Square root of 85
2026-02-28 23:30 Diff
https://brightchamps.com/en-us/math/algebra/square-root-of-85

864 Learners

Last updated on August 5, 2025

The square root of 85 is the inverse operation of squaring a value “y” such that when “y” is multiplied by itself → y ⤫ y, the result is 85. It contains both positive and a negative root, where the positive root is called the principal square root.

What Is the 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  
 

Finding the Square Root of 85

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

  • Prime factorization method
  • Approximation/Estimation method

Square Root of 85 By Prime Factorization 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

Explore Our Programs

Square Root of 85 By Long Division Method

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…

Common Mistakes and How to Avoid Them in the Square Root of 85

When we find the square root of 85, we often make some key mistakes, especially when we solve problems related to that. So, let’s see some common mistakes and their solutions.

Download Worksheets

Problem 1

Simplify √85/√102 ?

Okay, lets begin

√85/√102

= √(85/102)

= √(5/6) 

Answer : √(5/6)

Explanation

divided both denominator and numerator with 17 and simplified

Well explained 👍

Problem 2

What is √85 multiplied by 2?

Okay, lets begin

√85 ⤬ 2

=  2√85


Answer: 2√85 
 

Explanation

 just multiplied with 2
 

Well explained 👍

Problem 3

Which is greater √85 or √86 ?

Okay, lets begin

Approximate both values 


√85 ≅ 9.21,  

√86 ≅ 9.27


√85 < √86


Answer: √86 is greater than √85
 

Explanation

 It is obvious that √86 is greater than √85, but still we should find out the exact values and compare.
 

Well explained 👍

Problem 4

find √85 (√85 (√85))

Okay, lets begin

√85 (√85╳ √85)

= √85 ╳ 85

= 85√85


Answer: 85√85
 

Explanation

 multiplying √85 with itself, then again multiplying the product with √85
 

Well explained 👍

Problem 5

Find √85 / √ 170

Okay, lets begin

√85/√170

= √(85/170)

= √(1/2)


Answer : √(1/2)
 

Explanation

we simplified the division

Well explained 👍

FAQs on 85 Square Root

1.What is the square value of 85 ?

What is the square value of 85 ?
 

2.What is √85 to the nearest 10th ?

3.Is 85 a perfect square or non-perfect square?

85 is a non-perfect square, since 85 =(9.21954445729)2.
 

4.Is the square root of 85 a rational or irrational number?

The square root of 85 is ±9.21954445729. So, 9.21954445729 is an irrational number, since it cannot be obtained by dividing two integers and cannot be written in the form p/q, where p and q are integers.

5.How to represent √85 in a number line?

So, locate the value between 9 and 10 on the number line. It will be slightly nearer to 9, just between 9.1 and 9.3

6. What is the value of √85 by Prime Factorization Method?

√(17⤫5)= √85 is the value of √85 by Prime Factorization Method
 

Important Glossaries for Square Root of 85

  • 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

What Is Algebra? 🧮 | Simple Explanation with 🎯 Cool Examples for Kids | ✨BrightCHAMPS Math

▶

Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.