 # Square Root of 165 + Solution With Free Steps The square root of 165 represented as √165 is equivalent to 12.84 where the symbol is called the square root or under root symbol. The square root is equal to power 1/2, hence, when the answer of the square root is raised to power 2 the original digit is received. For example, the square root of 165 is 12.84

In this article, we will analyze and find the square root of 165 using various mathematical techniques, such as the approximation method and the long division method.

## What Is the Square Root Of 165?

The square root of the number 165 is 12.84.

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:

√165 = √(12.84 x 12.84)

√165 = √(12.84)$^2$

√165 = ±12.84

The square can be canceled with the square root as it is equivalent to 1/2; therefore, obtaining 12.84. Hence 12.84 is 165’s square root. The square root generates both positive and negative integers.

## How To Calculate the Square Root of 165?

You can calculate the square root of 165 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 165 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 165 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 165 using the long division method:

### Step 1

First, write the given number 165 in the division symbol, as shown in figure 1.

### Step 2

Starting from the right side of the number, divide the number 165 into pairs such as 65 and 1.

### Step 3

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

### Step 4

After this, bring down the next pair 65. Now the dividend is 65. 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 $\leq$ 65 when multiplied with 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 2 to the divisor and multiplying 22 with 2 results in 44 $\leq$ 65. The remainder obtained is 21. 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 12.84 is the square root of 165. 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 165 using the approximation technique.

### Step 1

Consider a perfect square number 144 less than 165.

### Step 2

Now divide 165 by 12.

165 ÷ 12 = 13.75

### Step 3

Now take the average of 12 and 13.75. The resulting number is approximately equivalent to the square root of 165.

(12 + 13.75) ÷ 2 = 12.87

### Important points

• The number 165 is not a perfect square.
• The number 165 is an irrational number.
• The number 165 can be split into its prime factorization.

## Is Square Root of 165 a Perfect Square?

The number 165 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 165 is concerned, it is not a perfect square. It can be proved as below:

Factorization of 165 results in 3 x 55.

Taking the square root of the above expression gives:

= √(3 x 55)

= (3 x 55)$^{1/2}$

= 12.84

This shows that 165 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 165 is equivalent to 12.84. Images/mathematical drawings are created with GeoGebra.