# Square Root of 85 + Solution With Free Steps

Using the symbol âˆšÂ , which stands for the square root, we can express the square root of 85 as âˆš85, which is equal to 9.21Â in decimal form. A number’s square root yields the number that, when multiplied by itself, yields the actual number. For example, the square root of 85 is 9.219, so 9.21 x 9.21 = 85.

In this article, we will analyze and find the square root of 85 using various mathematical techniques, such as the approximation method and the long division method.

## What Is the Square Root Of 85?

The square root of the number 85 is 9.21.

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:

âˆš85= âˆš(9.21 x 9.21)

âˆš85 = âˆš(9.21)$^2$

âˆš85 = Â±9.21

The square can be canceled with the square root as it is equivalent to 1/2; therefore, obtaining y. Hence 9.21 is 85‘s square root. The square root generates both positive and negative integers.

## How To Calculate the Square Root of 85?

You can calculate the square root of 85Â 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 as85Â raised to the powerÂ 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 85Â using the long division method:

### Step 1

First, write the given number 85Â in the division symbol, as shown in figure 1.

### Step 2

Starting from the right side of the number, divide the number 85 into pairs such as 00 and 85.

### Step 3

Now divide the digit a by a number, giving a number either 9 or less than 9. Therefore, in this case, the remainder is 4Â  whereas the quotient isÂ  9.

### Step 4

After this, bring down the next pair 400. Now the dividend is 400. 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$400 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 18 with 2 results in 364 $\leq$ 400. The remainder obtained is 36. 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 9.21 is the square root of 85. Figure 1 given below shows the long division process in detail:Â Â

Â

Â  Â  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 85Â using the approximation technique.

### Step 1

Consider a perfect square number 81 less than 85.

### Step 2

Now divide 85 by âˆš85.

85 Ã· 9 = 9.44

### Step 3

Now take the average of 9 and 9.44. The resulting number is approximately equivalent to the square root of 85.

(9+9.44) Ã· 2 = 9.22

### Important points

• The number 85 is not a perfect square.
• The number 85Â  is a rational number.
• The number 85 can be split into its prime factorization.

## Is Square Root of 85 a Perfect Square?

The number 85 is Â 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 85 is concerned, it is not a perfect square. It can be proved as below:

(If 85 is not a perfect square)

Factorization of 85 results in 5 x 17.

Taking the square root of the above expression gives:

= âˆš(5 x 17)

= (5 x 17)$^{1/2}$

= 9.21

This shows that 85 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 85 is equivalent to 9.21.

Images/mathematical drawings are created with GeoGebra.