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# Square Root of X + Solution With Free Steps

The **square root of X** can be expressed as **√X, **which is equivalent to y, where the symbol **√** is called the square root. The **square root** of a number produces the number that, when multiplied by itself, produces the actual number. For instance, the square root of X is y which means **y x y = x**.

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

## What Is the Square Root Of X?

**The square root of the number X is y.**

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:

**√X = √(y x y)**

**√X = √(y)$^2$**

**√X = ±y**

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

## How To Calculate the Square Root of X?

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

### Step 1

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

### Step 2

Starting from the right side of the number, divide the number X into **pairs** such as bc and a.

### Step 3

Now divide the digit a by a number, giving a number either a or less than a. Therefore, in this case, the remainder is zero or less than a (depending upon the situation, select according to the given number X), whereas the quotient is one or a.

### Step 4

After this, bring down the next pair bc. Now the **dividend** is bc. 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 $\leq$ bc 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 p to the divisor and multiplying 2p with p results in m $\leq$ bc. The remainder obtained is m1. 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 y is the square root of X. 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 X** using the approximation technique.

### Step 1

Consider a perfect square number x1 less than X.

### Step 2

Now divide X by x1.

**X ÷ 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) ÷ 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:

**= √(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:

**= √(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.*