Square Root

# What Is Square Root? What Are the Methods Used to Find the Square Root?

Written by Prerit Jain

Updated on: 26 May 2023

Contents

### What Is Square Root? What Are the Methods Used to Find the Square Root?

Square root calculation is used in finance, quadratic formula, standard deviation lengths and distance, and many more. With this blog, we will understand what is square root, the formula and symbol, and some solved examples.

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**Square root**

The square root of a value is a number when multiplied by itself gives the original number or the square root is the inverse of squaring the numbers. Which tells that both concepts are dependable on each other.

The square root of a number in the radical form is denoted as √ when they are in the exponential form they are denoted as the (number)^{1/2 }or the (number) ^{0.5}.

Let us take p and q. The square root of p = √q or p = q^{2} the number that is positive when positive or multiplied gives the exact number.

That is if we take the number 9 the square root of 9 is found by multiplying the number twice.

9 = 3^{2 }or √9 = 3, when the number three is multiplied it gives the number 9.

**Symbol of the square root**

The square root symbol is denoted in a radical way. Let us say we have to express x in the square root then it is denoted as x. The symbol in radical form is also called radicant. Here x = any number.

The square root of 4 is expressed as √4

**The formula for finding the square root**

The formula is simple **x = √y or x = y ^{2}**

**Rational and irrational numbers**

Rational numbers are expressed as a fraction of the integer. That is they are simple fractions and p/q ≠0. **Let us take the number 121 as an example:**

The square root of 121 = 11, when we divide the number 121/11 = 11 and not zero or a complicated fraction.

An irrational number leads to complicated fractions and is often difficult to solve. Any number that is not expressed as a fraction is called an irrational number.

The square root of 5 = 7.071 which cannot be further converted into a fraction and is also not equal to 0.

**Perfect square and not a perfect square**

A perfect square number is one whose square roots give a whole number. The square root of 4 is 2, the square root of 400 is 20, etc.,

A not-perfect square number is one whose roots are not whole numbers but rather decimals.

The square root of 5 = 7.071

**What are the methods to find the square root of a number?**

The numbers that end with 2,3, 7 0r 8 at the unit place are not perfect square numbers and are a little difficult to find. The numbers ending with 1,4,5,6 or 9 in the unit place are perfect square numbers and can be found easily using the prime factorization or the long division method.

The method to find the square root is

- Prime factorization method
- Long division method
- Repeated subtraction method

**Prime factorization method**

Prime factorization is one of the easiest methods to find the square root of any number. Let us take n as the prime number, by grouping the similar numbers we get n^{2}, by multiplying the similar n^{2. }The value we get is the square root of the number.

Let us take the number 144 as an example and solve them using the prime factorization method.

The prime factor of 144 = 12×12

By squaring them we get 12^{2}, since two is the only similar square we take the common one.

Here, the common square is 12.

Hence, the √**144 = 12**

**Long division method**

**Step 1:**Find the smallest integer that can divide the number.**Step 2:**Keep following the long division using divisor and dividend.**Step 3:**When the particular number of satisfaction is reached the quotient is the square root of the number.

**Find the square root of 2**

**Step 1:**Find the smallest integer that can divide the number. 1 is the perfect root number that is smaller and closest to 2.**Step 2:**Keep following the long division using divisor and dividend.

**Step 3**: When the particular number of satisfaction is reached the quotient is the square root of the number. Hence, the**√2 = 1.4142**

**Repeated subtraction method**

The repeated subtraction method first subtracts the given number from the odd numbers until you get zero. The step in which zero is obtained is the square root of the number. It is a time-consuming and lengthy process.

Find the square root of 36 using the repeated subtraction method

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

We got zero in the sixth step. Hence, √**36 = 6**.

**The square root of -1**

Understanding the square root of minus values is simple. There is no real square root for the negative integers which are often referred to as imaginary roots. Why? Because what happens when you multiply two negative values is you get a positive number and not a negative one. Let’s say we need to find the square root of -1. We know the definition of square root, thus multiplying or squaring these values we get 1 and not a negative result.

-1 = (-1) (-1) = 1.

Therefore they are referred to as imaginary roots and are denoted as I.

Find the square root of -2

√-2 = 1.4142i

**The square root of the numbers from 1 to 20**

Squaring number | Square root |

1 | 1 |

2 | 1.4142 |

3 | 1.732 |

4 | 2 |

5 | 2.236 |

6 | 2.449 |

7 | 2.6457 |

8 | 2.828 |

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 |

**Solved examples**

**Q1: Solve the equation √2a+9 = 5**

A1: √2a+9 = 5

Let us take square on both sides

2a+9 = 5^{2}

2a+9 = 25

2a = 25-9

2a = 16

a = 16/2**a=8**

**Q2: Square root of 125 using the prime factorization method**

A2: The prime factors of 125 = 5×5×5

√125 = 5×5

= 5 × 2.236

= 11.8

**Q3: Square root of 12 using the long division method**

A 3: √12 = 3.464

**Q4: Find the square root of pi**

A 4: √𝛑 = 1.77

**Q5: Find the square root of 64 using the repeated subtraction method**

- 64-1 = 63
- 63-3 = 60
- 60 – 5 = 55
- 55-7 = 48
- 48 – 9 = 39
- 39-11 = 28
- 28-13 = 15
- 15-15 = 0
- √64 = 8

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**Frequently asked question**

**What is a square root?**

The square root of a value is a number when multiplied by itself gives the original number or the square root is the inverse of squaring the numbers. Which tells that both concepts are dependable on each other

**Give an example of** **a perfect square number.**

The numbers that are perfect squares are 4,16 and 25 are some of the examples for perfect square numbers.

**What are irrational numbers?**

Any number that is not expressed as a fraction is called an irrational number.

**What is the square root of -1?**

We know the definition of square root, thus multiplying or squaring these values we get 1 and not a negative result.

-1 = (-1) (-1) = 1. The square root of -1 is i

**What is a radicand?**

The square root symbol is denoted as √ called a radicand.

**What is the formula for square root?**

The formula is simple x = y or x = y^{2}

**What is the square root of 100?**

The square root of 100 is 10

Written by

Prerit Jain