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Square Root of 13
The symbol of square root is written as (√) and is an integral part of mathematics. Once you understand the basics of finding the square root of a number, you can solve any square rootrelated problem.
Today, let us find the square root of 13, and explore the answer to questions like, is the square root of 13 a rational number and the square root of 13 in radical form.
 Square root of 13: √13 = 3.60555128
 Square of 13: 13^{2} = 169
1.  What Is the Square Root of 13? 
2.  Is Square Root of 13 Rational or Irrational? 
3.  How to Find the Square Root of 13? 
4.  FAQs on Square Root of 13 
What Is the Square Root of 13?
The square root of 13 is expressed as √ 13 in the radical form and as 13^{½} in the exponent form.
The square root of 13 rounded to 5 decimal places is ±3.60555.
Is the Square Root of 13 Rational or Irrational?
A number that cannot be expressed as a ratio of two integers is an irrational number. The decimal form of the irrational number is nonterminating (i.e., it never ends) and nonrecurring (i.e., the decimal part of the number never repeats a pattern). Now let us look at the square root of 13.
Is the square root of 13 rational or irrational?
 √13 = 3.60555128
Do you think the decimal part stops after 3.60555128? No, it is neverending and you cannot see a pattern in the decimal part.
 Square root of √13 is an irrational number.
How to Find the Square Root of 13?
Square Roots can be calculated using various methods:
 By simplifying the radical of the numbers that are perfect squares.
 By long division method for perfect and nonperfect squares
13 is a prime number and hence, it is not a perfect square. Therefore, the square root of 13 can only be calculated by the long division method.
Simplified Radical Form of Square Root of 13
To simplify the square root of 13, let us first express 13 as a product of its prime factors.
 Prime factorization of 13 = 1 × 13.
 √13 is in the lowest form and cannot be simplified further.
 We have expressed the square root of 13 in the radical form.
Square Root of 13 By Long Division
Let us follow the steps to find the square root of 13 by long division.
 Step 1: Group the digits into pairs from right to left by placing a bar over it.
 Step 2: Find the largest number such that when you multiply it with itself, the product is less than or equal to 13. We know that 3 × 3 is 9 and is less than 13. Now let us divide 13 ÷ 3.
 Step 3: Let us place a decimal point and pairs of zeros after that to continue our division.
 Now, multiply the quotient by 2 and the product becomes the starting digit of our next divisor.
 Step 4: Choose a number in the units place for the new divisor such that its product with a number is less than or equal to 400. We know that 6 is in the tens place and our product has to be 400 and the closest multiplication is 66 × 6 = 396
 Step 5: Bring down the next pair of zeros and multiply the quotient 36 (ignore the decimal) by 2, which is 72, and the starting digit of the new divisor.
 Step 6: Choose the largest digit in the units place for the new divisor such that the product of the new divisor with the digit at ones place is less than or equal to 400. We see that 721 when multiplied by 1, gives 721 which is greater than 400. Therefore, we will take 720 × 0 = 0 which is less than 400.
 Step 7: Add more pairs of zeros and repeat the process of finding the new divisor and product as done in step 2.
 Step 8: Choose the largest digit at the units place for the new divisor such that the product of the new divisor with the digit at ones place is less than or equal to 40000. We see that 7205 when multiplied by 5, gives 36025 which is less than 40000.
 Thus, the square root of 13 obtained by long division is 3.60555128.
Can you try and express the square root of 17 in a similar way? The answer of √17 should be 4.123
Think Tank:
 Can you think of a quadratic equation which has a root as √13?
 Since (√13)^{2 }= 13, can we say that √13 is also a square root of 13?
Important Notes:
 The square root of 13 in the radical form, is expressed as √13.
 The square root of 13 is expressed as 13^{½} in the exponent form.
 The real roots of √13 are ±3.605
Square Root of 13 Solved Examples

Example 1
James told his friends that the value of √13 is the same as √13. What do you think?
Solution
Negative square roots cannot be real numbers.
√13 = 3.60555 is a real number.
But √13 = 3.60555 √1 is an imaginary number.
Hence, they are not the same.
√13 is not the same as √13 
Example 2
Thomas had a question. He knew that 3.605 is the square root of 13 and wanted to know if 3.605 is also the square root of 13? Can you answer his question?
Solution
Let us take an example of a perfect square number and extend the same logic to answer his question.
We know that 2 is a square root of 4 because when 2 is multiplied to itself it gives 4.
But what about 2?
Let us multiply and check.
2 × 2 = 4 because () × () = (+)
Therefore, 2 is also a square root of 4.
Going by the same logic,
3.605 is also the square root of 13.
FAQs On Square Root of 13
What is the square root of 13?
The square root of 13 is √13 = 3.60555128.
What is the square of 13?
The square of 13 is 169.
How do you work out the square root of 13?
We can find the square root of 13 using various methods.
 Repeated Subtraction
 Prime Factorization
 Estimation and Approximation
 Long Division
If you want to learn more about each of these methods, click here.
What is the square root of 13 in the simplest radical form?
√13 is the simplest radical form.
Is the square root of 13 an irrational number?
√13 = 3.60555128. √13 is an irrational number.
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