Square Root of 360 + Solution With Free Steps

Some numbers are indicated as perfect square numbers while some are not. Let us see the number 360 is a  perfect square or not if we take the square root of this number 360 then it return √360=18.973. Here the square root of 360 includes decimal places so we cannot say 360 as the perfect square number.

In this article, we will analyze and find the square root of 360 using various mathematical techniques, such as the approximation method and the long division method.

What Is the Square Root Of 360?

The square root of the number 360 is 18.973.

The square root can be defined as the quantity that can be doubled to produce the square of that similar quantity. In simple words, it can be explained as:

√360 = √(18.973 x 18.973)

√360 = √(18.973)$^2$

√360 = ±18.973

The square can be canceled with the square root as it is equivalent to 1/2; therefore, obtaining 18.973. Hence 18.973 is 360’s square root. The square root generates both positive and negative integers.

How To Calculate the Square Root of 360?

You can calculate the square root of 360 using any of two vastly used techniques in mathematics; one is the Approximation technique, and the other is the Long Division method.

The symbol √ is interpreted as 360 raised to the power 1/2. So any number, when multiplied by itself, produces its square, and when the square root of any squared number is taken, it produces the actual number.

Let us discuss each of them to understand the concepts better.

Square Root of 360 by Long Division Method

The process of long division is one of the most common methods used to find the square roots of a given number. It is easy to comprehend and provides more reliable and accurate answers. The long division method reduces a multi-digit number to its equal parts.

Learning how to find the square root of a number is easy with the long division method. All you need are five primary operations- divide, multiply, subtract, bring down or raise, then repeat.

Following are the simple steps that must be followed to find the square root of 360 using the long division method:

Step 1

First, write the given number 360 in the division symbol, as shown in figure 1.

Step 2

Starting from the right side of the number, divide the number 360 into pairs such as 60 and 3.

Step 3

Now divide the digit 3 by a number, giving a number either 3 or less than 3. Therefore, in this case, the remainder is 2, whereas the quotient is 1.

Step 4

After this, bring down the next pair 60. Now the dividend is 260. To find the next divisor, we need to double our quotient obtained before. Doubling 1 gives 2; hence consider it as the next divisor.

Step 5

Now pair 2 with another number to make a new divisor that results in $\leq$ 260 when multiplied with the divisor. 

Step 6

Adding 8 to the divisor and multiplying 28 with 8 results in 224 $\leq$ 260. The remainder obtained is 36. Move the next pair of zeros down and repeat the same process mentioned above.

Step 7

Keep on repeating the same steps till the zero remainder is obtained or if the division process continues infinitely, solve to two decimal places.

Step 8

The resulting quotient 18.973 is the square root of 360. Figure 1 given below shows the long division process in detail:

Square Root by Approximation Method

The approximation method involves guessing the square root of the non-perfect square number by dividing it by the perfect square lesser or greater than that number and taking the average.

The given detailed steps must be followed to find the square root of 360 using the approximation technique.

Step 1

Consider a perfect square number 324 less than 360.

Step 2

Now divide 360 by √324.

360 ÷ 18 = 20

Step 3

Now take the average of 18 and 20. The resulting number is approximately equivalent to the square root of 360.

(18 + 20) ÷ 2 = 19

Important points

• The number 360 is not a perfect square.
• The number 360 is a rational number.
• The number 360 can be split into its prime factorization.

Is Square Root of 360 a Perfect Square?

The number 360 is not a perfect square. A number is a perfect square if it splits into two equal parts or identical whole numbers. If a number is a perfect square, it is also rational.

A number expressed in p/q form is called a rational number. All the natural numbers are rational. A square root of a perfect square is a whole number; therefore, a perfect square is a rational number.

A number that is not a perfect square is irrational as it is a decimal number. As far as 360 is concerned, it is not a perfect square. It can be proved as below:

Factorization of 360 results in 3 x 120.

Taking the square root of the above expression gives:

= √(3 x 120)

= (3 x 120)$^{1/2}$

= 18.973

This shows that 360 is not a perfect square as it has decimal places; hence it is an irrational number.

Therefore the above discussion proves that the square root of 360 is equivalent to 18.973.