## How to find Square Root of 289

Square of 289:

- In mathematics, finding of square of any number is mostly easy because when we multiply the same number with itself then we will get the square of that number.

For example:

- Let us suppose we have to find the square of any number say X, then we multiply X by itself i.e. X and we will get its square as Y. It can be written as (X)2 = X*X= Y
- In similar way we find the square of 9
- To find the square of 9,we multiply 9 by the number itself i.e. by 9 and we write it as follows.(289)2 = 17*17= 289

Square root of 289

- Now, in reverse manner if we have to find the square root of Y. The square root of Y is that single value which when multiplied with itself gives the value Y.
- That means, √Y = √(X*X) = X

Where √ is the symbol named as radical.

For example:

- The square root of 17 can be written as,

√289 = √ (17*17) = 17

Where √ is the symbol which is called as radical sign.

- In short, we remember square of 16 and square root of 256 as

Note:

- Every positive real number has two roots.
- The square of any negative number is always the positive number.

For example:

- 289 is the positive perfect square which has two roots +17 and -17 also.
- But, the positive square root value is taken mostly which is called as principal square root or non-negative square root.
- Hence, √289 = √(-17)*(-17) = -17 and √289 = √(17)*(17) = 17

Similarly,

- (-17)*(-17) = (-17)2 = +289 and (+17)*(+17) = (+17)2 = 289

### Methods to find square root of perfect square like 289:

There are many methods to find the square root of perfect squares out of which we see the following method in detail.

- Repeated Subtraction Method
- Prime factorization method

#### Repeated Subtraction Method:

- In repeated subtraction method, we have to subtract the consecutive odd numbers starting from 1, from the perfect square number whose square root we have to find.
- e. to find square root of 289, first we subtract 1 from it. 289 – 1 = 288

- Then next odd number is 3, so we have to subtract it from 288. 288– 3 = 285
- In this way, we subtract the consecutive odd numbers from the corresponding values obtained after subtraction continuously till we get final value as 0.
- And the value of number of odd numbers required to get 0 is the required square root.

For example:

- We find the square root of 289 by repeated subtraction method as follows:

289– 1 = 288

288– 3 = 285

285– 5 = 280

280– 7 = 273

273– 9 = 264

264– 11 = 253

253 – 13 = 240

240– 15 = 225

225 – 17 =208

208 – 19 =189

189 – 21 = 168

168 – 23 = 145

145 – 25 = 120

120 – 27 = 93

93 – 29. = 64

64 – 31 = 33

33 – 33 = 0

- Thus, here the total odd numbers used are 1, 3, 5, 7, 9, 11, 13, 15 ,17,19,21,23,25,27,29 ,31and 33which are 17 in numbers.
- Hence, the square root of 289 by repeated subtraction method is 17.

#### Prime Factorization method:

- In prime factorization method, we have to divide the perfect square number whose square root we have to find by prime number starting from 2, 3, 5… and so on till we get the remainder as 1.
- Initially we have to divide by prime number 2, if that number is not divisible by 2 then we have to take next prime number i.e. 3 and the process will be continued till we get remainder as 1.
- Finally, we have to make pairs of the prime numbers taken in the form of multiplication and then we have to take its square root.

For example:

- Following is the process to find the square root of 289 by prime factorization method.
- As 289 is odd number hence it must be divisible by only two be prime number 17.

289 ÷ 17= 17

17 ÷ 2 = 1

- Thus, the prime number 17 used to get remainder as 1 are 17

Thus, 289= 17*17= 17^2

And 289= (17*17)

- By taking square root on both sides, we get

√289 = √(17*17) = √(17*17)

√289 = (17*17)= 17

- Thus, we found the square root of 289 as 17 by using prime factorization method.

### Multiple Choice Questions:

1) What is the 5th multiple of 17

a) 511

b)85

c)1789

d)1534

Ans: b) 85

2) 289 is the type of number

a) whole number

b) natural number

c) rational number

d) all of these

Ans: d) all of these

3) The square root of 289 by prime factorization method is

a) 4

b) 9

c) 7

d) 17

Ans: d) 17