Square of a number: The Square of a number is that number raised to the power .

Examples: Square of and Square of

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

Example: We have

Some Properties of Squares of Numbers

(i) The square of an even number is always an even number.

Example: is even and , which is even.

(ii) The square of an odd number is always an odd number.

Example: is odd and , which is odd.

(iii) The square of a proper fraction is a proper fraction less than the given fraction.

(iv) The square of a decimal fraction less than 1 is smaller than the given decimal.

Example: and

(v) A number ending in is never a perfect square.

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

(vi) A number ending in an odd number of zeros is never a perfect square.

Examples: The numbers and 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 is that number which when multiplied by itself gives as the product. We denote the square root of a number by.

Example: Since , so , i.e., the square root of is

Methods of finding the square root 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.

Example: . Hence

*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 root of fractions*

For any positive real numbers and , we have:

i)

ii)

Cube of a number: The cube of a number is that number raised to the power 3.

Example: Cube of

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

Example: and so on.

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

Example: Since , therefore

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.

Example: Find Cube of

Therefore

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 |

Examples:

Cube roots of numbers, using cube root table

The table given below shows the values of where 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 |

Example: