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Square Root of 306
Taking the square root of a number is raising the number to the power half which is the inverse process of squaring the number. Hence, the square root of 306 is written as √306. In this mini lesson, let us learn about the square root of 306, find out whether the square root of 306 is rational or irrational, and see how to find the square root of 306 by long division method.
 Square Root of 306: √306 = 17.492
 Square of 306: 306^{2} = 93,636
What Is the Square Root of 306?
 If a = b^{2}, then √a = b, where a and b are natural numbers. √a is the square root of a and it is expressed in the exponential form as a^{½}.
 √306 = √(a number × a number).
 √306 = (17.492× 17.492) or ( 17.492 × 17.492) ⇒ √306= ±17.492.
Is Square Root of 306 Rational or Irrational?
Irrational numbers are the real numbers that cannot be expressed as the ratio of two integers. √306 = 17.4928556845359 and hence the square root of 306 is an irrational number where the numbers after the decimal point go up to infinity.
How to Find the Square Root of 306?
The square root of 306 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 306 in its Simplest Radical Form
 The square root of 306 is expressed in the radical form as √306. This can be simplified using the prime factorization.
 Let us 306 as a product of its prime factors as 306 = 2 × 3 × 3 × 17.
 √306 = √(2 × 3 × 3 × 17) = (2 × 3 × 3 × 17)^{½} = (2 × 3^{2} × 17)^{½} = 3 (34)^{½ }= 3√34.
Square Root of 306 by the Long Division Method
The long division method helps us to find a more accurate value of square root of any number. The following are the steps to evaluate the square root of 306 by the long division method.
 Step 1: Write 306.000000. Take the number in pairs from the right. 1 stands alone. Now divide 3 by a number such that (number × number) gives ≤ 3.
 Step 2: Obtain quotient = 1 and remainder = 1. Double the quotient. We get 2. Have 20 as our new divisor. Bring down 06 for division.
 Step 3: Find a number such that (20 + that number) × that number gives the product ≤ 2 43. We find that 27 × 7 = 1 89.
 Step 4: Subtract from 2 06 and get 17 as the remainder. Bring down the pair of zeros. 17 00 is our new divisor.
 Step 5: 17 is our quotient and on doubling it becomes 34 and 340 is our new divisor. Find a number such that (340 + the number) × number gets 17 00 or less than that. We find 344 × 4 = 13 76.
 Step 6: Subtract this from 17 00 and get the remainder and bring down the zeros. 3 24 00 becomes the new dividend.
Repeat the steps until we approximate the square root to 3 decimal places.
Hence, √306 = 17.492.
Explore Square roots using illustrations and interactive examples:
Important Notes
 The square root of 306 is 17.492 approximated to 3 decimal places.
 The simplified form of 306 in its radical form is 3√34.
 √306 is an irrational number.
Challenging Questions
 Evaluate √3.06 ÷ √306.
 Evaluate the square root of √306 up to 3 decimal places.
Square Root of 306 Solved Examples

Example 1: Sara covers 15 feet North and 9 feet East. How far has she covered now? Approximate it to the nearest tenth.
Solution:
This forms the right triangle. The shortest distance covered by Sara will be the hypotenuse.
(Distance covered)^{2} = (Distance towards North)^{2 }+ (Distance towards East)^{2}
⇒ (Distance covered)^{2 }= 15^{2} + 9^{2} = 225 + 81 = 306
Total distance covered^{2} = 306 feetHence, the distance covered by Sara is √306 = 17.5 feet.

Example 2: Evaluate: The area of the table top is 612 sq inches. If the length is twice its width, find the length of the table top.
Solution:
Length × Width = Area
⇒ Length = 2 × width
Area = 2 × width × width = 612
⇒ 2 width^{2}= 612
⇒ width^{2 }= 612 ÷ 2 = 306
width = √306 = 17.492 inches
length = 2 × width = 2 × 17.492 = 34.984 inches.Hence, the length of the table top is 34.984 inches.
FAQs on Square Root of 306
What is the square root of 306?
The square root of 306 is √306 = 17.492.
What is the square root of 306 simplified?
The square root of 306 is simplified as 3√34.
How to get the square root of 306?
The square root of 306 is evaluated by the long division method.
Is √306 a rational number?
No, √306 is not a rational number, because the value of √306 is not a whole number and it is a nonterminating decimal.
How to find the square root of 306 to the nearest hundredth?
The square root of 306 is evaluated using the division method and rounded off to the nearest hundredth.√306 =17.492 ⇒ √306 = 17.49.
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