Square Root of 514
The square root of 514 is expressed as √514 in the radical form and as (514)^{½} or (514)^{0.5} in the exponent form. The square root of 514 rounded up to 5 decimal places is 22.67157. It is the positive solution of the equation x^{2} = 514.
 Square Root of 514: 22.67156809750927
 Square Root of 514 in exponential form: (514)^{½} or (514)^{0.5}
 Square Root of 514 in radical form: √514
1.  What is the Square Root of 514? 
2.  How to find the Square Root of 514? 
3.  Is the Square Root of 514 Irrational? 
4.  FAQs 
What is the Square Root of 514?
The square root of 514, (or root 514), is the number which when multiplied by itself gives the product as 514. Therefore, the square root of 514 = √514 = 22.67156809750927.
☛ Check: Square Root Calculator
How to Find Square Root of 514?
Value of √514 by Long Division Method
Explanation:
 Forming pairs: 05 and 14
 Find a number Y (2) such that whose square is <= 5. Now divide 05 by 2 with quotient as 2.
 Bring down the next pair 14, to the right of the remainder 1. The new dividend is now 114.
 Add the last digit of the quotient (2) to the divisor (2) i.e. 2 + 2 = 4. To the right of 4, find a digit Z (which is 2) such that 4Z × Z <= 114. After finding Z, together 4 and Z (2) form a new divisor 42 for the new dividend 114.
 Divide 114 by 42 with the quotient as 2, giving the remainder = 114  42 × 2 = 114  84 = 30.
 Now, let's find the decimal places after the quotient 22.
 Bring down 00 to the right of this remainder 30. The new dividend is now 3000.
 Add the last digit of quotient to divisor i.e. 2 + 42 = 44. To the right of 44, find a digit Z (which is 6) such that 44Z × Z <= 3000. Together they form a new divisor (446) for the new dividend (3000).
 Divide 3000 by 446 with the quotient as 6, giving the remainder = 3000  446 × 6 = 3000  2676 = 324.
 Bring down 00 again. Repeat above steps for finding more decimal places for the square root of 514.
Therefore, the square root of 514 by long division method is 22.6 approximately.
Is Square Root of 514 Irrational?
The actual value of √514 is undetermined. The value of √514 up to 25 decimal places is 22.67156809750926778624096. Hence, the square root of 514 is an irrational number.
☛ Also Check:
 Square Root of 900  √900 = 30
 Square Root of 14  √14 = 3.74166
 Square Root of 256  √256 = 16
 Square Root of 320  √320 = 17.88854
 Square Root of 34  √34 = 5.83095
 Square Root of 512  √512 = 22.62742
 Square Root of 15  √15 = 3.87298
Square Root of 514 Solved Examples

Example 1: Solve the equation x^{2} − 514 = 0
Solution:
x^{2}  514 = 0 i.e. x^{2} = 514
x = ±√514
Since the value of the square root of 514 is 22.672,
⇒ x = +√514 or √514 = 22.672 or 22.672. 
Example 2: If the area of an equilateral triangle is 514√3 in^{2}. Find the length of one of the sides of the triangle.
Solution:
Let 'a' be the length of one of the sides of the equilateral triangle.
⇒ Area of the equilateral triangle = (√3/4)a^{2} = 514√3 in^{2}
⇒ a = ±√2056 in
Since length can't be negative,
⇒ a = √2056 = 2 √514
We know that the square root of 514 is 22.672.
⇒ a = 45.343 in 
Example 3: If the surface area of a cube is 3084 in^{2}. Find the length of the side of the cube.
Solution:
Let 'a' be the length of the side of the cube.
⇒ Area of the cube = 6a^{2} = 3084 in^{2}
⇒ a = ±√514 in
Since length can't be negative,
⇒ a = √514
We know that the square root of 514 is 22.672.
⇒ a = 22.672 in
FAQs on the Square Root of 514
What is the Value of the Square Root of 514?
The square root of 514 is 22.67156.
Why is the Square Root of 514 an Irrational Number?
Upon prime factorizing 514 i.e. 2^{1} × 257^{1}, 2 is in odd power. Therefore, the square root of 514 is irrational.
What is the Square of the Square Root of 514?
The square of the square root of 514 is the number 514 itself i.e. (√514)^{2} = (514)^{2/2} = 514.
Is the number 514 a Perfect Square?
The prime factorization of 514 = 2^{1} × 257^{1}. Here, the prime factor 2 is not in the pair. Therefore, 514 is not a perfect square.
Evaluate 19 plus 9 square root 514
The given expression is 19 + 9 √514. We know that the square root of 514 is 22.672. Therefore, 19 + 9 √514 = 19 + 9 × 22.672 = 19 + 204.044 = 223.044
What is the Square Root of 514?
The square root of 514 is an imaginary number. It can be written as √514 = √1 × √514 = i √514 = 22.671i
where i = √1 and it is called the imaginary unit.