Exponent – where 2 is the base & 6 is the exponent, and read as “two raised to the power six”
Negative integral exponent of a rational number when a is the non zero R.N ,  (there is and multiplication inverse)  where m > n   Express the numbers in exponantal form where e.g  Square
The square of a number is the product of the number with the number itself
e.g. , , 1,4,9,16,25……… are called perfect square.

To find the perfect square– factorize the given number, if their factors are in pair, then the number is perfect square,
e.g. is a perfect square.

Number perfect square.

No perfect square will end with 2,3,7,or 8 at the unit place.
A number having 0,1,4,5,6,9, in the unit place may or may not be a perfect number

(1) If the number has 1 or 9 at the unit place than its square and with 1
(2) The square of a number which has 4 or 6 at the units place will and in 6
(3) A number ending in an odd number of zero is never a perfect square.
E.g. 640, 5000, 44000
Between the square of the numbers n and n+1 there are 2n non perfect square numbers , ,   , ,  , , The square of an even number is always an even number & the square of an odd number is always an odd number. , The secure of a natural number (expert ) is either or multiple of 3 or exceeds a multiple of 3 & 41  Similarly- with 4  The square of natural number ending with five follows a delimit pattern
52 = (0×1) hundred + 25 = 25
152 = (1×2) hundred + 25 = 225
252 = (2×3) hundreds + 25 = 625
352 = (3×4) hundred + 25 = 1225
452 = (4×5) hundred + 25 = 2025

The sum of first nodal natural numbers in n2
Sum of first odd number = 1 =12
Sum of first two odd number = 1+3 = 4 = 22
Sum of first three odd number = 1+3+5 = 9 = 32
The sum of rfirst four oddd number = 1+3+5+> = 16 =42

Look at this pattern whose number include only one.

12 = 1 ___________(= 12)
112 = 121 (1+2+1 = 4 = 22)
1112 = 12321 (1+2+3+2+1 = 9 = 32)
11112 = 1234321 (1+2+3+4+3+2+1 = 16 = 42)
111112 = 121 (1+2+3+4+5+4+3+2+1 = 25 = 52)

Square of these numbers is serially overdraft 1,2,3 …………..equal to the number of its digit & decreases vice verse ………3,2,1
The sum of the digit of their product is also a perfect square.

72 = 49 pattern- when number of digit = 1 than number of 4 = n, & number of 8 = n time +1
672 = 4489
6672 = 444889
666722 = 44448889

Pythagorean triplet
In right angled triangle:-
For any numbewr m>1, (2m, m2-1, m2 +1) is a Pythagorean triplet
If 3, 4, 5
M=2 22-1, 2×2, 22+1

Square root
The square root of the number , is that number, which when multiplied by itself , gives the number as the product.
√x×√x=x we denote the square root of x, by √x , square root is a inverse process, of square.
2×2 = 4 & √(4 )= √(2×2) = 2
Note:- if a number has a natural numbewr as square root then its units digit must be 0,1,4,5,6 or 9. Negative numbers have no square root in the system of natual numbers
e.g √25= ≠5
to find the square root by factorisation method
√16= √(4×4)=4
The number being the perfect square, will have one or more pairs of ther prime factor, write one factor from each pair & multiplied these factors , the product will be the square root of the number e.g√81= √(3×3×3×3)=3×3=9
To find square root by successive substraction method
The sum of the first n odd natural numbers is n2
This method is useful to find the square root of smaller natural numbers.
81-1 = 80
80-3 = 77
77-5 = 72
72-7 = 65
65-9 = 56
56-11 = 45
45-13 = 32
32-15 = 17
17-17 = 0
Square root by division method
e.g.
to find the number of digit in the square root
√((81) ̅ ) = 1 digit = 9
√(2 ̅(25) ̅ )=2 digit=25
√((20) ̅(25) ̅ )=2 digit=48
√(2 ̅8 ̅(224) ̅ )=3 digit=168
Square root of rational numbers(fraction)
√(a/b)= √a/√b
Square root of decimal
We see that 0.2 ×0.2 = 0.04 ∴ √(0.(04) ̅ ) = 0.2 number of digit 1 & root of 4 = 2
Approximate value of square root
We get the square of that number is multiplied by itself x×x = x2
Similarly if a number is multiplied by itself 3 times we get cube of that number x×x×x = x3
Perfect cube:- a natural number is said to be a perfect cube if it is the cube or a natural number.
e.g 13 = 1, 23 = 8, 33 = 27 thus 1,8,27 are perfect cube.
Properties:-
The cube of even numbers are even & odd number are odd.
In a perfect cube each prime numbers appears three times in its prime factorization √27= √(3×3×3)=
Cube of negative numbers is negative
Cube of number ending with 0,1,4,5,6 & 9 also end with the same digit ending with 8 will end with similarly cubes of number ending 3 &7 will end with 7 & 3 respectively.
Smallest number:- some numbers are expressed as the same of two square & sum of two cubes also.
c.g. :-
1729 = 1728 + 1 = 123 +1
1729 = 1000 + 729 = 103 + 93
4104 = 8 + 4096 = 23 + 163, 4104 = 729 + 3375 = 92 + 153
13832 = 5832 + 8000 = 183 + 203, 13832 = 8+13824 = 23 + 243
Cube root of decimal numbers
To find the cube root of aq decimal numbers, write the number in the form of p/q and then find their cube root

Algebraic expression

A combination of constant and variables connected by some or all at the the four fundamental operations, additions, substraction, multiplication & division is called an algabric expression
e.g- 3x + zy

Terms:- the different part of an algebraic expression separated by sign+ or – are called the terms of an expression.
e.g. 3x+2y (term – 3x & 2y)

factor of terms :- we can factroized all terms all terms.-
e.g 3x + 2y
3x = 3×x
2y = 2×y

Types of algebaric expressions.

Monomial– which contains only one term is said a monimial.
Ginomial– which contains two terms e.g. 3x + 2y
Trinomials– which contains three terms – e.g- 3x +2y +z
Quadrinomials – which contains four terms e.g. 2x + 3y + z-6
Polynomials– which contains one or more terms.

Degree of polynomials The highest power of the variable in a polynomials is called its degree. Here the degree of the polynomial is 3

Linear polynomial– a polynomial of degree is called a linear polynomials. E.g.- x+3

Quadratic polynomials– a polynomials of degree 2 is called a quadratic polynomials.

E.g. (x+2)(x+3) = x2 + 5x + 6

Cubic polynomial– A polynomial of degree 3 is called cubic polynomials.
e.g. degree of the term- 3x =1
2xy = 1+1 = 2
3x2g = 2 = 1 = 3

Algebraic expressions contains one or more forms and each terms contain variable and numerical coefficient, we find the value of terms to put the value of variables.
Equation– a statement of equality which invader one or more variable is called an equation. Terms of left hand side is equal to right hand side e.g – 3x+5 = 8

Solution of an equation e.g. 3x+5=8

(1 )Trial & error method– put the value of variable x, so that L.H.S= R.H.S.
Put the value of x= 1,2,3 who satisfy by the equation
3(1)=5 =8 the value of x= 1 is satisfied equation

(2) (A) if same number or terms is added, substract multiply or divide to both side of equation , the equation remain same (elemination method)
3x+5=8
3x=3(deduct 5 in both side)
X=1 (divid from 3 in both side)
(B) change in the side of required terms.
3x+5= 8
3x= 8-5
3x=3
x = 3/3 = 1

Passing object from jsp page to action using model driven

Project Explorer struts.xml

StudentAction.java

Student.java

showstudent.jsp

OUTPUT  Passing parameter from jsp page to action

Project Explorer Library files struts.xml

StudentAction.java

showstudent.jsp

web.xml

Output  