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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.
Square Roots and Cube Roots | Quantitative Aptitude (Quant) - CAT
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,

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

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 Square Roots and Cube Roots | Quantitative Aptitude (Quant) - CAT. 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.

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

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:
Square Roots and Cube Roots | Quantitative Aptitude (Quant) - CAT

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 Square Roots and Cube Roots | Quantitative Aptitude (Quant) - CAT. 
There are two main methods to simplify a square root.

Method 1: Factor the number under the Square Roots and Cube Roots | Quantitative Aptitude (Quant) - CAT 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 Square Roots and Cube Roots | Quantitative Aptitude (Quant) - CATinto prime factors and then simplify by bringing out any factors that came in pairs.

Example 2
Simplify √32.

Square Roots and Cube Roots | Quantitative Aptitude (Quant) - CAT
In Example , the largest perfect square is easy to see, and Method 1 probably is a faster method.

Example 3
Simplify √2016.

Square Roots and Cube Roots | Quantitative Aptitude (Quant) - CAT
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
x2
Cube
x3
Square Root x1/2Cubic Root x1/3
11111
2481.4141.26
39271.7321.442
4166421.587
5251252.2361.71
6362162.4491.817
7493432.6461.913
8645122.8282
98172932.08
1010010003.1622.154
1112113313.3172.224
1214417283.4642.289
1316921973.6062.351
1419627443.7422.41
1522533753.8732.466
16256409642.52
1728949134.1232.571
1832458324.2432.621
1936168594.3592.668
2040080004.4722.714
2144192614.5832.759
22484106484.692.802
23529121674.7962.844
24576138244.8992.884
256251562552.924
26676175765.0992.962
27729196835.1963
28784219525.2923.037
29841243895.3853.072
30900270005.4773.107
31961297915.5683.141
321024327685.6573.175
331089359375.7453.208
341156393045.8313.24
351225428755.9163.271
3612964665663.302
371369506536.0833.332
381444548726.1643.362
391521593196.2453.391
401600640006.3253.42
411681689216.4033.448
421764740886.4813.476
431849795076.5573.503
441936851846.6333.53
452025911256.7083.557
462116973366.7823.583
4722091038236.8563.609
4823041105926.9283.634
49240111764973.659
5025001250007.0713.684
5126011326517.1413.708
5227041406087.2113.733
5328091488777.283.756
5429161574647.3483.78
5530251663757.4163.803
5631361756167.4833.826
5732491851937.553.849
5833641951127.6163.871
5934812053797.6813.893
6036002160007.7463.915
6137212269817.813.936
6238442383287.8743.958
6339692500477.9373.979
64409626214484
6542252746258.0624.021
6643562874968.1244.041
6744893007638.1854.062
6846243144328.2464.082
6947613285098.3074.102
7049003430008.3674.121
7150413579118.4264.141
7251843732488.4854.16
7353293890178.5444.179
7454764052248.6024.198
7556254218758.664.217
7657764389768.7184.236
7759294565338.7754.254
7860844745528.8324.273
7962414930398.8884.291
8064005120008.9444.309
81656153144194.327
8267245513689.0554.344
8368895717879.114.362
8470565927049.1654.38
8572256141259.224.397
8673966360569.2744.414
8775696585039.3274.431
8877446814729.3814.448
8979217049699.4344.465
9081007290009.4874.481
9182817535719.5394.498
9284647786889.5924.514
9386498043579.6444.531
9488368305849.6954.547
9590258573759.7474.563
9692168847369.7984.579
9794099126739.8494.595
9896049411929.8994.61
9998019702999.954.626
100100001000000104.642
The document Square Roots and Cube Roots | Quantitative Aptitude (Quant) - CAT is a part of the CAT Course Quantitative Aptitude (Quant).
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FAQs on Square Roots and Cube Roots - Quantitative Aptitude (Quant) - CAT

1. What are square roots and cube roots?
Ans. Square roots and cube roots are mathematical operations used to find the original number that was squared or cubed. The square root of a number is the value that, when multiplied by itself, gives the original number. Similarly, the cube root of a number is the value that, when multiplied by itself twice, gives the original number.
2. How can I find the square root of a number?
Ans. To find the square root of a number, you can use a calculator or use manual methods such as the prime factorization method or the long division method. The prime factorization method involves breaking down the number into its prime factors and then taking the square root of each factor. The long division method involves dividing the number into smaller parts and iteratively finding the square root.
3. How can I find the cube root of a number?
Ans. Similar to finding the square root, you can use a calculator or manual methods to find the cube root of a number. One common method is the estimation method, where you make an initial guess and refine it using an iterative process. Another method is using prime factorization, where you break down the number into its prime factors and then find the cube root of each factor.
4. Are square roots and cube roots always positive?
Ans. Square roots and cube roots can be both positive and negative. When we talk about square roots, we usually refer to positive square roots. However, every positive number has two square roots - one positive and one negative. Similarly, every number has one real cube root, which can be positive or negative.
5. Can I find the square root or cube root of a negative number?
Ans. Yes, you can find the square root or cube root of a negative number. When dealing with negative numbers, the square root will always be a complex number, involving the imaginary unit "i." For example, the square root of -9 is 3i. Similarly, the cube root of a negative number will also involve complex numbers.
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