Exponent


{2}^{6} – 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
a^{(-n)}=  (\frac{1}{a})^n , (\frac{2}{3})^{-4}=  \frac{1}{\frac{3}{4}}^4
(\frac{a}{b}){-n}= (\frac{b}{a})^n (there is \frac{a}{b} and \frac{b}{a} multiplication inverse)

a^m×a^n=a^{(m+n)} a^m/(a^n  ) = a^{(m-n)} where m > n (a^m )^n=a^{mn}

(ab)^n= a^n*b^n (\frac{a}{b})^n=\frac{a^n}{b^n}

Express the numbers in exponantal form
(a*10)^b where 1<a<10 e.g .008 = \frac{8}{1000} = {8×10}^3 400000 = 4*100000 = (4×10)^5 Square
The square of a number is the product of the number with the number itself
e.g. 1^{2}=1, 2^2  =4, 3^2=9
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. 36 = 2*2*3*3 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
1^2=1 , 2^2  =4 , n=1 ∴2n=2
2^2=4 , 3^2=9 , n=2     ∴2n=4
3^2=9 , 4^2=16 , n=3     ∴2n=6

The square of an even number is always an even number & the square of an odd number is always an odd number.
2^2=4, 3^2=9
The secure of a natural number (expert ) is either or multiple of 3 or exceeds a multiple of 3 & 41
3^2=9=3×3
4^2=16=3×5+1
Similarly- with 4
4^2=16=4×4
5^2=25=4×6+1

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