SQUARE OF A NUMBER: The Square of a number is that number raised to the power 2.

Examples:

PERFECT SQUARE: A natural number is called a perfect square, if it is the square of some natural number.

Example

Some Properties of Squares of Numbers

- The square of an even number is always an even number.

- The square of an odd number is always an odd number.

- The square of a proper fraction is a proper fraction less than the given fraction.

- The square of a decimal fraction less than 1 is smaller than the given decimal.

Example:

- A number ending in is never a perfect square.

Example: The numbers respectively. So, none of them is a perfect square.

- A number ending in an odd number of zeros is never a perfect square.

Examples: The numbers 690, 87000 and 4900000 end in one zero, three zeros and five zeros respectively. So, none of them is a perfect square

SQUARE ROOT: The square root of a number *x* is that number which when multiplied by itself gives as the product. We denote the square root of a number by.

METHODS OF FINDING THE SQUARE ROOTS OF NUMBERS

To Find the Square Root of a Given Perfect Square Number Using Prime Factorization Method:

- Resolve the given number into prime factors
- Make pairs of similar factors
- The product of prime factors, chosen one out of every pair, gives the square root of the given number.

Test for a number to be a Perfect Square: A given number is a perfect square, if it can be expressed as the product of pairs of equal factors.

To Find the Square Root of a given number By Division Method

- Mark off the digits in pairs starting with the unit digit. Each pair and remaining one digit (if any) is called a period.
- Think of the largest number whose square is equal to or just less than the first period. Take this number as the divisor as well as quotient.
- Subtract the product of divisor and quotient from first period and bring down the next period to the right of the remainder. This becomes the new dividend.
- Now, new divisor is obtained by taking twice the quotient and annexing with it a suitable digit which is also taken as the next digit of the quotient, chosen in such a way that the product of new divisor and this digit is equal to or just less than the new dividend.

Repeat steps 2, 3 and 4 till all the periods have been taken up. Now, the quotient so obtained is the required square root of the given number.

SQUARE ROOT OF NUMBERS IN DECIMAL FORM

Method: Make the number of decimal places even, by affixing a zero, if necessary. Now, mark periods (starting from the right most digit) and find out the square root by the long-division method. Put the decimal point in the square root as soon as the integral part is exhausted.

Square root of numbers which are not perfect squares

SQUARE ROOTS OF FRACTIONS: For any positive real numbers *a *and *b, *we have:

i)

Example: Find the square root of

ii)

Example:

CUBE OF A NUMBER: The cube of a number is that number raised to the power 3.

PERFECT CUBE: A natural number is said to be a perfect cube, if it is the cube of some natural number.

CUBE ROOT: The cube root of a number *x* is that number which when multiplied by itself three times gives* x *as the product*. *We denote the cube root of a number

METHOD OF FINDING THE CUBE ROOT OF NUMBERS: Cube Root of a Given Number by Prime Factorization Method

- Resolve the given number into prime factors.
- Make groups in triplets of similar
- The product of prime factors, chosen one out of ever triplet, gives the cube root of the given number.

Cube Roots of Fractions and Decimals

Example: Find cube root of:

SQUARE ROOTS BY USING TABLES

A table showing the square roots of all natural numbers from 1 to 100 has been given, each approximating to 3 places of decimal using this table, we can find the square roots of numbers, larger than 100, as illustrated in the following examples.

1 | 1.000 | 21 | 4.583 | 41 | 6.403 | 61 | 7.810 | 81 | 9.000 |

2 | 1.414 | 22 | 4.690 | 42 | 6.481 | 62 | 7.874 | 82 | 9.055 |

3 | 1.732 | 23 | 4.796 | 43 | 6.557 | 63 | 7.937 | 83 | 9.110 |

4 | 2.000 | 24 | 4.899 | 44 | 6.633 | 64 | 8.000 | 84 | 9.165 |

5 | 2.236 | 25 | 5.000 | 45 | 6.708 | 65 | 8.062 | 85 | 9.220 |

6 | 2.449 | 26 | 5.099 | 46 | 6.782 | 66 | 8.124 | 86 | 9.274 |

7 | 2.646 | 27 | 5.196 | 47 | 6.856 | 67 | 8.185 | 87 | 9.327 |

8 | 2.828 | 28 | 5.292 | 48 | 6.928 | 68 | 8.246 | 88 | 9.381 |

9 | 3.000 | 29 | 5.385 | 49 | 7.000 | 69 | 8.307 | 89 | 9.434 |

10 | 3.162 | 30 | 5.477 | 50 | 7.071 | 70 | 8.367 | 90 | 9.487 |

11 | 3.317 | 31 | 5.568 | 51 | 7.141 | 71 | 8.426 | 91 | 9.539 |

12 | 3.464 | 32 | 5.657 | 52 | 7.211 | 72 | 8.485 | 92 | 9.592 |

13 | 3.606 | 33 | 5.745 | 53 | 7.280 | 73 | 8.544 | 93 | 9.644 |

14 | 3.742 | 34 | 5.831 | 54 | 7.348 | 74 | 8.602 | 94 | 9.695 |

15 | 3.873 | 35 | 5.916 | 55 | 7.416 | 75 | 8.660 | 95 | 9.747 |

16 | 4.000 | 36 | 6.000 | 56 | 7.483 | 76 | 8.718 | 96 | 9.798 |

17 | 4.123 | 37 | 6.083 | 57 | 7.550 | 77 | 8.775 | 97 | 9.849 |

18 | 4.243 | 38 | 6.164 | 58 | 7.616 | 78 | 8.832 | 98 | 9.899 |

19 | 4.359 | 39 | 6.245 | 59 | 7.681 | 79 | 8.888 | 99 | 9.950 |

20 | 4.472 | 40 | 6.325 | 60 | 7.746 | 80 | 8.944 | 100 | 10.000 |

CUBE ROOTS OF NUMBERS, USING CUBE ROOT TABLE

The table given below shows the values of where x is a natural number. Using this table, we may find the cube root of any given natural number.

1 | 1.000 | 2.154 | 4.642 | 35 | 3.271 | 7.047 | 15.183 | 69 | 4.102 | 8.837 | 19.038 |

2 | 1.260 | 2.714 | 5.848 | 36 | 3.302 | 7.114 | 15.326 | 70 | 4.121 | 8.879 | 19.129 |

3 | 1.442 | 3.107 | 6.694 | 37 | 3.332 | 7.179 | 15.467 | 71 | 4.141 | 8.921 | 19.220 |

4 | 1.587 | 3.420 | 7.368 | 38 | 3.362 | 7.243 | 15.605 | 72 | 4.160 | 8.963 | 19.310 |

5 | 1.710 | 3.684 | 7.937 | 39 | 3.391 | 7.306 | 15.741 | 73 | 4.179 | 9.004 | 19.399 |

6 | 1.817 | 3.915 | 8.434 | 40 | 3.420 | 7.368 | 15.874 | 74 | 4.198 | 9.045 | 19.487 |

7 | 1.913 | 4.121 | 8.879 | 41 | 3.448 | 7.429 | 16.005 | 75 | 4.217 | 9.086 | 19.574 |

8 | 2.000 | 4.309 | 9.283 | 42 | 3.476 | 7.489 | 16.134 | 76 | 4.236 | 9.126 | 19.661 |

9 | 2.080 | 4.481 | 9.655 | 43 | 3.503 | 7.548 | 16.261 | 77 | 4.254 | 9.166 | 19.747 |

10 | 2.154 | 4.642 | 10.000 | 44 | 3.530 | 7.606 | 16.386 | 78 | 4.273 | 9.205 | 19.832 |

11 | 2.224 | 4.791 | 10.323 | 45 | 3.557 | 7.663 | 16.510 | 79 | 4.291 | 9.244 | 19.916 |

12 | 2.289 | 4.932 | 10.627 | 46 | 3.583 | 7.719 | 16.631 | 80 | 4.309 | 9.283 | 20.000 |

13 | 2.351 | 5.066 | 10.914 | 47 | 3.609 | 7.775 | 16.751 | 81 | 4.327 | 9.322 | 20.083 |

14 | 2.410 | 5.192 | 11.187 | 48 | 3.634 | 7.830 | 16.869 | 82 | 4.344 | 9.360 | 20.165 |

15 | 2.466 | 5.313 | 11.447 | 49 | 3.659 | 7.884 | 16.985 | 83 | 4.362 | 9.398 | 20.247 |

16 | 2.520 | 5.429 | 11.696 | 50 | 3.684 | 7.937 | 17.100 | 84 | 4.380 | 9.435 | 20.328 |

17 | 2.571 | 5.540 | 11.935 | 51 | 3.708 | 7.990 | 17.213 | 85 | 4.397 | 9.473 | 20.408 |

18 | 2.621 | 5.646 | 12.164 | 52 | 3.733 | 8.041 | 17.325 | 86 | 4.414 | 9.510 | 20.488 |

19 | 2.668 | 5.749 | 12.386 | 53 | 3.756 | 8.093 | 17.435 | 87 | 4.431 | 9.546 | 20.567 |

20 | 2.714 | 5.848 | 12.599 | 54 | 3.780 | 8.143 | 17.544 | 88 | 4.448 | 9.583 | 20.646 |

21 | 2.759 | 5.944 | 12.806 | 55 | 3.803 | 8.193 | 17.652 | 89 | 4.465 | 9.619 | 20.724 |

22 | 2.802 | 6.037 | 13.006 | 56 | 3.826 | 8.243 | 17.758 | 90 | 4.481 | 9.655 | 20.801 |

23 | 2.844 | 6.127 | 13.200 | 57 | 3.849 | 8.291 | 17.863 | 91 | 4.498 | 9.691 | 20.878 |

24 | 2.884 | 6.214 | 13.389 | 58 | 3.871 | 8.340 | 17.967 | 92 | 4.514 | 9.726 | 20.954 |

25 | 2.924 | 6.300 | 13.572 | 59 | 3.893 | 8.387 | 18.070 | 93 | 4.531 | 9.761 | 21.029 |

26 | 2.962 | 6.383 | 13.751 | 60 | 3.915 | 8.434 | 18.171 | 94 | 4.547 | 9.796 | 21.105 |

27 | 3.000 | 6.463 | 13.925 | 61 | 3.936 | 8.481 | 18.272 | 95 | 4.563 | 9.830 | 21.179 |

28 | 3.037 | 6.542 | 14.095 | 62 | 3.958 | 8.527 | 18.371 | 96 | 4.579 | 9.865 | 21.253 |

29 | 3.072 | 6.619 | 14.260 | 63 | 3.979 | 8.573 | 18.469 | 97 | 4.595 | 9.899 | 21.327 |

30 | 3.107 | 6.694 | 14.422 | 64 | 4.000 | 8.618 | 18.566 | 98 | 4.610 | 9.933 | 21.400 |

31 | 3.141 | 6.768 | 14.581 | 65 | 4.021 | 8.662 | 18.663 | 99 | 4.626 | 9.967 | 21.472 |

32 | 3.175 | 6.840 | 14.736 | 66 | 4.041 | 8.707 | 18.758 | 100 | 4.642 | 10.000 | 21.544 |

33 | 3.208 | 6.910 | 14.888 | 67 | 4.062 | 8.750 | 18.852 | ||||

34 | 3.240 | 6.980 | 15.037 | 68 | 4.082 | 8.794 | 18.945 |