Square Root of 313

The square root of 313 is irrational. The approximate value of the square root of 313 is 17.69. When we say square root of a number, the square root number multiplied by itself gives a number that will be a perfect square. A number “s” is a square root, when s × s or -s × -s gives a product p. “p” will be called the perfect square number and “s” its square root. 

Consider the number 17 × 17 = 289 and 18 × 18 = 324. 313 lies between 289 and 324. So the square root of 313 lies between 17 and 18 as 313 is not a perfect square number. Hence, the square root of 313 is also not a perfect integer. Therefore, √313 is an irrational number. You can refer to Finding Square roots.

Note the Following:

• The Square root of 313 = √313 where ‘√’ is the radical, and 313 is the radicand.
• Exponential Form of Square root of 313 = 313^1/2. 
• Solution for √313 ≈ 17.69
• The square root of 313 is Irrational = True.

What is the Square root of 313?

The square root of 313 is 17.69, as the square root of 313 is irrational.

How to Find the Square root of 313?

There are three methods to find the Square root of 313 

• Prime Factorisation method
• Long Division method
• Repeated Subtraction method

Square root of 313 by Prime Factorisation Method

In the prime factorization method, the steps are: Divide the given number with prime factors, group the same prime factors and then multiply the groups to find the square root of the given number.

The given number, 313 will be expressed as;

313 = 313 × 1

313 is a prime number and it doesn’t have any other prime factors or factors. Hence it cannot be grouped and therefore it does not have a square root. 

Therefore the square root of 313 = √313= 17.69

Square root of 313 by Long Division Method

To understand the long division method, follow the below mentioned detailed steps;

Step 1:  Grouping the given number into pairs. 

Step 2: Consider the first number, use the divisor such that the product is less than or equal to the first number. The divisors should be:

1 × 1 = 1 

2 × 2 = 4

3 × 3 = 9

Step 3: Continue the division, by bringing the next pair.

Step 4: The first divisor in the next division to be  (Divisor of first division + quotient of first division) such that:

d1 × 1 =

d2 × 2 =

d3 × 3 =

 And so on. 

Step 5: If the division is complete with the remainder as zero, then the quotient will be the square root. Else the number is not a perfect square number.

Applying the above steps to 313.

Groups are 3 and 13.

The squares are: 

1 × 1 = 1

2 × 2 = 4 

No further division is possible and the remainder is non- zero. Hence 313 does not have a perfect square. Therefore 313 is an irrational number whose root lies between 17 and 18.

Square root of 313 by Repeated Subtraction Method.

In the repeated Subtraction method, 313 is subtracted with odd numbers starting with 1 and the results with, 3, 5, 7 and so on. If the result at any point becomes zero, then the number is a perfect square and the step at which it becomes zero is the square root. 

The table below shows repeated subtraction for the given number 313. 

The result at step 18 becomes negative and not zero. Hence 313 is not a perfect square number. The square root of 313 lies between 17 and 18.

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Solved Examples

1. Which number that is lesser than or greater than 313, that has a perfect square root?

289, which is less than 313 and 324 which is greater than 313, are the perfect square numbers. 17 and 18 are their perfect square roots respectively.

2. What is the square root of 324?

18 is the square root of 324.