Suppose your friend asks you to find a number which when multiplied by itself gives 25. You can easily answer it. 5 multiply 5 gives 25. It is similar to finding the length of the side of a square when its area is known. We know that 25 is a perfect square. What is 5 known as? 5 is the square root of the square number 25. You can say finding square roots are the inverse operations of squaring. In this section, we will study about square root in more details.

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## Square Roots

When a number multiplies itself the product is the square number. The number is the square root. We get perfect square roots for a perfect square number. A square root is represented by a √ sign. If x^{2} is a square number then x is a square root of it.

Consider 1^{2} = 1, therefore square root of 1 is 1. 2^{2} = 4, therefore square root of 4 is 2. Similarly, 9^{2} = 81, therefore square root of 81 is 9 and so on. It is also interesting to know that when 9 square, it gives 81 and when −9 squares it also gives 81.

A perfect square has a negative and a positive square root both. But a negative square number cannot have a square root.

### Real Life Applications of Square Roots

Square roots have application in almost every known filed. The most basic amongst them are calculating the sides of a square when its area is known.

- To calculate the length of the diagonal of a square or rectangle.
- Pythagoras theorem for calculating the third side when any of the two sides of a right-angled triangle is known.
- In calculating the standard deviation from the variance of a data set.
- In solving a quadratic equation.

### Properties of Square Root

- A perfect square root exists for a perfect square number only.
- The square root of an even perfect square is even.
- An odd perfect square will have an odd square root.
- A perfect square cannot be negative and hence the square root of a negative number is not defined.
- Numbers ending with (having unit’s digit) 1, 4, 5, 6, or 9 will have a square root.
- If the unit digit of a number is 2, 3, 7, or 8 then a perfect square root is not possible.
- If a number ends with an odd number of zeros, then it cannot have a square root. A square root is only possible for even number of zeros.
- Two square roots can be multiplied. √5, when multiplied by √2, gives √10 as a result.
- Two same square roots are multiplied to give a non- square root number. When √25 is multiplied by √25 we get 25 as a result.

## Basic Methods of Finding a Square Root

### Repeated Subtraction Method

In this method, a perfect square number is subtracted repeatedly by the successively odd numbers i.e., 1, 3, 5, 7 etc. till we get zero. The number of times the subtraction is performed to get zero is counted. This count is the square root of the perfect square number. We know that 25 is a perfect square. Let us calculate its square root.

25 − 1 | = 24 |

24 − 3 | = 21 |

21 − 5 | = 16 |

16 − 7 | = 9 |

9 − 9 | = 0 |

The number of times the subtraction is performed is 5. Hence 5 is the square root of 25.

### Prime Factorization Method

The perfect square is factorized into its prime factors by successive division. The pairs of the prime factors are paired. Taking the product of one factor from each pair will result in the square root of the perfect square. Let us find the square root of 144.

The prime factorization of 144 is 144 = 2 × 2 × 2 × 2 × 3 × 3. Pairing the prime factors and selecting one from each pair gives 2 × 2 × 3 = 12. So, the square root of 144 = √144 = 12.

### Division Method

It is quite time-consuming and difficult to calculate the square root of a large number. To overcome this problem, a new method for finding the square root is developed. This method basically uses the division operation by a divisor whose square is either less than or equal to the dividend.

- Take the number whose square root is to find.
- Place a bar over every pair of the digit of the number starting from that in unit’s place (rightmost side).
- We divide the leftmost number by the largest number whose square is less than or equal to the number under the leftmost bar.
- Take this number as the divisor and the quotient. The number under the leftmost bar is considered to be the dividend.
- Divide and get the number.
- Bring down the number under the next bar to the right of the remainder.
- Double the divisor (or add divisor to itself).
- To the right of this divisor find a suitable number which together with divisor forms a new divisor for the new dividend. The new number in the quotient will have the same number as selected in the divisor. The condition is the same as being either less or equal to that of the dividend.

This process continues till we get zero as the remainder. The quotient thus obtained will be the square root of the number.

#### Example for Division Method

Let’s find out the square root of 225. Placing bar over the pair of digits in 225 starting from the unit place

Start the division from the leftmost side. Here 1 is the number whose square is less than 2. Putting it in the divisor and the quotient and then doubling it will give

Now we need to find a number for the blanks in divisor and quotient. Let that number be x. We need to check when 2x multiplies by x gives a number which is either less than or equal to 125. Take x = 1, 2, 3 and so on and check.

In this case, 24 × 4 = 96 and 25 × 5 = 125. So we choose x = 5 as the new digit to be put in divisor and in the quotient. The remainder here is 0 and hence 15 is the square root of 225.

## Solved Example for You

**Question 1: Find the number which should be subtracted to make 9609 a perfect square.**

**Answer :** Finding the square root of 9609 by long division method leaves a remainder 5. It shows that 98^{2} is less than 9609 by 5. This means that if we subtract 9609 by 5 (remainder), we get a perfect square.

**Question 2: What are square roots?**

**Answer: **When a number is multiplying itself the product refers to the square number. Further, the number is the square root. Thus, you get perfect square roots for a perfect square number. We represent a square root by a √ sign. In other words, if a2 is a square number then a is a square root of it.

**Question 3: What is square of square root?**

**Answer:** Finding the square root of a number is basically the inverse operation of squaring that number. Thus, the square of a number is that number times itself. Moreover, the perfect squares are the squares of the whole numbers.

**Question 4: What are the 4 properties of the square root?**

**Answer:** The four properties are that firstly, a perfect square root exists for a perfect square number only. Then, the square root of an even perfect square is even. Moreover, an odd perfect square will have an odd square root.

**Question 5: Who invented the square root?**

**Answer:** Babylonians invented the square root method. It dates back to 1900 BC. They had an accurate plus simple way of finding out the square roots of numbers. We also refer to it as Heron’s method.

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