Square Root of 82
Square root is denoted with the symbol √. If n is an integer, then square of n is equal to m, which is also an integer. If n² = m, then n=√m. The square root of 82 is written as √82. Let us explore the square root of 82 in detail. 82 is a composite number, as it has more than 2 factors. √82 an irrational number. In this chapter, we will calculate the square root of 82 by long division method and see why it is an irrational number along with some solved examples. Let us see what the square root of 82 is.
 Square Root of 82:√82 = 9.05538
 Square of 82: 82^{2} = 6,724
1.  What Is the Square Root of 82? 
2.  Is Square Root of 82 Rational or Irrational? 
3.  How to Find the Square Root of 82? 
4.  Thinking Out of the Box! 
5.  FAQs on Square Root of 82 
What Is the Square Root of 82?
Square root is just an inverse operation of square. The number whose square gives 82 is the square root of 82. Square root of 82 in the radical form is represented as √82, while in the exponential form, it is expressed as (82)^{½}. Nonsquare numbers also have a square root, but they are not whole numbers. The square root of 82, rounded to 5 decimal places, is 9.05538.
Is the Square Root of 82 Rational or Irrational?
A rational number is a number that is expressed in the form of p/q where p and q are integers and q is not equal to 0. A number that cannot be expressed as a ratio of two integers is an irrational number. Nonterminating decimals having repeated numbers after the decimal point are rational numbers. √82 = 9.05538. Square root of 82 cannot be written in the form of p/q, where p, q are integers and q is not equal to 0. The value of √82 is 9.05538. Hence, √82 is not a rational number.
How to Find the Square Root of 82?
There are different methods to find the square root of any number. Click here to know more about the different methods. 82 is an even composite number that can be obtained by product of prime numbers 2 and 41. Since it has unique prime numbers, the simplified radical form of √82 is √82.
We can find the square root of 82 by the following two methods:
 Prime Factorization Method
 Long Division Method
Square Root of 82 by Prime Factorization
To find the square root of 82 by prime factorization method, we need to find the prime factors of 82.
82 = 2 × 41
√82 = √2 × √41
Value of square root of 2 is 1.1412
Value of square root of 41 is 6.4031
√82 = 9.05538
Square Root of 82 by Long Division
The value of square root of 82 by long division method consists of following steps:
 Step 1: Place a bar over every pair of digits of the number starting from the unit’s place (rightmost side). We will have 1 pair, i.e., 82. After 82, we can add zeroes with a decimal point as shown in image.
 Step 2: We divide the leftmost number by the largest number whose square is less than or equal to the number in the leftmost pair. (9 × 9 = 81)
 Step 3: Bring down the number under the next bar to the right of the remainder. Add the last digit of the quotient to the divisor (9 + 9 = 18). To the right of the obtained sum, find a suitable number which together with the result of the sum forms a new divisor (180) for the new dividend (100) that is carried down.
 Step 4: The new number in the quotient will have the same number as selected in the divisor (180 × 0 = 0). The condition is the same as being either less or equal to that of the dividend (180 > 100).
 Step 5: Now, we will continue this process further using a decimal point and adding zeros in pairs to the remainder.
 Step 6: The quotient thus obtained will be the square root of the number.
On repeating the above steps we will obtain value of square root of 82 is √82 = 9.0553 up to 4 decimal places
Explore square roots using illustrations and interactive examples
Think Tank:
 Can you find any quadratic equation which has a root as √82?
 As (√82)^{2 }= 82, can we say that √82 is also a square root of 82?
Square Root of 82 Solved Examples

Example 1: Help Ron in finding square root of 82 up to 3 decimal places?
Solution
Following the same steps as discussed above we will find the square root of 82 up to 3 decimal places.
 Step 1: Place a bar over every pair of digits of the number starting from the unit’s place (rightmost side). We will have 1 pair, i.e., 82. After 82 we can add zeroes with a decimal point as shown in image.
 Step 2: We divide the leftmost number by the largest number whose square is less than or equal to the number in the leftmost pair. (9 × 9 = 81)
 Step 3: Bring down the number under the next bar to the right of the remainder. Add the last digit of the quotient to the divisor (9 + 9 = 18). To the right of the obtained sum, find a suitable number which together with the result of the sum forms a new divisor (180) for the new dividend (100) that is carried down.
 Step 4: The new number in the quotient will have the same number as selected in the divisor (180 × 0 = 0). The condition is the same as being either less or equal to that of the dividend (180 > 100).
 Step 5: Now, we will continue this process further using a decimal point and adding zeros in pairs to the remainder.
 Step 6: The quotient thus obtained will be the square root of the number.
 Step 7: Repeat the process till 3 decimal places.

Example 2: What is the sum of lengths of radii of circles having areas 82π and 81π square inches?
Solution
For finding the length of radius of a circle having area 82π, we use the formula: area = πr^{2} = 82π.
Here, r = √82 = 9.055 inches
For finding the length of radius of a circle having area 81π, we use the formula: area = πr^{2} = 81π.
Here, r = √81 = 9 inches
Hence, the sum of lengths of radii of circles having areas 82π and 81π square inches is (9.055 + 9) = 18.055 inches.
FAQs on Square Root of 82
How do I calculate square root of 82?
To calculate the square root of perfect square numbers, we can use prime factorization and repeated subtraction method.
To calculate the square root of a nonperfect square number, we can use approximation method and long division method. 82 is a nonperfect square number.
Using approximation method and long division method we have √82 = 9.0553.
What is the square root of 82 rounded to its nearest tenth?
The square root of 82 rounded to its nearest tenth is √82 = 9.05
Why is √82 an irrational number?
A number with decimal expansion as nonterminating and nonrepeating is always an irrational number. So, √82 is an irrational number.
Is the square root of 82 rational or irrational?
The square root of 82 is irrational.
Is square root of 82 a real number?
Yes, the square root of 82 is a real number.
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