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Square roots and cube roots, while being a very basic part of mathematics, make for an important part of the syllabus for competitive exams. Questions on square roots and cube roots are expected as a part of the Quantitative Aptitude for exams for Bank PO, NABARD Grade A, NABARD Grade B, SBI-PO, IBPS SO, RBI, Railways and SSC. In this chapter let us try to understand what squares and square roots are and in the next chapter, we will look at cubes and cube roots.

Questions from this topic will not be directly asked in the exams. However, this topic will help you in improving your calculation speed.

WHAT ARE SQUARE ROOTS?

In mathematics, the square root of a positive original number (say, x) is a unit when multiplied by itself gives the original number. While Arab mathematicians used the Arabic word “*jadhr*” to denote this term, medieval Europeans used the Latin word radix, a word that has been adopted universally in the mathematical discipline. Thus, a square root of a number is called its radix. The original number, x, is a perfect square. Take a look at the simple equation below.

√16= 4.

√16= 4x4

Here, 16 is the original non-integer number (x) which is a perfect square having a square root.

4 is the square root of the original number, which when multiplied by itself gives us the perfect square. Alternately, the same equation may be expressed as;

42= 16

A square number can be expressed as a two-dimensional square.

It must be remembered that although positive original numbers have positive square roots even negative original numbers have positive square roots because when two negative numbers are multiplied the answer achieved is positive.

√-49=x

√-49= -7x -7

√-49= 7

∴ The square root of √-49 is 7.

These square roots have some properties and patterns which can be remembered for quick and efficient calculations.

- The square of even numbers is always even and that of odd numbers is always odd.

e.g., 182= 324; 192= 361

- If the original number ends with a 1 or 9, its square root will end with a 1.

e.g., √121= 11

- If the square number ends with 5, its square root will also always end with a 5

e.g., √225= 15

- If the square number ends in 6, the square root will end with 4 or 6

e.g., √196= 14

- If a square has an even number of zeroes at the end of it, the square root will have half the number of zeroes.

e.g., √90000= 300

- Similarly, the number of zeroes in a perfect square is always even

e.g., 102= 100

- If the unit’s place ends with 2, 3, 7, and 8, then the square root is not a natural number.

Provided below is the square root table for numbers 1-30 from the Decimal number system (as we learned in the chapter on number systems). Memorizing this table containing square roots 1 to 30. It will prove helpful for all mathematics questions at large.

SQUARE ROOT | ORIGINAL NUMBER/ SQUARE | EXPANDED FORM |
---|---|---|

1 |
1 |
1x1 |

2 |
4 |
2x2 |

3 |
9 |
3x3 |

4 |
16 |
4x4 |

5 |
25 |
5x5 |

6 |
36 |
6x6 |

7 |
49 |
7x7 |

8 |
64 |
8x8 |

9 |
81 |
9x9 |

10 |
100 |
10x10 |

11 |
121 |
11x11 |

12 |
144 |
12x12 |

13 |
169 |
13x13 |

14 |
196 |
14x14 |

15 |
225 |
15x15 |

16 |
256 |
16x16 |

17 |
289 |
17x17 |

18 |
324 |
18x18 |

19 |
361 |
19x19 |

20 |
400 |
20x20 |

21 |
441 |
21x21 |

22 |
484 |
22x22 |

23 |
529 |
23x23 |

24 |
576 |
24x24 |

25 |
625 |
25x25 |

26 |
676 |
26x26 |

27 |
729 |
27x27 |

28 |
784 |
28x28 |

29 |
841 |
29x29 |

30 |
900 |
30x30 |

How to find square roots?

It is understood that squares and square roots of all the numbers cannot be committed to memory. As such, there are a few easy methods that we can use to quickly derive the square roots of large numbers.

Subtraction method: In this method, we will repeatedly subtract the number with consecutive odd numbers.

√25= x

25-1=24

24-3=21

21-5=16

16-7=9

9-9=0

As we can see, we have had to subtract the number 25 a total of 5 times to get 0. Thus, the square root is 5. Therefore, x=5.

Prime factorization method: Firstly, find the prime factors of the original number. Pair the prime numbers such that both factors are equal. Take 1 number from each pair and multiply them with each other. The result is the square root.

√25= x

Prime factorization numbers of 25 are 1 and 5

(1x1) x (5x5) = x

1x5= x

∴ x = 5.

Find the square root of 225.

√225=x

Prime factorization of 225= 3x3x5x5

(3x3)x(5x5)= x

3x5=x

∴ x = 15.

Estimation method: This method of estimation and approximation is generally applied to numbers that are not perfect squares and therefore do not have perfect square roots. The nearest perfect square roots (both higher than and lower than the given number) are identified and then the estimated number is placed in between the two. Let us study with an example for better clarity.

Find the square root of √29.

The nearest perfect squares are 25 and 36 the square roots of which are 5 and 6 respectively. Therefore, the square root of √29 lies between 5 and 6. Let us now assume that the root lies between 5 and 5.5. 5 squared is 25 and 5.5 squared is 30.25. Since 29 is closer to 30.25 we can assume that the square root is closer to 5.5. The same process can be repeated between 5.3 and 5.4 till the answer is derived. The square root of 29 to two places is 5.38 which can be rounded off as 5.4 if brought to one place after the decimal. Thus, the square root is 5.4. However, this method is avoided since it is inefficient and time-consuming.

Square root by Long Division method: This method is widely used when calculating the square roots of large numbers. Let us try to understand the workings of this method as we illustrate with an example for better understanding.

Find the square root of 1296.

- Begin making pairs from the right of the number.
- Thus, 9 and 6 will be paired as will 1 and 2.
- We have 12 and 96
- Find the perfect square closest to 12, i.e., 9.
- The square of 9 is 3. So, we will select 3 as one of the first quotients. We need to find the second half of the quotient, let’s call it y. The quotient is 3y.
- By carrying over the remainder, (12-9=3), the dividend now becomes 396.
- We will multiply the first number above with 2. Thus, 3x2= 6.
- For y, we will think of a suitable number that could give us the square root. Since 1, 3, 5 and 7 are not probably options (odd numbers) we will begin with a middle place even number like 4.
- 34x34= 1156
- Since this is not the correct answer, we will try the next even number, i.e., 6
- 36x36= 1296.
- ∴ The square root of 1296 is 36.

Tips and Tricks to find Square Roots:

- Try to memorize the squares of numbers between 1 and 20.
- Once you have memorized them, finding the square roots of bigger numbers like 5-digit and 6-digit numbers is easy.
- Remember, earlier we learned how the last digits of squares of numbers between 1 to 10 always result in the same number.
- You can easily find square roots of larger numbers by using this simple trick. let us study with an example:

Find the square root of : 2916.

Pair the given numbers from the right hand side. so now we have 16 and 29.

Since we know that 16 ends with a 6 the root number in the unit’s place will end with either 4 or 6 .

For 29 we will find the closest squared number which is less than 29. The closest square is 25 which is the square of 5.

Now we will multiply this square root number by number +1which is = 6.

5x6=30. Since 29 is less than 30 we will pick the smaller number for the unit’s place.

So the square root of 2916 is = 54.

- This same method can be applied to 5-digit numbers as well.

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