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Square Roots and Cube Roots
To find the square root of a number, you want to find some number that when multiplied by itself gives you the original number. In other words, to find the square root of 25, you want to find the number that when multiplied by itself gives you 25. The square root of 25, then, is 5. The symbol for square root is √0. Following is a list of the first eleven perfect (whole number) square roots.
Special note: If no sign (or a positive sign) is placed in front of the square root, then the positive answer is required. Only if a negative sign is in front of the square root is the negative answer required. This notation is used in many texts and is adhered to in this book. Therefore,
Cube roots
To find the cube root of a number, you want to find some number that when multiplied by itself twice gives you the original number. In other words, to find the cube root of 8, you want to find the number that when multiplied by itself twice gives you 8. The cube root of 8, then, is 2, because 2 × 2 × 2 = 8. Notice that the symbol for cube root is the radical sign with a small three (called the index) above and to the left . Other roots are defined similarly and identified by the index given. (In square root, an index of two is understood and usually not written.) Following is a list of the first eleven perfect (whole number) cube roots.
Approximating square roots
To find the square root of a number that is not a perfect square, it will be necessary to find an approximate answer by using the procedure given in Example.
Example 1
Approximate √42.
Since 6^{2 }= 36 and 7^{2} = 49, then √42 is between √36 and √49.
Therefore, √42 is a value between 6 and 7. Since 42 is about halfway between 36 and 49, you can expect that √42 will be close to halfway between 6 and 7, or about 6.5. To check this estimation, 6.5 × 6.5 = 42.25, or about 42.
Square roots of nonperfect squares can be approximated, looked up in tables, or found by using a calculator. You may want to keep these two in mind:
Simplifying square roots
Sometimes you will have to simplify square roots, or write them in simplest form. In fractions, 2/4 can be reduced to 1/2. In square roots, √32 can be simplified to .
There are two main methods to simplify a square root.
Method 1: Factor the number under the into two factors, one of which is the largest possible perfect square. (Perfect squares are 1, 4, 9, 16, 25, 36, 49, …)
Method 2: Completely factor the number under the into prime factors and then simplify by bringing out any factors that came in pairs.
Example 2
Simplify √32.
In Example , the largest perfect square is easy to see, and Method 1 probably is a faster method.
Example 3
Simplify √2016.
In Example , it is not so obvious that the largest perfect square is 144, so Method 2 is probably the faster method.
Many square roots cannot be simplified because they are already in simplest form, such as √7, √10, and √15.
Square, Cube, Square Root and Cubic Root for Numbers Ranging 0  100
Number x  Square x^{2}  Cube x^{3}  Square Root x^{1/2}  Cubic Root x^{1/3} 
1  1  1  1  1 
2  4  8  1.414  1.26 
3  9  27  1.732  1.442 
4  16  64  2  1.587 
5  25  125  2.236  1.71 
6  36  216  2.449  1.817 
7  49  343  2.646  1.913 
8  64  512  2.828  2 
9  81  729  3  2.08 
10  100  1000  3.162  2.154 
11  121  1331  3.317  2.224 
12  144  1728  3.464  2.289 
13  169  2197  3.606  2.351 
14  196  2744  3.742  2.41 
15  225  3375  3.873  2.466 
16  256  4096  4  2.52 
17  289  4913  4.123  2.571 
18  324  5832  4.243  2.621 
19  361  6859  4.359  2.668 
20  400  8000  4.472  2.714 
21  441  9261  4.583  2.759 
22  484  10648  4.69  2.802 
23  529  12167  4.796  2.844 
24  576  13824  4.899  2.884 
25  625  15625  5  2.924 
26  676  17576  5.099  2.962 
27  729  19683  5.196  3 
28  784  21952  5.292  3.037 
29  841  24389  5.385  3.072 
30  900  27000  5.477  3.107 
31  961  29791  5.568  3.141 
32  1024  32768  5.657  3.175 
33  1089  35937  5.745  3.208 
34  1156  39304  5.831  3.24 
35  1225  42875  5.916  3.271 
36  1296  46656  6  3.302 
37  1369  50653  6.083  3.332 
38  1444  54872  6.164  3.362 
39  1521  59319  6.245  3.391 
40  1600  64000  6.325  3.42 
41  1681  68921  6.403  3.448 
42  1764  74088  6.481  3.476 
43  1849  79507  6.557  3.503 
44  1936  85184  6.633  3.53 
45  2025  91125  6.708  3.557 
46  2116  97336  6.782  3.583 
47  2209  103823  6.856  3.609 
48  2304  110592  6.928  3.634 
49  2401  117649  7  3.659 
50  2500  125000  7.071  3.684 
51  2601  132651  7.141  3.708 
52  2704  140608  7.211  3.733 
53  2809  148877  7.28  3.756 
54  2916  157464  7.348  3.78 
55  3025  166375  7.416  3.803 
56  3136  175616  7.483  3.826 
57  3249  185193  7.55  3.849 
58  3364  195112  7.616  3.871 
59  3481  205379  7.681  3.893 
60  3600  216000  7.746  3.915 
61  3721  226981  7.81  3.936 
62  3844  238328  7.874  3.958 
63  3969  250047  7.937  3.979 
64  4096  262144  8  4 
65  4225  274625  8.062  4.021 
66  4356  287496  8.124  4.041 
67  4489  300763  8.185  4.062 
68  4624  314432  8.246  4.082 
69  4761  328509  8.307  4.102 
70  4900  343000  8.367  4.121 
71  5041  357911  8.426  4.141 
72  5184  373248  8.485  4.16 
73  5329  389017  8.544  4.179 
74  5476  405224  8.602  4.198 
75  5625  421875  8.66  4.217 
76  5776  438976  8.718  4.236 
77  5929  456533  8.775  4.254 
78  6084  474552  8.832  4.273 
79  6241  493039  8.888  4.291 
80  6400  512000  8.944  4.309 
81  6561  531441  9  4.327 
82  6724  551368  9.055  4.344 
83  6889  571787  9.11  4.362 
84  7056  592704  9.165  4.38 
85  7225  614125  9.22  4.397 
86  7396  636056  9.274  4.414 
87  7569  658503  9.327  4.431 
88  7744  681472  9.381  4.448 
89  7921  704969  9.434  4.465 
90  8100  729000  9.487  4.481 
91  8281  753571  9.539  4.498 
92  8464  778688  9.592  4.514 
93  8649  804357  9.644  4.531 
94  8836  830584  9.695  4.547 
95  9025  857375  9.747  4.563 
96  9216  884736  9.798  4.579 
97  9409  912673  9.849  4.595 
98  9604  941192  9.899  4.61 
99  9801  970299  9.95  4.626 
100  10000  1000000  10  4.642 
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