Understand Multiplying Radicals

Introduction: 
                   A radical is an expression that expresses the square roots, cube roots etc. Radical is the word that comes from the Latin word Radix. The meaning of the radical is ROOT. The word radical was used to refer to an irrational number. The radical symbol was introduced by John Bell in 1668. Radical form having the same properties as the properties of the numbers. When we use the radical without a sign or index number, it mentions the positive square root of the radicand.

Learn Rules for Multiplying Radical:

                 The following rule can be used in multiplying radicals:
`root(n)(a)` . `root(n)(b)` = `root(n)(ab)`
The most important thing when we multiply radicals is, both radical must have the common index.
For example,
`root(2)(8)` . `root(2)(4)` = `root(n)(32)`
           
Simplification:
               After we multiply radicals we need to make sure that the radical we end up with is simplified. Sometimes we can simplify before we multiply. We never have to simplify before we multiply but we must simplify after we multiply radicals (if possible)
For example,
               `sqrt(2)` . `sqrt(8)` = `sqrt(16)`
                               = `sqrt(4^2)`
                               =  4                              
                `sqrt(2)` . `sqrt(8)` 

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Explain Factorising

Introduction:

             Factorising is the reverse of expanding brackets, so it is putting 2x² + x - 3 into the form (2x + 3)(x - 1). This is an important way of solving quadratic equations.
The first step of factorising an expression is to 'take out' any common factors which the terms have. So if you were asked to factorise x² + x, since x goes into both terms, you would write x(x + 1) .

There are various methods of factorising a polynomial. They are,

1. Factorisation by dividing the expression by the HCF of the terms of the given expression.
2. Factorisation by grouping the terms of the expression and expression calculator.
3. Factorisation using identities.

Methods of factorising polynomials:

There are various methods of factorising a polynomial. They are,
1. Factorisation by dividing the expression by the HCF of the terms of the given expression.
2. Factorisation by grouping the terms of the expression.
3. Factorisation using identities.
Let us study these methods one after the other in detail.

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What is Postulates?

Introduction:

             A statement whose validity is accepted without proof is called a postulate.
In addition to point, line plane etc, it is also necessary to start with certain other basic statements that are accepted without proof. In Basic geometry these are called postulates.
A postulate, though itself is an unproved statement, can be cited as a reason to support a step in a proof. Postulates are just like axioms in arithmetic sequence and algebra, that they are accepted without proof.

Some of the postulates we use often are:

• The line containing any two points in a plane lies wholly in that plane.
• An angle has only one and only one bisector.
• Through any point outside a line, one and only one perpendicular lines can be drawn to the given line.
• A segment has one and only one mid point.
• Linear pair postulate: If a ray stands on a line, then the sum of the two adjacent angles so formed is 180^o.

Activity 1:

Mark two distinct points A and B on the plane of your note book. Can you draw a line passing through A and B?

By experience we find that we can draw only one line through two distinct points A and B.
Hence "Given any two distinct points in a plane, there exists one and only line containing them".
This is a self-evident truth. Hence it is a postulate.

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Explain Polynomial Expressions

Introduction:

The monomials in a polynomial are called the terms and the highest degree of the terms, the degree of the polynomial. The coefficient of the highest power of x in a polynomial is called the leading coefficient of the polynomial. In this article we shall discus about polynomial expression and degree of polynomial expression in detail.

Polynomial Expressions:

Consider the algebraic expression 6x³ – 8x² + 3.it has three terms, namely 6x^3, -8x² and 3

The power of x in the term 6x² is 3

The power of x in the term -8x² is 2

The third term 3 is a constant term or an independent term.

We note that in this expression the power of the value x are whole numbers.

Hence we call this algebraic expressions as polynomial expressions.

In the above expression the power of x is 3. The power should be a constant. it should be a variable. It must be an independent term. But we can write 3 as 3 * x⁰ ( since the value of x⁰ is 1).hence the power x in the term 3 is 0.

Note:

The power of any variable in the independent term is always 0.

Now consider another algebraic expression 2x³ – 3x^3/2 + 7x + 10.

It has 4 terms namely 2x³, -3x3^/2, 7x and 10.

In the second term -3x3^/2, the power of x is 3/2.the power is not a whole number. Therefore this algebraic expression cannot be called as a polynomial expression.

In an algebraic expression if the power of each variable is a whole number, it is called polynomial expression or simply a polynomial.

Progressions and Types

Introduction:

       Progression is very interesting since we can find the nth term of the particular sequence in much easier way. Moreover we will see in detail about different types involved in progression. 

There are three types of Progression in math,

• Arithmetic progression

• Geometric progression

• Harmonic progression
  

Let us see these progression and their properties in the following section.

Arithmetic Progression:

        It is a sequence of numbers in which each term except the first term can be calculated by adding constant number (common difference) to the immediately preceding number.

Geometric Progression:

        It is a sequence of numbers in which each term of the sequence except the first term can be calculated by multiplying the preceding term by means of a constant factor (common ratio).

Harmonic Progression:

       In mathematics, a harmonic progression is a progression formed by taking the reciprocals of an arithmetic progression. Equivalently, a sequence is a harmonic progression when each term is the harmonic mean of the neighboring terms.

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30 60 90 Right Triangles

Introduction to 30 60 90 right triangles:

                             The triangle consists of 3 sides and these three lines are enclosed to form the triangle shape. The triangle can be classified according to its side measurement and angles measurement. The 30 60 90 right triangles are one of the special kind of the triangle. This shows the degree measurements. Now we are going to see about the 30 60 90 right triangle.

                                                     

Construction of 30 60 90 Right Triangle:

     Now we are going to see about the construction steps for 30 60 90 degree right triangle. It is a special kind of right triangle with the angles 30 60 and 90 degrees.

Steps of construction:

                                      

1. Draw a line segment of any length EF.

2. After we have to first construct the 30 degree angle.

3. With the help of compass take a small radius of length and Draw an arc at s in EF. Then draw an arc from S which cuts the arc and mark it as T

4. With T as center draw an arc that cuts the arc at R.

5. Thus, the angle 30 is drawn. Repeat the same procedure from the other end F. Thus, it will make 60 degree construction.

6. Now extend the line which meets the 30 degree line at G. Thus, 90 will be formed at the angle G.

Please leave the comments if you have any doubts.

Permutations with example problem

Introduction:

         The process of selecting things is called as combination and that of arranging things is called as permutation.

A few problems related to the permutations are,

1. Arrangement of books on a shelf.
2. Formation of numbers with the given digits;
3. Formation of words with the given letters.

To find the number of permutations of n object taken r at a time denoted by ⁿP_r. The letter P is an abbreviation of the word ‘permutations’.

Permutations Example Problem

Example 1:

Suppose the six digits 1, 2, 4, 5, 6, 7 are given to us and we have to find the total number of number which can be formed under different condition.

Condition 1.      There is no restriction:

The number of 6-digits numbers = ⁶P₆ = 6! = 6x5x4x3x2x1=7200

Condition 2.      The numbers in which a particular digit occupies a particular place.

Suppose we have to form a number in which 5 always occur in the ten’s place. In this case the ten’s place is fixed and reaming five place can be filled in by the reaming 5 digit in ⁵P_5, i.e. 5! = 5x4x3x2x1 = 120.

Condition 3.      Number divisible by a particular number

Suppose we have to form a numbers which may be divisible by 2. These numbers will have 2 or 4 or 6 in unit place. Thus the unit’s place can be filled in 3 ways. After having filled up the unit’s place in any one of the above ways, the remaining five place can be filled in ⁵P₅ = 5! = 120.

The total number of numbers divisible by 2 = 120 x 3 = 360.