Square root of 85

The square root of 85 is ±9.21954445729.The positive value, 9.21954445729 is the solution of the equation x² = 85. As defined, the square root is just the inverse of squaring a number, so, squaring 9.21954445729 will result in 85.  The square root of 85 is expressed as √85 in radical form, where the ‘√’  sign is called “radical”  sign. In exponential form, it is written as (85)^1/2  
 

We can find the square root of 85 through various methods. They are:

• Prime factorization method

• Long division method

• Approximation/Estimation method

The prime factorization of 85 involves breaking down a number into its factors. Divide 85 by prime numbers, and continue to divide the quotients until they can’t be separated anymore. After factorizing 85, make pairs out of the factors to get the square root. If there exists numbers which cannot be made pairs further, we place those numbers with a “radical” sign along with the obtained pairs

So, Prime factorization of 85 = 5 × 17    

 for 28,no pairs of factors can be obtained but a single 5 and a single 17 are obtained

So, it can be expressed as  √85 =√(5 × 17) = √85

 √85 is the simplest radical form of √85

This is a method used for obtaining the square root for non-perfect squares, mainly. It usually involves the division of the dividend by the divisor, getting a quotient and a remainder too sometimes.

Follow the steps to calculate the square root of 85:

Step 1 : Write the number 85, and draw a bar above the pair of digits from right to left.

               
Step 2 : Now, find the greatest number whose square is less than or equal to 85. Here, it is 9, Because 9²=81 < 85.

Step 3 : Now divide 85 by 9 (the number we got from Step 2) such that we get 9 as quotient, and we get a remainder. Double the divisor 9, we get 18 and then the largest possible number A1=2 is chosen such that when 2 is written beside the new divisor, 18, a 3-digit number is formed →182 and multiplying 2 with 182 gives 364 which is less than 400.

Repeat the process until you reach remainder 0

We are left with the remainder, 1759 (refer to the picture), after some iterations and keeping the division till here, at this point 

             
Step 4 : The quotient obtained is the square root. In this case, it is 9.21…

• Exponential form: An algebraic expression that includes an exponent. It is a way of expressing the numbers raised to some power of their factors. It includes continuous multiplication involving base and exponent.Ex: 3 ⤬ 3 ⤬ 3 ⤬ 3 = 81 or, 3⁴ = 81, where 3 is the base, 4 is the exponent 

• Factorization: Expressing the given expression as a product of its factors.Ex: 52=2 ⤬ 2 ⤬ 13 

• Prime Numbers: Numbers which are greater than 1, having only 2 factors as →1 and Itself. Ex: 1,3,5,7,....

• Rational numbers and Irrational numbers: The Number which can be expressed as p/q, where p and q are integers and q not equal to 0 are called Rational numbers. Numbers which cannot be expressed as p/q, where p and q are integers and q not equal to 0 are called Irrational numbers. 

• perfect and non-perfect square numbers:Perfect square numbers are those numbers whose square roots do not include decimal places. Ex: 4,9,25 Non-perfect square numbers are those numbers whose square roots comprise decimal places. Ex :2, 8, 18