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Square Root of 512
Square root is described with the symbol √. If n is an integer, the square of n is equal to m which is also an integer. If n² = m, then n=√m. The square root of 512 is written as √512. Let us explore the square root of 512 in detail in this lesson. 512 is a composite number, as it has more than 2 factors. √512 an irrational number. In this lesson, we will calculate the square root of 512 by long division method and see why 512 is an irrational number. Let us now find the square root of 512.
 Square Root of 512: √512 = 22.62741
 Square of 512: 512^{2} = 262,144
1.  What Is the Square Root of 512? 
2.  Is Square Root of 512 Rational or Irrational? 
3.  How to Find the Square Root of 512? 
4.  Thinking Out of the Box! 
5.  FAQs on Square Root of 512 
What Is the Square Root of 512?
Square root is just an inverse operation of square. The number whose square gives 512 is the square root of 512. Square root of 512 in the radical form is represented as √512. It is expressed as (512)^{½ }in the exponent form. Nonsquare numbers also have a square root, but they are not whole numbers. The square root of 512 rounded to 5 decimal places is 22.62741.
Is the Square Root of 512 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 with repeated numbers after the decimal point are rational numbers. √512 = 22.62741. Square root of 512 cannot be written in the form of p/q, where p, q are integers and q is not equal to 0. The value of √512 is 22.62741. Hence, √512 is not a rational number.
How to Find the Square Root of 512?
There are different methods to find the square root of any number. Click here to know more about the different methods.
Simplified Radical Form of Square Root of 512
512 is a composite number obtained by the product of the prime number 2. Hence, the simplified radical form of √512 is 16√2.
We can find the square root of 512 by the following two methods:
 Prime Factorization Method
 Long Division Method
Square Root of 512 by Prime Factorization
To find the square root of 512 by prime factorization method, we need to find the prime factors of 512.
512 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2^{9}
512 = 16√2
Square Root of 512 by Long Division
The value of the square root of 512 by long division method consists of the 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 two pairs, i.e. 5 and 12.
 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. (2 × 2 = 4)
 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 (2 + 2 = 4). To the right of the obtained sum, find a suitable number which together with the result of the sum forms a new divisor (42) for the new dividend (112) that is carried down.
 Step 4: The new number in the quotient will have the same number as selected in the divisor (42 × 2 = 84). The condition is the same as being either less than or equal to that of the dividend (84 < 112).
 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 the value of the square root of 512 which is √512 = 22.62741 up to 5 decimal places.
Explore square roots using illustrations and interactive examples
Think Tank:
 Can you find a quadratic equation with root as √512?
 As (√512)^{2 }= 512, can we say that √512 is also a square root of 512?
Square Root of 512 Solved Examples

Example 1: Help Ron find the square root of 512 up to 3 decimal places.
Solution
Following the same steps as discussed above, we will find the square root of 512 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 two pairs, i.e. 5 and 12.
 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. (2 × 2 = 4)
 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 (2 + 2 = 4). To the right of the obtained sum, find a suitable number which together with the result of the sum forms a new divisor (42) for the new dividend (112) that is carried down.
 Step 4: The new number in the quotient will have the same number as selected in the divisor (42 × 2 = 84). The condition is the same as being either less than or equal to that of the dividend (84 < 112).
 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 difference between the lengths of the radii of circles having areas 512π and 100π square inches?
Solution
The length of the radius of a circle with area 512π is to be calculated.
Area = πr^{2} = 512π
Here, r = √512 = 22.62 inches
Next, the length of the radius of a circle with area 100π is to be calculated.
Area = πr^{2} = 100π
Here, r = √100 = 10 inches
Hence, the difference between the lengths of the radii of circles having areas 512π and 100π square inches is (22.62  10) = 12.62 inches.
FAQs on Square Root of 512
How do I calculate the square root of 512?
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. 512 is a nonperfect square number.
Using approximation method and long division method, we get √512 = 22.62.
What is the square root of 512 rounded to its nearest tenth?
The square root of 512 rounded to its nearest tenth is √512 = 22.62.
Why is √512 an irrational number?
A number with decimal expansion as nonterminating and nonrepeating is always an irrational number. Hence, √512 is an irrational number.
Is the square root of 512 rational or irrational?
The square root of 512 is irrational.
Is the square root of 512 a real number?
Yes, the square root of 512 is a real number.
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