Square of a number: The Square of a number is that number raised to the power $2$.

Examples: Square of $9=9^2=9\times 9=81$ and Square of $0.2=(0.2)^2=0.2\times 0.2=0.04$

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

Example: We have $1^2=1, \ 2^2=4, \ 3^2=9$

Some Properties of Squares of Numbers

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

Example: $2$ is even and $2^2=4$, which is even.

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

Example: $3$ is odd and $3^2=9$,  which is odd.

(iii) The square of a proper fraction is a proper fraction less than the given fraction. $\displaystyle \text{ Example: Square of } \frac{1}{2} = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \text{ and we see that } \frac{1}{4}<\frac{1}{2}$

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

Example: $0.1<1$ and $(0.1)^2=0.1 \times 0.1=0.001<0.1$

(v) A number ending in $2, 3, 7 \ or\ 8$ is never a perfect square.

Example: The numbers $72, 243, 567$  and $1098$ end  in $2, 3, 7$  and $8$ 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 $690, 87000$ 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 $x$ as the product. We denote the square root of a number $\sqrt{x}$ by.

Example: Since $7 \times 7 = 49$ , so $\sqrt{49}=7$, i.e., the square root of $49$ is $7$

Methods of finding the square root of numbers:

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

1. Resolve the given number into prime factors
2. Make pairs of similar factors
3. 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: $\ 1296=2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3$. Hence $\sqrt{1296}=2\times 2\times 3\times 3=36$

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

1. Mark off the digits in pairs starting with the unit digit. Each pair and remaining one digit (if any) is called a period.
2. 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.
3. 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.
4. 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 $a$ and $b$, we have:

i) $\sqrt{ab}=\sqrt{a}\times \sqrt{b}$ $\displaystyle \text{ Example: Find the square root of } \frac{\sqrt{441}}{\sqrt{1849}} = \frac{21}{43}$

ii) $\displaystyle \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ $\displaystyle \text{ Example: } \sqrt{\frac{9}{16}} = \frac{\sqrt{9}}{\sqrt{16}} = \frac{3}{4}$

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

Example: Cube of $2=2^3=2 \times 2 \times 2=8$

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

Example: $1^3=1,\ 2^3=8,\ 3^3=27$ and so on.

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 $\sqrt{x}$

Example: Since $\ 5 \times 5 \times 5=125$, therefore $\sqrt{125}=5$

Method of finding the cube root of numbers: Cube Root of a Given Number by Prime Factorization Method

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

Example: Find Cube of $17576$ $17576 = 2 \times 2 \times 2 \times 13 \times 13 \times 13$

Therefore $\sqrt{17576} = 2 \times 13 = 26$

Cube Roots of Fractions and Decimals

Example: Find cube root of: $\displaystyle \sqrt{\frac{2197}{343}} = \frac{\sqrt{2197}}{\sqrt{343}} = \frac{\sqrt{13\times 13\times 13}}{\sqrt{7\times 7\times 7}}$ $\displaystyle \sqrt{19.683}= \frac{\sqrt{19686}}{\sqrt{1000}} = \frac{27}{10}$

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. ${x}$ ${x}$ $\sqrt{x}$ $\sqrt{x}$ ${x}$ ${x}$ $\sqrt{x}$ $\sqrt{x}$ ${x}$ ${x}$ $\sqrt{x}$ $\sqrt{x}$ ${x}$ ${x}$ $\sqrt{x}$ $\sqrt{x}$ ${x}$ ${x}$ $\sqrt{x}$ $\sqrt{x}$ 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: $\sqrt{20}=4.472,\ \sqrt{54}=7.348,\ \sqrt{83}=9.110,\ \sqrt{99}=9.950$

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. ${x}$ ${x}$ $\sqrt{x}$ $\sqrt{x}$ $\sqrt{10x}$ $\sqrt{10x}$ $\sqrt{100x}$ $\sqrt{100x}$ ${x}$ ${x}$ $\sqrt{x}$ $\sqrt{x}$ $\sqrt{10x}$ $\sqrt{10x}$ $\sqrt{100x}$ $\sqrt{100x}$ ${x}$ ${x}$ $\sqrt{x}$ $\sqrt{x}$ $\sqrt{10x}$ $\sqrt{10x}$ $\sqrt{100x}$ $\sqrt{100x}$ 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: $\sqrt{5}=1.710;\ \sqrt{50}=3.684;\ \sqrt{500}=7.937$