#mathematics class

Introduction:-

What are Powers in Math or we can say what are exponents in math is our  today's main discussion which is a very nice and an important topic of mathematics.

Explanation:- 

To start this discussion let us quick remember in general we find four operations in algebra and they are addition,subtraction,multiplication and division. Powers are associated with multiplication.Let us know, What are Powers in Math. Let,  us suppose,  aⁿ is an expression ,here 'a' is called base and n is  called exponent and aⁿ is the power of a in the expression.In another way we can explain that the total number of times when any number is multiplied itself is called its power .

 e.g;        3 × 3 × 3 × 3 × 3 = 3⁵ where, 5 is the exponent of base 3 and  3⁵ is the power of 3

Now we will know about formula of power.

The formulae of exponents :-

   (i) aⁿ × aⁱ = aⁿ⁺ⁱ

Rule(1):-

    In case of multiplication of exponents, If the bases are equal then powers are added.

(ii) aⁿ ÷ aⁱ = aⁿ⁻ⁱ

 Rule(2):-

In case of division of exponents,If the bases are equal then powers are subtracted.

(iii) a⁰ = 1

Rule(3):-

If any number has power zero, then result will be 1

(iv) a⁻ⁿ = 1/aⁿ

Rule(4):-

 If any number has power negative,then it will be 1 divides the number withe power to make it positive.

(v) (aⁿ)ⁱ = aⁿⁱ

Rule(5):-

 In case of exponents if any expression has the power of power, then simply powers are multiplied themselves.

(vi)√a = a¹⁽²

Rule(6):-

A surd or root number can be converted into exponent by dividing the power with root value. Let, us now we will prove some above formulas of  exponents.

Prove  that,aⁿ × aⁱ = aⁿ⁺ⁱ

Answer:-

L.H.S = aⁿ × aⁱ   a       =(a.a.a…..n number of factors) × (a.a.a…..i number of factors)          = aⁿ × aⁱ          = aⁿ⁺ⁱ [∵According to law of exponents, if bases are equal then in case of multiplication powers are added]          = R.H.S   Proved.

Prove  that,aⁿ ÷ aⁱ = aⁿ⁻ⁱ

Answer:-

        L.H.S = aⁿ ÷ aⁱ   (a.a.a…..n number of factors) = —-----------------------------------------     (a.a.a…..i number of factors)           aⁿ =  —---------           aⁱ =   aⁿ⁻ⁱ [∵According to law of exponents, if bases are equal then in case of division powers are subtracted]  = R.H.S Proved.

Prove that, a⁰ = 1

Answer:-

           L.H.S = a⁰                       = aⁿ⁻ⁿ                         aⁿ                    = —-------                         aⁿ   [ ∵ Exponents are in subtraction, so they are will be in division individually.]                   =  1                   = R.H.S Proved.

 Prove that, a⁻ⁿ = 1/aⁿ

 Answer:- 

L.H.S = a⁻ⁿ              = a⁰⁻ⁿ                   a⁰              = —--                  aⁿ     [ ∵ Exponents are in subtraction, so they are will be in division individually.]                    1             = —----- [∵using formula(iii)                    aⁿ              = R.H.S Proved.

Some problems related on exponents:

 Express the following numbers in exponents or indices form.

  (i) 64  (ii) 343 Answer:- (i)  64                 = 4³ [∵ 4 × 4 × 4 = 64 ]                = (2²)³ [ ∵ 4 = 2 × 2 ]                = 2⁶ [ ∵ According to the law of exponents power of power is multiplied themselves]           (ii) 343            =  7³ [∵ 7 × 7 × 7 = 343]

Simplify:-

   (i) 125⁻¹⁽³  (ii) 64¹⁽⁶ Answer:-      (i) 125⁻¹⁽³      =  {(5)³}⁻¹⁽³ [ ∵5×5×5 = 125]      = 5 ³×⁻¹⁽³ [ ∵power of power will be multiplied]      = 5⁻¹ [ ∵3×(- ⅓ )= -1]      = ⅕ [ using formula (iv)      (ii) 64¹⁽⁶      = (2⁶)¹⁽⁶      =  2⁶ ×¹⁽⁶ [ ∵power of power will be multiplied]       =  2¹ [∵ 6 ×1/6 =1]       = 2

Find the value of n from the given expression:-

Answer:-   27ⁿ ÷ ⅓  = 3⁵   => (3³)ⁿ ÷ 3⁻¹ = 3⁵    => 3³ⁿ ÷ 3⁻¹ =3⁵[∵using formula(v)] => 3³ⁿ ⁻ ¹ = 3⁵ [∵using formula(ii)]   => 3n - 1 = 5 [ ∵ bases in both sides are equal, so their powers are also equal]   =>3n = 5 + 1   => n = 6/3     => n =2 

 Conclusion:-

From the above discussion we have learnt What are Powers in Math.  Generally exponents ..........

Introduction:-

All of we know what is light? Light is a form of energy,which gives us sensation to look anything clearly,without light in our life we cann't imagine a second.Today I will discuss on a small niche of light and that is what is the index of refraction of water.

Introduction:-

All of we know what is light? Light is a form of energy,which gives us sensation to look anything clearly,without light in our life we cann't imagine a second.Today I will discuss on a small niche of light and that is what is the index of refraction of water.

Table Of Contents

 Explanation:-

 To go further our discussion, let us first know

 what is  reflection?

Answer:-Light ray which is travelling from a source through a medium (say air) in a straight line if it is obstructed by another medium or a surface then it bounce back from that medium or surface to the its 1st medium, this property of light is called reflection of light.

In the picture,       MN = A plane mirror       AO = Incident ray       OB = Reflected ray       OC = Normal        O = Point of incidence

     <AOC = i = Angle of incidence

     <BOC = r = Angle of reflection Now we will learn,

What is refraction?

Answer:- Light ray which is coming from a source through a medium in a straight line  when it enters in another medium,then it slightly change the direction of its path on the contact point of two medium, this property of light is called refraction of light.

In the picture,             Ao = Incident ray.             OB = Refracted ray             CD = Normal             O = Point of contact             <i = Angle of incidence             <r = Angle of Refraction

Now we will know  

what is refractive index or index of refraction of light?

Answer:- According to the second law or Snell's law of refraction, the ratio of the sin of angle of incidence to the sin of angle of refraction is always a constant, which is termed as the refractive index or index of refraction of second medium with respect to first medium. In another way refractive index or index of refraction can be explained as, the ratio of the velocity of light in first medium to the velocity of light in second medium is also a constant,which is know as the refractive index or index of refraction of second medium with respect to first medium. Mathematically, Refractive index can be expressed as.   ¹μ₂ = v₁/v₂ Now I will reflect on our main topic 

Introduction:

Today  I will explain the topic Adding of Polynomials which is a very important topic of mathematics for school going children.

except coefficients all variables present in a term are called literal factors. e.g; 3xyz ,this is a monomial and literal factors if this term is xyz.    The terms which have same literal factors may have different coefficients are known as like terms. e.g; 5xy²z , -7xy²z are like terms.     The terms which have different literal factors are known as unlike terms. e.g; 6pqr, 6qrp² are unlike terms.

Find coefficients, literal factors, like terms and unlike terms.

Answer:- From the picture,

Coefficients :- 3,-2,5,1                 Literal factors:-xyz,xyp,mnl                 Like terms:- 3xyz,-2xyz               Unlike terms:- xyp,mnl

Add the following polynomials.

Question(1) 

3xyz , - xyz    Answer:-        3xyz                         - xyz                   —--------------                        2xyz

  Rule(1):- 

              (i) Polynomials are written vertically one after another.            (ii) Both like polynomials and coefficients of upper and lower are 3 and -1 respectively. So simply subtracting 3 - 1 = 2 [ ∵ Here grater number is + 3, so after subtraction the results sign will be +, here it is +2]

What is a linear equation?

Answer:- An equation which has one or two variables and highest power of the variables are one and when plotted on the graph forms a straight line is called a linear equation. e.g; 3x +2y = 0 is a linear equation of two variables, where x, y are two variables. again,ax + 5 = 0 is a linear equation of one / single variable of x.