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Square Root of 500
The square root of 500 is expressed as √500 in the radical form and as (500)^{½} or (500)^{0.5} in the exponent form. The square root of 500 rounded up to 8 decimal places is 22.36067977. It is the positive solution of the equation x^{2} = 500. We can express the square root of 500 in its lowest radical form as 10 √5.
 Square Root of 500: 22.360679774997898
 Square Root of 500 in exponential form: (500)^{½} or (500)^{0.5}
 Square Root of 500 in radical form: √500 or 10 √5
1.  What Is the Square Root of 500? 
2.  Is Square Root of 500 Rational or Irrational? 
3.  How to Find the Square Root of 500? 
4.  FAQs on Square Root of 500 
What is The Square Root of 500?
 The square root of 500 is written as √500 and can be simplified to 10√5.
 The square root of 500 in decimal form is 22.3606.
Is Square Root of 500 Rational or Irrational?
The square root of 500 is not an integer and a rational number is a number that can be represented in the form of p/q where q ≠ 0. As the square root of 500 cannot be written in the form of p/q. Hence, the square root of 500 is an irrational number.
How to Find the Square Root of 500?
We will now find the square root of 500 using the below two methods:
Square Root of 500 Using Approximation Method
 The two consecutive perfect squares among which 500 lies are 484 (22^{2}) and 529 (23^{2}).
So, the whole number part of the square root of 500 is 22.  The decimal part can be determined by the formula: (Given number – Smaller perfect square) / (Greater perfect square – smaller perfect square) = (500 – 484)/(529 – 484) = 16/45 = 0.35.
 So, the estimated value of the square root of 500 by the approximation method is 22.35.
Square Root of 500 By Long Division
Now we will calculate the square root of 500 by the long division method.
 Pair the digits from the right side in pairs of two by putting a bar on top of them. In the case of 500, we will have two pairs 00 and 5(pairing from right).
 Now we have to find a number(y) whose square is ≤ 5. The value of z will be 2 as 2 × 2 = 4≤ 5.
 We get the quotient (2) and the remainder (1). Now, add the divisor z with itself and get the new divisor 2z (4).
 Drag the pair of zero (new dividend becomes 100) and find a number (a) such that 4a × a ≤ 100. The value of a comes out to be 2.
 Now, add a decimal in the dividend part (100) and quotient part (22) simultaneously. Also, add 3 pairs of zero in the dividend part after the decimal and repeat the above step for the other three pairs of zero as well.
So, we get the value of the square root of √500 = 22.360 by the long division method.
Explore square roots using illustrations and interactive examples
Important Notes:
 The number 500 is not a perfect square.
 The square root of 500 is an imaginary number.
 The square root of 500 is an irrational number.
Square Root of 500 Solved Examples

Example 1: Find the square root of 500 using the prime factorization method. (Given √5 = 2.236)?
Solution:
The prime factorization of 500 is: 2 × 2 × 5 × 5 × 5 = 2^{2} × 5^{3}
The prime factors of 500 in pairs = (2 × 2) × (5 × 5) × 5
The square root of 500 = √((2 × 2) × (5 × 5) × 5) =√(2^{2} × 5^{2} × 5) = 10√5 = 10 × 2.236 = 22.36 
Example 2: What is the smallest multiple of 500 which is a perfect square?
Solution:
A perfect square is a number whose factors is a square of a number. The prime factorization of 500 is 2^{2} × 5^{3}. To make it a perfect square we have to make the power of every prime factor an even number. Therefore the multiple of 500 which is a perfect square will be 500 × 5. That is the square of 50 = 2500. 
Example 3: Batman is chasing the Riddler his Batmobile at an unknown speed. He covers √500 miles in 50√5 secs. Find the speed of Batmobile in Miles/hour?
Solution:
The speed of Batmobile = (Distance/time) = (√500) / (50√5)
We know that √500 = 10√5
Therefore, the speed of Batmobile = (10√5)/(50√5) = 0.2 Miles/sec and the speed in Miles/hour = 0.2 × 3600 = 720 Miles/hour
FAQs on the Square Root of 500
What is the Value of the Square Root of 500?
The square root of 500 is 22.36067.
Why is the Square Root of 500 an Irrational Number?
Upon prime factorizing 500 i.e. 2^{2} × 5^{3}, 5 is in odd power. Therefore, the square root of 500 is irrational.
What is the Square of the Square Root of 500?
The square of the square root of 500 is the number 500 itself i.e. (√500)^{2} = (500)^{2/2} = 500.
What is the Square Root of 500 in Simplest Radical Form?
We need to express 500 as the product of its prime factors i.e. 500 = 2 × 2 × 5 × 5 × 5. Therefore, √500 = √2 × 2 × 5 × 5 × 5 = 10 √5. Thus, the square root of 500 in the lowest radical form is 10 √5.
Evaluate 12 plus 12 square root 500
The given expression is 12 + 12 √500. We know that the square root of 500 is 22.361. Therefore, 12 + 12 √500 = 12 + 12 × 22.361 = 12 + 268.328 = 280.328
Is the number 500 a Perfect Square?
The prime factorization of 500 = 2^{2} × 5^{3}. Here, the prime factor 5 is not in the pair. Therefore, 500 is not a perfect square.
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