = 2

x + 6 = 0

x = -6

Thus, the domain value of the given function is all real numbers except  for x, x = 2 or - 6

Example - Define rational function 2:

Define the domain of the function f(x) = `(7x)/(18x - 9)`

Solution:

Step 1: Given rational function is,

f(x) = `(7x)/(18x - 9)`

Step 2: Set the denominator of the given function f(x) = `(7x)/(18x - 9)` to zero and solve for x

18x - 9 = 0

By adding 9 on both sides, we get

18x =  9

Divide 18 on both sides, therefore

x = `9/18 `

x = `1/2`

x = 0.5

Therefore, the domain of a given rational function is all real numbers except for x = 0.5.