Square Root of 369
The square root of a number is the value which when multiplied by itself gives the original number. For example, 19 × 19 = 361. So, the square root of 361 is 19. In this minilesson, we will learn about the square root of 369. We will do so using methods like long division. The square root of 369 is a number which when multiplied by itself, results in the number 369. 369 is an odd number and a composite number. Let us determine the square root of 369 in this article. We will learn how to calculate the square root of 369, find out if the square root of 369 is rational or irrational, and look at a few problems to help us understand this topic.
 Square Root of 369: √369 = 19.2
 Square of 369: 369^{²} = 136161
1.  What Is the Square Root of 369? 
2.  Is Square Root of 369 Rational or Irrational? 
3.  How to Find the Square Root of 369? 
4.  Important Notes 
5.  FAQs on Square Root of 369 
What Is the Square Root of 369?
If a number is a perfect square, it is easy to evaluate the square root using the inverse operation of the squaring operation. 369 is not a perfect square. √369 can be evaluated using the long division method. The square root of a number n is written as √n. This number when squared or multiplied by itself results in the original number n. The square root of 369 can be written as:
 Radical form: √369
 Decimal form: 19.2
 Exponent form: (369)^{½}
Is Square Root of 369 Rational or Irrational?
The square root of 369 is a nonrepeating and nonterminating number. So, it cannot be expressed in the form of p/q where q ≠ 0. Hence, the square root of 369 is an irrational number. 369 cannot be broken into two equal factors which on multiplying give 369. It can be approximately written as a square of 19.2, which is a nonrecurring and nonterminating decimal number. This shows it isn't a perfect square, which also proves that the square root of 369 is an irrational number.
How to Find the Square Root of 369?
There are 2 ways to find the square root of 369:
 Long Division Method
 Prime Factorization
Long Division Method
The square root of 369 by long division method consists of the following steps:
 Pair the digits of the number from one's digit by placing a bar over them as shown in the image.
 The first pair just consists of 1 number, i.e., 3. Square of 1 gives a product less than 3. The number is subtracted from the first pair and subsequently, the next pair is added as the divisor, i.e., 69.
 The double of 1 is 2. Now a number is to be placed along with 2, i.e., 9. The product of 29 by 9 gives 261. When the difference between 269 and 261 is taken, the remainder is 8.
 Now double of the quotient is taken. Now a decimal is added to the quotient, hence letting us add a pair of zeros to the original dividend. A number along with 38 is added in the blank and it is multiplied by the same number in the quotient. Product of 382 with 2 gives 764.
 The difference obtained from the above step is 800  764 = 36.
 The process is repeated the same way.
Therefore, the square root of 369 = 19.2
The square root of 369 by prime factorization:
Prime factorization of 369 = 3 × 3 × 41
Simplified form of √369 is = 3√41
Square root of 41 is 6.4031.
√369 = 3 × 6.4031 = 19.2
Therefore, the square root of √369 = 19.209
Important Notes:
 369 is a number that is not a perfect square, meaning it does not have a natural number as its square root.
 Square root of 369 is simlified in decimal form as,√369 = 19.2
 Square root of 369 cannot be expressed as a fraction if the form p/q which tells us that the square root of 369 is an irrational number.
Explore square roots using illustrations and interactive examples
 Square Root of 361
 Square Root of 324
 Square Root of 289
 Square Root of 300
 Square Root of 250
 Square Root of 288
 Square Root of 69
Challenging Questions
 What is the negative root of 3690?
 Find the square root of 369 up to 5 decimal places?
 What is the square root of:
a) 36996
b) 369469
Solved Examples

Example 1: The area of a squareshaped plot is 369 square units. Calculate the measurement of one side of a land.
Solution:
The area of land = 369 sq. units
To find the side of a squareshaped land, let us take square root of 369 by prime factorization method.
Prime factorization of 369 = 3 × 3 × 41
Simplified form of √369 is = 3√41
Square root of 41 is 6.4031.
√369 = 3 × 6.4031 = 19.2
Therefore, the square root of √369 = 19.209
Therefore, side of square plot is 19.209 units. 
Example 2: What number Tim should be multiplying with 369 to obtain a perfect square number. If the obtained product is 400, then what is the square root of 400?
Solution:
To find the number let us assume number = a
On multiplying 369 times a = 400
a = 400 ÷ 369
a = 1.0840
369 × 1.0840 = 399.99 ≅ 400
400 is a perfect square number. Hence, √400 = 20.
The number is 1.0840. 
Example 3: What number should Amy subtract from 369 to obtain a perfect square number. If the square root of the obtained number is 19, find the number. Find out whether the obtained number is also a perfect square.
Solution:
To find the number, let us assume number = a
On subtracting 369  a = (19)²
a = 369  361
a = 8
8 is not a perfect square number √8 = 2.8284
The number is 8, and it is not a perfect square.
FAQs on Square Root of 369
What is the square root of 369 using prime factorization?
Prime factorization of 369 = 3 × 3 × 41
Simplified form of √369 is = 3√41
Square root of 41 is 6.4031.
√369 = 3 × 6.4031 = 19.2
Therefore, the square root of √369 = 19.209
Is 369 a perfect square root?
369 is not a perfect square. 369 is a natural number, but since there is no other natural number that can be squared to result in the number 369, it is NOT a perfect square.
What is the decimal value of the square root of 369?
The decimal value of the square root of 369 is √369 = 19.209.
Is the square root of 369 a rational number?
No, the square root of 369 is not a rational number since the square root of 369 is nonterminating and cannot be represented in the form of p/q where q ≠ 0.
Can we find the square root of 369 by the repeated subtraction method?
No, we can’t find the square root of 369 by repeated subtraction method as it can be used only for perfect squares. 369 is not a perfect square.
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