# Square Root of 164 + Solution With Free Steps

The **square root of 164**Â can also be written in the form of **âˆš164. **It is equivalent to 12.80. **âˆš** is the square root symbol. The **square root** of a particular digit produces the number that produces the required number when multiplied by the same number. Take an example that the **square root of 144 is 12** which means **12 x 12 = 144**.Â

In this article, we will analyze and find the **square root of 164**Â using various mathematical techniques, such as the approximation method and the long division method.

## What Is the Square Root Of 164?

**The square root of the number 164 is 12.80.**

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:

**âˆš164 = âˆš(12.80 x 12.80)**

**âˆš164 = âˆš(12.80)$^2$**

**âˆš164 = Â±12.80**

The square can be canceled with the square root as it is **equivalent to 1/2**; therefore, obtaining 12.80. Hence 12.80 is 164’s square root. The square root generates both **positive** and **negative integers**.

## How To Calculate the Square Root of 164?

You can calculate the **square root of 164** using any of two vastly used techniques in mathematics; one is the **Approximation technique**, and the other is the **Long Division method**.

The symbolÂ **âˆšÂ **is interpreted asÂ **164**Â raised to the power**Â 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 **164** using the long division method:

### Step 1

First, write the given **number 164** in the division symbol, as shown in figure 1.

### Step 2

Starting from the right side of the number, divide the number 164 into **pairs** such as 64 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 64. Now the **dividend** is 64. 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 $\leq$ 64 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 p results in 44 $\leq$ 64. The remainder obtained is 20. 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 of 12.80 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 164** using the approximation technique.

### Step 1

Consider a perfect square number x1 less than X.

### Step 2

Now divide X by x1.

**X Ã· x1 = y1**

### Step 3

Now take the average of x1 and y1. The resulting number is approximately equivalent to the square root of X.

**(x1 + y1) Ã· 2 = y**

### Important points

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

## Is Square Root of X a Perfect Square?

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

**(If X is a perfect square)**

Factorization of X results in a x a that can also be expressed as a$^2$.

Taking the square root of the above expression gives:

**= âˆš(a$^2$)**

**= (a$^2$)$^{1/2}$**

**= a**

This shows that X is a perfect square and a rational number.

**(If X is not a perfect square)**

Factorization of X results in a x b.

Taking the square root of the above expression gives:

**= âˆš(a x b)**

**= (a x b)$^{1/2}$**

**= a.a1a2**

This shows that X 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 X is equivalent to y.

*Images/mathematical drawings are created with GeoGebra.*