# Square Root of 11 + Solution With Free Steps

When the Square root of 11 is taken, the output screen displays 3.3166. The square root of 11 is represented as √11. After taking the square root of a number, the result must satisfy a condition. That is,  if the square of the resultant value is taken, you should get the actual number. This can be arithmetically expressed as, √81 = 9 and 9×9=81.

In this article, we will analyze and find the square root of 11 using various mathematical techniques, such as the approximation method and the long division method.

## What Is the Square Root Of 11?

The square root of the number 11 is 3.3166.

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:

√11 = √(3.3166 x 3.3166)

√11 = √(3.3166)$^2$

√11 = ±3.3166

The square can be canceled with the square root as it is equivalent to 1/2; therefore, obtaining 3.3166. Hence 3.3166 is 11’s square root. The square root generates both positive and negative integers.

## How To Calculate the Square Root of 11?

You can calculate the square root of 11 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 11 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 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 11 using the long division method:

### Step 1

First, write the given number 11 in the division symbol, as shown in figure 1.

### Step 2

Starting from the right side of the number, make a pair of the number 11.

### Step 3

Now divide the digit 11 by a number, giving a number either 11 or less than 11. Therefore, in this case, the remainder is 2, whereas the quotient is 3.

### Step 4

After this, bring down the next pair 00. Now the dividend is 200. To find the next divisor, we need to double our quotient obtained before. Doubling 3 gives 6; hence consider it as the next divisor.

### Step 5

Now pair 6 with another number to make a new divisor that results in $\leq$ 200 when multiplied by the divisor. If the number is not a perfect square, add pair of zeros to the right of the number before starting division.

### Step 6

Adding 3 to the divisor and multiplying 63 with 3 results in 189 $\leq$ 200. The remainder obtained is 11. 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 3.3166 is the square root of 11. 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 11 using the approximation technique.

### Step 1

Consider a perfect square number 9 less than 11.

### Step 2

Now divide 11 by √9 .

11 ÷ 3 = 3.67

### Step 3

Now take the average of 3 and 3.67. The resulting number is approximately equivalent to the square root of 11.

(3 + 3.67) ÷ 2 = 3.33

### Important points

• The number 11 is a perfect square/ not a perfect square.
• The number 11 is a rational number/ irrational number.
• The number 11 can be split into its prime factorization.

## Is Square Root of 11 a Perfect Square?

The number 11 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 11 is concerned, it is not a perfect square. It can be proved as below:

Factorization of 11 results in 11 x 1.

Taking the square root of the above expression gives:

= √(11 x 1)

= (11 x 1)$^{1/2}$

= 3.3166

This shows that 11 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 11 is equivalent to 3.3166.

Images/mathematical drawings are created with GeoGebra.