# Square Root of 89 + Solution With Free Steps

The square root of **89** is **9.43**. Â 89 is the square of **âˆš89**. 89 is not a perfect square as its square root, **9.43**, is not an integer but an irrational number. As square and square roots are opposite to each other so we can also say that **9.43 x 9.43 = 89.**

In this article, we will analyze and find the **square root of 89**Â using various mathematical techniques, such as the approximation method and the long division method.

## What Is the Square Root Of 89?

**The square root of the number 89 is 9.43.**

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:

**âˆš89 = âˆš(9.43 x 9.43)**

**âˆš89 = âˆš(9.43)$^2$**

**âˆš89 = Â±9.43**

The square can be canceled with the square root as it is **equivalent to 1/2**; therefore, obtaining 9.43. Hence 9.43 is 89’s square root. The square root generates both **positive** and **negative integers**.

## How To Calculate the Square Root of 89?

You can calculate the **square root of 9.43**Â 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 as 89Â 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 89 using the long division method:

### Step 1

First, write the given **number 89**Â in the division symbol, as shown in figure 1.

### Step 2

Starting from the right side of the number, divide the number 89 into **pairs** such as 89.

### Step 3

Now divide the digit 89 by a number, giving a number either a or less than 89. Therefore, in this case, the remainder 8, whereas the quotient is 8.

### Step 4

After this, bring down the next pair 00. Now the **dividend** is 800. To find the next divisor, we need to double our quotient obtained before. Doubling 9 gives 18; hence consider it as the next divisor.

### Step 5

Now pair 18 with another number to make a new divisor that results in $\leq$ 800 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 4 to the divisor and multiplying 184 with 4 results in 736 $\leq$ 800. The remainder obtained is 64 . 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 9.43 is the square root of 89. 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 89**Â using the approximation technique.

### Step 1

Consider a perfect square number 81 less than 89.

### Step 2

Now divide 89 by âˆš81.

**89 Ã· 9 = 9.88**

### Step 3

Now take the average of 9 and 9.88. The resulting number is approximately equivalent to the square root of 89.

**(9 + 9.88) Ã· 2 = 9.44**

### Important points

- The number 89 is not a perfect square.
- The number 89 is a rational number.
- The number 89 can be split into its prime factorization.

## Is Square Root of 89 a Perfect Square?

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

Factorization of 89 results in 89 x 1.

Taking the square root of the above expression gives:

**= âˆš(89 x 1)**

**= (89 x 1)$^{1/2}$**

**= 9.43**

This shows that 89 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 89 is equivalent to 9.43.

*Images/mathematical drawings are created with GeoGebra.*