Square Root of 208
208 is an even composite number that has 10 factors. 208 = 2^{4} × 13. In this mini lesson, let us learn about the square root of 208, find out whether the square root of 208 is rational or irrational, and see how to calculate the square root of 208 by long division method.
 Square Root of 208 = √208 = 14.422
 Square of 208: 208^{2} = 43,264
What Is the Square Root of 208?
 The square root of 208 can be written as √208
 It means then there is a number a such that: a × a = 208. Now it can also be written as: a^{2} = 208
 a = √208 . a is the 2nd root of 208 and a = ± 14.422
 In the exponential form, we denote √208 as (208) ^{½}
Is Square Root of 208 Rational or Irrational?
 The square root of 208 is an irrational number where the numbers after the decimal point go up to infinity. √208 = 14.42220510185596
 Thus √208 cannot be written in the form of p/q, hence it is an irrational number.
How to Find the Square Root of 208?
The square root of 208 or any number can be calculated in many ways. Two of them are the prime factorization method and the long division method.
Square Root of 208 in the Simplest Radical Form
 208 = 2 × 2 × 2 × 2 × 13
 √208 = √( 2 × 2 × 2 × 2 × 13)
 √208 = √(4 × 4 × 13)
 √208 = 4√13
Square Root of 208 by Long Division Method
The long division method helps us find a more accurate value of square root of any number. Let's see how to find the square root of 208 by the long division method.
 Write 208 as 2 08 00 00 00. Take numbers as pairs from the right. 2 is left without pair. 2 is the number to be divided first.
 Find a quotient × divisor that results in the product < or = 2, wherein quotient = divisor. We find 1 × 1 = 1.
 Subtract 1 from 2. Get the remainder as 1 and bring the next pair of digits 08.
 Double the quotient. 1 × 2 = 2. Have 2 as 2x. Find a number in the quotient that multiplied with 2x gives 108 or less than that. We determine 4 × 24 = 96.
 Subtract this from 108 and obtain the remainder as 12. Bring down the next pair of zeros down.
 12 00 is the new dividend and doubling the quotient 14, we get 28x as the new divisor.
 Find a number in the quotient such that the number × 28x is 12 00 or less than that. We find 4 × 284 = 11 36.
 Subtract this from 1200 and get the remainder as 64. Bring down the next pair of zeros and double the quotient.
 Repeat the process until we approximate the value of square root to 3 decimal places.
Thus √208 = 14.422
Explore square roots using illustrations and interactive examples.
Tips and Tricks
 Find the two whole numbers which are closer to the 208 such that they are perfect squares, one less than 208 and the other greater than 208.
 √196 = √208 = √225 ⇒ 14 = √208 = 15
 Divide 208 by 15. 208 ÷ 15 = 13.86
 Find the average of 15 and 13.86. We obtain (13.86+ 15) ÷ 2 = 28.86 ÷ 2 = 14.43
 Thus √208 ≈ 14.43
Important Notes
 The square root of 208 is irrational. √208 = ± 14.422
 √208 is simplified as 4√13
Square Root of 208 Solved Examples

Example 1: Find the value of √208 ÷ √13
Solution:
√208 = 4√13
√208 ÷ √13 = 4√13 ÷ √13 = 4
√208 ÷ √13 = 4

Example 2: Evaluate: √0.00208
Solution: √0.0208 = √(208 ÷ 10000)
√208 = 4√13
√10000 = 100
√(208 ÷ 10000) = 4√13 ÷ 100 = √13 / 25
√0.00208 = √13 / 25
FAQs on Square Root of 208
What is the square root of 208?
The square root of 208 is ±14.422
Can the square root of 208 be simplified?
The square root of 208 is simplified as 4√13
Is 208 a perfect square?
208 is not a perfect square as √208 = 14.422 × 14.422 and not a product of whole number × whole number.
Is the square root of 208 a rational number?
√208 is an irrational number, because the value of √208 = 14.42220510185596, which is a nonterminating decimal.
How to find the square root of 208?
The accurate value of 208 can be evaluated using the long division method.