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Square Root of 360

The square of 19 is 361. Since 360 is not a perfect square, the square root of 360 is a decimal number and not a whole number. In this mini lesson, let us learn about the square root of 360, find out whether the square root of 360 is rational or irrational, and see how to find the accurate value of the square root of 360 by long division method. 

  • Square Root of 360: √360 = 18.973
  • Square of 360: 3602 = 129, 600
1. What Is the Square Root of 360?
2. Is Square Root of 360 Rational or Irrational?
3. How to Find the Square Root of 360?
4. Important Notes on Square Root of 360
5. Challenging Questions
6. FAQs on Square Root of 360

What Is the Square Root of 360?

  • If a = b2, 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 a½.  
  • √360 = √(a number × a number). 
  • √360 = (18.973 × 18.973) or (- 18.973 ×  -18.973) ⇒ √360 = ±18.973.

Is Square Root of 360 Rational or Irrational?

Irrational numbers are the real numbers that cannot be expressed as the ratio of two integers. √360 = 18.97366596101028 and hence the square root of 360 is an irrational number where the numbers after the decimal point go up to infinity.

How to Find the Square Root of 360?

The square root of 360 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 360 in its Simplest Radical Form

The square root of 360 is expressed in the radical form as √360. This can be simplified using the prime factorization. Let us express 360 as a product of its prime factors. 360= 5 ×  5 × 13. We can express √360 = √(6 × 6 × 10). Hence, √360 = 6√10.

Square Root of 360 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 360 by the long division method.

  • Step 1: Write 360.000000. Take the number in pairs from the right. 3 stands alone. Now divide 3 by a number such that (number × number) gives  ≤ 3.
  • Step 2: Obtain quotient = 1 and remainder = 2. Double the quotient. We get 2. Have 20 as our new divisor. Bring down 60 for division.
  • Step 3: Find a number such that (20 + that number) × that number gives the product ≤ 260.  We find that 28 × 8 = 224. Subtract from 260 and get 36 as the remainder. Bring down the pair of zeros. 36 00 is our new divdend.
  • Step 4: 18 is our quotient. Double it and get 36 and thus 360 becomes the new divisor. Find a number such that (360 + the number) × number gets ≤ 36 00. We find that (360 + 9) × 9 = 369 × 9 = 33 21.
  • Step 5: Subtract and get 279 as the remainder and bring down the zeros. 2 79 00 becomes the new dividend. 18.9 is the quotient.
  • Step 6: Double the quotient. 189 × 2 = 378. Have 3780  in the place of the new divisor. Find a number such that (3780 + that number) × number ≤ 2 79 00.
  • Step 7: We find 3780 × 7 = 2 65 09. Subtract this from 2 79 00 and get the remainder as  13 91. Repeat the steps until we approximate the square root to 3 decimal places. 

Hence, √360 =  18.973.

square root of 360 by long division

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Important Notes

  • The square root of 360 is 18.973 approximated to 3 decimal places.
  • The simplified form of 360 in its radical form is 6√10.
  • √360 is an irrational number. 

Challenging Questions

  • What will be the least number multiplied and divided to 360 to make it perfect square?
  • Evaluate √(√360).
 

Square Root of 360 Solved Examples

  1. Example 1: What is the length of the ladder that is placed on a wall at a height of 18 feet and kept at 6 feet away from the wall? Approximate the length to the nearest tenth.

    Solution: 

    Applying the Pythagorean theorem, we have
    The length of the ladder2 = the height of the wall2 + Distance between the wall and the ladder2
    ⇒ 182 + 62 = 324 + 36 = 360
    The length of the ladder2 = 360 feet
    The length of the ladder is given by, √360 = 18.973 feet = 19 feet
    Hence, the length of ladder is 19 feet when approximated to nearest tenth.

  2. Example 2: Evaluate: √360 ÷ √3600

    Solution: 

    As √360 = 6√10 and √3600 = 60
    √360 ÷ √3600 =  6√10 ÷ 60 = √10/10

    Hence, √360 ÷ √3600 = √10/10.

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FAQs On Square Root of 360

What is the square root of 360?

The square root of 360 is 18.973.

What is the square root of 360 simplified?

The square root of 360 is simplified as 5√13.

How to get the square root of 360?

The square root of 360 is evaluated by the long division method. 

Is √360 a rational number?

No, √360 is not a rational number, because the value of √360 is not a whole number and it is a non-terminating decimal.

How to find the square root of 360 to the nearest hundredth?

The square root of 360 is evaluated using the division method and rounded off to the nearest hundredth.√360 = 18.027 ⇒ √360 = 18.97.

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