Square Root of 86
Did you know that 86 is the product of two distinct prime numbers, 2 and 43? In this lesson, we will learn that √86 is a nonterminating, nonrepeating decimal. The square root of 86 is written as √86. We call this a representation of square root of 86 in radical form. 86 can be called a perfect square only if the square root of 86 gives a perfect whole number. Let us see how to find the square root of 86 and check whether 86 is a perfect square or not.
 Square Root of 86: √86 = 9.2736
 Square of 86: 86^{2} = 7396
1.  What Is the Square Root of 86? 
2.  Is Square Root of 86 Rational or Irrational? 
3.  How to Find the Square Root of 86? 
4.  FAQs on Square Root of 86 
What Is the Square Root of 86?
Square root of a number is the value of power 1/2 of that number. In other words, it is the number that we multiply by itself to get the original number. It is represented using the symbol '√ '. The square root of a number n is written as √n. The square root of 86 can be represented in the following ways:
 Radical form: √86
 Decimal form: 9.2736
 Exponent form: (86)^{½}
Is Square Root of 86 Rational or Irrational?
 86 is not a perfect square, which means that it does not have a natural number as its square root.
 Square root of 86 in the decimal form is √86 = 8.88819.
 Square root of 86 cannot be expressed as a fraction of the form p/q. This indicates that the square root of 86 is an irrational number.
How to Find the Square Root of 86?
There are two ways to find the square root of 86:
 Long Division Method
 Prime Factorization Method
Long Division Method
The square root of 86 by long division method consists of the following steps:
 Step 1: Starting from the right, we will pair up the digits 86 by placing a bar above 86. We will also pair the 0s in the decimal from left to right.
 Step 2: Find a number that, when multiplied to itself, gives a product less than or equal to 86. The number 9 fits here as 9 square gives 81. Dividing 86 by 9 with quotient as 9, we get the remainder as 5.
 Step 3: Drag a pair of 0’s down and place them next to 5 to make the dividend 500.
 Step 4: Double the divisor 9, and enter 18 below with a blank digit on its right. Think of a number which is greater than or equal to the dividend, i.e., 500. 182 is the perfect number to divide 500.
 Step 5: Multiply 182 by 2 (182 × 2 = 364 < 500) and write the remainder, i.e. 136.
 Step 6: Repeat this process until you get the desired number.
Therefore, the square root of 86 = 9.273
Prime Factorization Method
 To find the square root of 86, we will first express it in terms of its prime factors. The prime factorization of 86 is given as 86 = 2 × 43.
 Since all the prime factors of 86 are unique, none of these factors are perfect squares. The square root of 86 cannot be simplified.
 The square root of 2 is 1.4142 and square root of 43 is 6.5574.
 Therefore, the square root of √86 = √2 × √43 = 9.273.
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Important Notes:
 The square root of 86 can be written as √86.
 a × a = 86. It can also be written as: a^{2} = 86.
 a = √86. a is the 2^{nd} root of 86 and a = 9.273.
 The square root of any number has two values; one is positive and the other is negative. So, √86 = +9.273 or  9.273.
 In the exponential form, we denote √86 as (86) ^{½}.
 In the simplest radical form square root of 86 is written as √86.
Square Root of 86 Solved Examples

Example 1: Jack is multiplying a number by itself. If the product is 86, help Jack find the number.
Solution:
To find the number, let us assume the number to be z
On multiplying z times z = z × z = 86
z² = 86
z = √86
z = 9.273
(9.273 × 9.273 = 85.9885 ≅ 86)
The number is 9.273. 
Example 2: The area of a squareshaped mat is 7396 square units. Calculate the length of one side of the mat.
Solution:
The area of the mat = 7396 sq. units
To find the side of a squareshaped mat, let us take square root of 7396 by prime factorization method.
√7396 = (2 × 2 × 43 × 43) = 2 × 43 = 86
Square root of 7396 is 86.
Therefore, the length of the side of the square mat is 86 units. 
Example 3: Help Joy to check whether the square of 86 is 7396 using long division method..
Solution:
To check whether the square of 86 is 7396 we will find out the square root of 7396 and observe the obtained quotient:
Step 1: Starting from the right, we will pair up the digits 73 96 as shown in image.
Step 2: Find a number that, when multiplied to itself, gives a product less than or equal to 73. The number 8 fits here as 8 square gives 64. Dividing 73 by 8 with quotient as 8, we get the remainder as 9.
Step 3: Drag a pair of 96 down and place them next to 9 to make the dividend 996.
Step 4: Double the divisor 8, and enter 16 below with a blank digit on its right. Think of a number which is greater than or equal to the dividend, i.e., 996. 166 is the perfect number to divide 996.
Step 5: Multiply 166 by 6 (166 × 6 = 996 = 996) and write the remainder, i.e. 0.
Hence, Square root of 7396 is 86 or square of 86 (86 × 86) is 7396.
FAQs on Square Root of 86
What is the square root of 86 using prime factorization?
To find the square root of 86, we will first express it in terms of its prime factors. The prime factorization of 86 is 86 = 2 × 43. On taking the square root both sides we get, √86 = √2 × √43. The value of square root of 2 is 1.4142 and the value of square root of 43 is 6.5574. Therefore, the square root of √86 = √2 × √43 = 9.273.
Is 86 a perfect square root?
86 is not a perfect square. 86 is a natural number, but since there is no other natural number that can be squared to result in the number 86, it is NOT a perfect square.
What is the decimal value of the square root of 86?
The decimal value of the square root of 86 is √86 = 9.273.
Is the square root of 86 a rational number?
No, the square root of 86 is not a rational number since the square root of 86 is nonterminating and cannot be represented in the form of p/q.
Can we find the square root of 86 by the repeated subtraction method?
No, we can’t find the square root of 86 by repeated subtraction method as it can be used only for perfect squares. 86 is not a perfect square.
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