Square root of 360

The square root is a number that, when multiplied by itself, results in the original number whose square root is to be found. Know that the square root of 360 is ±18.973665961.

We will see here more about the square root of 360. As defined, the square root is just the opposite (inverse) of squaring a number, so, squaring 18.973… will result in 360. The positive value, 18.973… is the solution of the equation x² = 360.

It contains both positive and a negative root, where the positive root is called the principal square root. The square root of 360 is expressed as √360 in radical form. In exponential form, it is written as (360)^1/2  .

The simplest radical form of √360 is 6√10. We will see how to obtain the simplest radical form further in this article.

We now came to a point where we can say that:

• 360 is a non-perfect square.

• √360 is an irrational number.

Let us now find how we got this value of 18.973… as a square root of 360.

We will use these methods below to find.

• Prime factorization method

•  Long division method

• Approximation/Estimation method

The prime factorization of 360 involves breaking down a number into its factors.

Factorize 360 by prime numbers, and continue to divide the quotients until they can’t be separated anymore. 

• Find the prime factors of 360.

• After factoring 360, make pairs out of the factors to get the square root.

•  If there exist numbers that cannot be made pairs further, we place those numbers with a “radical” sign along with the obtained pairs.

Prime factorization of 360 = 3×3×2×2×2×5

For 360, two pairs of factors 3 and 2 are obtained, but a single 5 is remaining.

So, it can be expressed as √360 = √(3×3×2×2×2×5) = 6√10.

6√10 is the simplest radical form of √360.

Long Division method is used for obtaining the square root for non-perfect squares, mainly. It usually involves the division of the dividend by the divisor, getting a quotient and a remainder for non-perfect squares. To make it simple it is operated on divide, multiply, subtract, bring down and do-again.

To calculate the square root of 360:

Step 1: On the number 360.000000, draw a horizontal bar above the pair of digits from right to left.

Step 2 :Find the greatest number whose square is less than or equal to 3. Here, it is 1, Because 1²=1 < 3.

Step 3 : Now divide 3 by 1 such that we get 1 as a quotient and then multiply the divisor with the quotient, we get 1.

Step 4: Subtract 1 from 3. Bring down 6 and 0 and place it beside the difference 2.

Step 5: Add 1 to the same divisor, 1. We get 2.

Step 6: Now choose a number such that when placed at the end of 2, a 2-digit number will be formed. Multiply that particular number by the resultant number to get a number less than 260. Here, that number is 8. 

28×8=224<260. In quotient’s place, we also place that 8.

Step 7: Subtract 260-224=36. Add a decimal point after the new quotient 18, again, bring down two zeroes and make 36 as 3600. Simultaneously add the unit’s place digit of 28, i.e., 8 with 28. We got here, 36. Apply Step 5 again and again until you reach 0.

 
We will show two places of precision here, and so, we are left with the remainder, 25271 (refer to the picture), after some iterations and keeping the division till here, at this point 

             
Step 8 : The quotient obtained is the square root. In this case, it is 18.973….

Estimation of square root is not the exact square root, but it is an estimate, or you can consider it as a guess.

Follow the steps below:

Step 1: Find the nearest perfect square number to 360. Here, it is 324 and 361.

Step 2: We know that, √324=±18 and √361=±19. This implies that √360 lies between 18 and 19.

Step 3: Now we need to check √360 is closer to 18.5 or 19. Since (18.5)²=342.25 and (19)²=361. Thus, √361 lies between 18.5 and 19.

Step 4: Again considering precisely, we see that  √360 lies close to (19)²=361. Find squares of (18.7)²=349.69 and (18.99)²= 360.6201.

We can iterate the process and check between the squares of 18.9 and 18.98 and so on.

We observe that √360 = 18.973…

• Negative Square root - The negative square root of 360 is √-360.

• GPS - GPS or Global Positioning System is basically a network of satellites used to determine the location of a point on Earth.

• Exponential form - An algebraic expression that includes an exponent. It is a way of expressing the numbers raised to some power of their factors, or a way to write a number that is multiplied by itself more than one time.  Ex: 5× 5 × 5 × 5 = 625 Or, 5⁴ = 625, where 5 is the base, 4 is the exponent 

• Prime Factorization - Expressing the given expression as a product of its factors Ex: 52=2 × 2 × 13 

• Prime Numbers - Numbers which are greater than 1, having only 2 factors as →1 and Itself. Ex: 1,3,5,7,....

• Rational numbers - The Number which can be expressed as p/q, where p and q are integers and q not equal to 0 are called Rational numbers

• Irrational numbers - Numbers which cannot be expressed as p/q, where p and q are integers and q not equal to 0 are called Irrational numbers. 

• Perfect square numbers - Perfect square numbers are those numbers whose square roots do not include decimal places. Ex: 4,9,25.

• Non-Perfect square numbers -  Those numbers whose square roots comprise decimal places. Ex : 3, 10, 29