Square Root of 57
The square root of 57 is expressed as √57 in the radical form and as (57)^{½} or (57)^{0.5} in the exponent form. The square root of 57 rounded up to 6 decimal places is 7.549834. It is the positive solution of the equation x^{2} = 57.
 Square Root of 57: 7.54983443527075
 Square Root of 57 in exponential form: (57)^{½} or (57)^{0.5}
 Square Root of 57 in radical form: √57
1.  What Is the Square Root of 57? 
2.  Is Square Root of 57 Rational or Irrational? 
3.  How to Find the Square Root of 57? 
4.  FAQs on Square Root of 57 
5.  Important Notes on Square Root of 57 
What Is the Square Root of 57?
 The square root of 57 is √57 = 7.5498.
 The number 57 has only two different prime factors with power 1. So, √57 cannot be simplified further using prime factorization.
 The number 57 is not a perfect square as its square root is not an integer.
Is Square Root of 57 Rational or Irrational?
The square root of 57 is a noninteger number. So, the square root of 57 is an irrational number because it cannot be expressed in the form of p/q where q ≠ 0.
How to Find the Square Root of 57?
Now we will find the square root of 57 using the following methods:
Square Root of 57 Using Approximation Method
 First, find two consecutive perfect squares among which the number 57 will lies. The two perfect squares are 49 (7^{2}) and 64 (8^{2}). Therefore, the whole number part of the square root of 57 will be 7.
 Now, for the decimal part, we will use the belowgiven formula:
(Given number – Smaller perfect square) / (Greater perfect square – smaller perfect square) = (57 – 49)/(64 – 49) = 8/15 = 0.533.  Hence, the square root of 57 via the approximation method is 7.533.
Square Root of 57 By Long Division
We will now calculate the square root of 57 by the long division method with the help of the belowgiven steps.
 Start grouping the digits from the unit’s place in pairs of two by adding a bar on top of them. We will get only one pair in this case (57).
 Find a number(t) which when multiplied with itself t × t ≤ 57. So, t will be 7 as 7 × 7 = 49.
 Now we get 8 as the remainder (5749) and the quotient as 7. Also, we have to add the divisor t with itself (t + t) to get the new divisor(2t). The new divisor here will be 14.
 Put a decimal in the dividend and quotient part simultaneously. Also, add 3 pairs of zero in the dividend part (57. 00 00 00).
 Bring down the pair of zero. So, our new dividend is 800. Now find a number(m) such that 14m × m ≤ 800. The number m will be 5 as 145 × 5 = 725 ≤ 800.
 Repeat the above step for all the pairs of zero.
So, we get the square root of √57 = 7.549 by the long division method.
Explore square roots using illustrations and interactive examples
Important Notes:
 The number 57 is not a perfect square.
 The square root of 57 is an irrational number.
 The square root of 57 is an imaginary number.
Square Root of 57 Solved Examples

Example 1: What is the value of (3√57)/(5√228)?
Solution:
As, √228 = 2√57.
Therefore, (3√57)/(5√228) = (3√57) / (5 × 2√57) = 3/10 = 0.3. 
Example 2: Joshua wants to find out the square root of 57. Can you help Joshua?
Solution:
The square root of all negative numbers are imaginary numbers. Because the square of any number (positive or negative) will result in a positive number. So, the square of 57 is written as √57 = ±7.549i (where i = √1).

Example: Solve the equation x^{2} − 57 = 0
Solution:
x^{2}  57 = 0 i.e. x^{2} = 57
x = ±√57
Since the value of the square root of 57 is 7.550,
⇒ x = +√57 or √57 = 7.550 or 7.550.
FAQs on the Square Root of 57
What is the Value of the Square Root of 57?
The square root of 57 is 7.54983.
Why is the Square Root of 57 an Irrational Number?
Upon prime factorizing 57 i.e. 3^{1} × 19^{1}, 3 is in odd power. Therefore, the square root of 57 is irrational.
What is the Value of 11 square root 57?
The square root of 57 is 7.550. Therefore, 11 √57 = 11 × 7.550 = 83.048.
Is the number 57 a Perfect Square?
The prime factorization of 57 = 3^{1} × 19^{1}. Here, the prime factor 3 is not in the pair. Therefore, 57 is not a perfect square.
What is the Square Root of 57?
The square root of 57 is an imaginary number. It can be written as √57 = √1 × √57 = i √57 = 7.549i
where i = √1 and it is called the imaginary unit.
If the Square Root of 57 is 7.550. Find the Value of the Square Root of 0.57.
Let us represent √0.57 in p/q form i.e. √(57/100) = 0.57/10 = 0.755. Hence, the value of √0.57 = 0.755
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