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# Square Roots and Cube Roots Notes | Study Quantitative Aptitude (Quant) - CAT

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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= 36 and 72 = 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

 Numberx Squarex2 Cubex3 Square Root x1/2 Cubic Root x1/3 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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