– where 2 is the base & 6 is the exponent, and read as “two raised to the power six”

The negative integral exponent of a rational number when a is the nonzero R.N

,

(there is and multiplication inverse)

where m > n

Express the numbers in exponential form

where e.g=”” =”” =”” <strong=””>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 squares.</a<1036 = 2*2*3*31^2=12^2 =4n=1∴2n=22^2=43^2=9n=2 ∴2n=43^2=94^2=16n=3 ∴2n=62^2=43^2=93^2=9=3×34^2=16=3×5+14^2=16=4×45^2=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 a right-angled triangle:-

For any number 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, the square root is an inverse process, of the square.

2×2 = 4 & √(4 )= √(2×2) = 2

Note:- if a number has a natural number as a 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 natural numbers

e.g √25= ≠5

to find the square root by factorization method

√16= √(4×4)=4

The number being the perfect square, will have one or more pairs of the 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 subtraction 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

The square root of rational numbers(fraction)

√(a/b)= √a/√b

The 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 multiplied by itself x×x = x2

Similarly, if a number is multiplied by itself 3 times we get the cube of that number x×x×x = x3

Perfect cube:- a natural number is said to be a perfect cube if it is a 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 numbers are odd.

In a perfect cube, each prime number appears three times in its prime factorization √27= √(3×3×3)=

The Cube of negative numbers is negative

Cubes of numbers ending with 0,1,4,5,6 & 9 also end with the same digit ending with 8 will end with Similar cubes of numbers ending in 3 &7 will end with 7 & 3 respectively.

Smallest number:- some numbers are expressed as the same of two squares & 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

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