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${2}^{6}$ – 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 $1Square

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