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# Square Roots Definition, Methods and Cube Roots List

Square Roots: The square root of a number is multiplying factor of a number which when multiplied by itself hives the actual number. The square root is just the opposite method of squaring a number. In squaring of a number same number is multiplied by itself giving the square of a number while in square root the original number is obtained on squaring the number. The square of any number is always a positive number while the square root of a number has both positive and negative values. We can understand the square and square root easily by taking a simple example. Suppose if ‘x’ is a square of the number ‘y’ means x × x = y but if ‘x’ is a square root of ‘y’ means x2= y.

## Square Roots

A Square root is a number that when multiplying gives the original number. It is like a factor, both square and square roots are a type of exponent. In other square roots can be defined as the number whose value has the power of 1/2 of a number. The symbol to represent the square root is √ which is called radical in mathematics and the number inside or called the radicand. For example, suppose a number 16 which can be obtained by multiplying 4 by 4 i.e. 4 × 4 = 16 me, and 16 is a square of 4. And suppose a number √9 on squaring the number √32 = 3 so the square root of √9 is 3. Here we will discuss the square root in detail. So refer to the below content to understand the square root in a good manner.

## How to find Square Root?

The square root of a number is the square of a number that gives the original number. When finding a square root of a number firstly we need to find whether the number is a perfect square or imperfect. A perfect square is a number that can be expressed in the form of power 2. The square root of the perfect square can be found easily by factoring it into the prime factors. If a number is an imperfect square then the long division method needs to be performed on it which is discussed below in the article.

To find the square root of a number following four methods are used:

• Square Root by Prime Factorization
• Square Root by Repeated Subtraction Method
• Square Root by Long Division Method
• Square Root by Estimation Method

We will discuss all the above four methods in detail with a suitable example.

### 1. Square Root by Prime Factorisation Method

The square root of a perfect square number can be found easily by Prime Factorisation Method. In this method, some steps need to be followed to find the square root of a number listed below:

• Divide the given number into its prime factors.
• Form pairs of similar factors such that both factors in each pair are equal.
• Take one factor from the pair.
• Find the product of the factors obtained by taking one factor from each pair.
•  The product gives the square root of the given number. 64 = 2×2×2×2×2×2×2

On pairing, we get 2×2×2 = 8

So the square root of 64 is 8.

### 2. Square Root by Repeated Subtraction Method

In the Repeated Subtraction Method, the square root of a number can be found by following the below steps if a given number is a perfect square.

• Repeatedly subtract the consecutive odd numbers from the given number
• Subtract till the difference remains zero
• The number of times we subtract will be the required square root

For example, let us find the square root of the number 36

• 36 – 1 = 35
• 35 – 3 = 32
• 32 – 5 = 27
• 27 – 7 = 20
• 20 – 9 = 11
• 11 – 11 = 0

Since, the subtraction is done 6 times, hence the square root of 36 is 6

### 3. Square Root by Long Division Method

Using the long division method the square roots of imperfect squares can be found easily. The method can be understood by the below steps.

• Place a bar over every pair of digits of the number starting from the units’ place (right-side).
•  Then we divide the left-most number by the largest number whose square is less than or equal to the number in the left-most pair.

Calculate the square root of 17.64 The square root of 17.64 by the long division method is found to be 4.2

### 4. Square Root by Estimation Method

This method used approximation to find the square root of a number. The square root can be found by guessing the approximate value.

For example, we know that the square root of 4 is 2 and the square root of 9 is 3, thus we can guess, that the square root of 5 will lie between 2 and 3.

But, we need to check the value of √5 is nearer to 2 or 3. Let us find the squares 2.2 and 2.8.

• 2.22 = 4.84
• 2.82 = 7.84

Since the square of 2.2 gives 5 approximately, thus we can estimate the square root of 5 is equal to 2.2 approximately.

Square Root 4

The square root of 4 is denoted by √4. We know that 4 is a perfect square whose square root can be found easily by the prime factorization method or repeated subtraction method. The square root of 4 is 2. But the square roots have two values always positive and negative as we already discussed. So √4 has two values +2 and -2.

Square roots 1 to 30

As we discussed the square riots can be found easily using the four methods mentioned above. The list of square roots from 1 to 30 is mentioned here.

 Square Roots 1 to 30 √1 = 1 √2 = 1.4142 √3 = 1.732 √4 = 2 √5 = 2.236 √6 = 2.4494 √7 = 2.6457 √8 = 2.8284 √9 = 3 √10 = 3.1622 √11 = 3.3166 √12 = 3.4641 √13 = 3.6055 √14 = 3.7416 √15 = 3.8729 √16 = 4 √17 = 4.1231 √18 = 4.2426 √19 = 4.3588 √20 = 4.4721 √21 = 4.5825 √22 = 4.6904 √23 = 4.7958 √24 = 4.8989 √25 = 5 √26 = 5.099 √27 = 5.1961 √28 = 5.2915 √29 = 5.3851 √30 = 5.4772

## Square Roots and Squares

The square root is a factor multiplying with a given number that gives the original number while the square is a number that is multiplied by itself. Both roots and squares are considered exponents. They are both opposite to each other. The examples of both square and square roots are discussed above already.

## Square Roots and Cube Roots

To find the square root of any number, we need to find a number that when multiplied twice by itself gives the original number. Similarly, to find the cube root of any number we need to find a number that when multiplied three times by itself gives the original number.

Notation: The square root is denoted by the symbol ‘√’, whereas the cube root is denoted by ‘∛’.

Examples:

• 4 = √(2 × 2) = 2
•  ∛27 = ∛(3 × 3 × 3) = 3

## Square Roots and Cube Roots List

The list of square roots is discussed below which will be helpful for the students to make their preparation boost for upcoming exams. So refer to the list below for the same.

Number
x
Square
x2
Cube
x3
Square Root
x1/2
Cubic Root
x1/3
1 1 1 1.000 1.000
2 4 8 1.414 1.260
3 9 27 1.732 1.442
4 16 64 2.000 1.587
5 25 125 2.236 1.710
6 36 216 2.449 1.817
7 49 343 2.646 1.913
8 64 512 2.828 2.000
9 81 729 3.000 2.080
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 2.410
15 225 3375 3.873 2.466
16 256 4096 4.000 2.520
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 4.690 2.802
23 529 12167 4.796 2.844
24 576 13824 4.899 2.884
25 625 15625 5.000 2.924
26 676 17576 5.099 2.962
27 729 19683 5.196 3.000
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 3.240
35 1225 42875 5.916 3.271
36 1296 46656 6.000 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 3.420
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 3.530
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 7.000 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 7.280 3.756
54 2916 157464 7.348 3.780
55 3025 166375 7.416 3.803
56 3136 175616 7.483 3.826
57 3249 185193 7.550 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 7.810 3.936
62 3844 238328 7.874 3.958
63 3969 250047 7.937 3.979
64 4096 262144 8.000 4.000
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 4.160
73 5329 389017 8.544 4.179
74 5476 405224 8.602 4.198
75 5625 421875 8.660 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 9.000 4.327
82 6724 551368 9.055 4.344
83 6889 571787 9.110 4.362
84 7056 592704 9.165 4.380
85 7225 614125 9.220 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 4.610
99 9801 970299 9.950 4.626
100 10000 1000000 10.000 4.642

### Square Roots: FAQs

Que.1 What are squares?

Ans – The squares are a twice multiplication of the number itself. For example, 25 is a square of 5 i.e. 5×5 = 25

Que.2 What are square roots?

Ans – Square root is just opposite to square. Is the 1/2 power of a number. For example, the square root of 9 is 3 i.e. √9 = 3 #### Congratulations! General Awareness & Science Capsule PDF 