 # Square Root of 84 + Solution With Free Steps The square of the number 84 is 9.16. we can clearly see that 9.16 is a decimal number which indicates that 84 is not a perfect square and hence, √84 is an irrational number, As the square root is equal to power ½ so we can mathematically write it as (84) ^1/2 = 9.16.

In this article, we will analyze and find the square root of 84 using various mathematical techniques, such as the approximation method and the long division method.

## What Is the Square Root Of 84?

The square root of the number 84 is 9.16.

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:

√84 = √(9.16 x 9.16)

√84 = √(9.16)$^2$

√84 = ±9.16

The square can be canceled with the square root as it is equivalent to 1/2; therefore, obtaining 9.16. Hence 9.16 is 84’s square root. The square root generates both positive and negative integers.

## How To Calculate the Square Root of 84?

You can calculate the square root of 84 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 84 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 84 using the long division method:

### Step 1

First, write the given number 84 in the division symbol, as shown in figure 1.

### Step 2

Starting from the right side of the number, divide the number 84 into pairs such as 84.

### Step 3

Now divide the digit 84 by a number, giving a number either 84 or less than 84. Therefore, in this case, the remainder 3, whereas the quotient is 9.

### Step 4

After this, bring down the next pair 00. Now the dividend is 300. 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 $\leq$ 300 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 1 to the divisor and multiplying 181 with 1 results in 181 $\leq$ 300. The remainder obtained is 119. 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 9.16 is the square root of 84. Figure 1 given below shows the long division process in detail: Figure 1

### 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 84 using the approximation technique.

### Step 1

Consider a perfect square number 81 less than 84.

### Step 2

Now divide 84 by 81.

84 ÷ 9 = 9.33

### Step 3

Now take the average of 9 and 9.33. The resulting number is approximately equivalent to the square root of 84.

(9 + 9.33) ÷ 2 = 9.16

### Important points

• The number 84 is not a perfect square.
• The number 84 is a rational number.
• The number 84 can be split into its prime factorization.

## Is Square Root of 84 a Perfect Square?

The number 84 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 84 is concerned, it is not a perfect square. It can be proved as below:

Factorization of 84 results in 7 x 12.

Taking the square root of the above expression gives:

= √(7 x 12)

= (7 x 12)$^{1/2}$

= 9.16

This shows that 84 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 84 is equivalent to 9.16.

Images/mathematical drawings are created with GeoGebra.