Square Root of 51
The square root of 51, also known just as root 51, is the number which when multiplied by itself results in the number 51. To find the length of a side of a square with area equal to 51, we need to know its square root. We will now see how to calculate the square root of 51 and discover few other interesting facts.
Square Root of 51: √51= 7.14142842…
Square of 51: 51² = 2601
1.  What Is the Square Root of 51? 
2.  Is Square Root of 51 Rational or Irrational? 
3.  How to Find the Square Root of 51? 
4.  Challenging Questions 
5.  Important Notes 
6.  FAQs on Square Root of 51 
What Is the Square Root of 51?
The square root of a number is a number which when multiplied by itself, results in the original number. For example, the square root of 25 is 5, as 5 times 5 results in 25 which makes it a perfect square. However, you can also have square roots of some numbers that do not result in whole numbers, such as 51. We can express the square root of 51 in different ways
 Decimal form: 7.141.
 Radical form: √51
 Exponent form: 51^{1/2}
Is the Square Root of 51 Rational or Irrational?
 The decimal part of the square root 51 is nonterminating. This is the definition of an irrational number.
 Looking at the decimal form of the root 51, we see that it is neverending.
√51= 7.14142842…….  Therefore, we can conclude that square root of 51 is irrational.
How to Find the Square Root of 51?
Since we concluded that the square root of 51 is nonterminating, we can only use 2 methods to calculate the value of its square root:
 Long Division Method
 Estimation and Approximation Method
Long Division
 Step 1: Starting from the right, we will pair up the digits 51 by putting a bar above them. We also pair the 0s in decimals in pairs of 2 from left to right.
 Step 2: Think of a number whose square is less than or equal to 51. In this case, that number would be 7.
 Step 3: Dividing by 51 by 7 with quotient set as 7, we get a remainder of 2.
 Step 4: Drag a pair of 0’s down and fill it next to 2 to make the dividend 200.
 Step 5: The divisor, in this case 7 is doubled and written below. Now, we have 14X as the new divisor, and we need to find a value of X that makes the product of 12X × X less than or equal to 200. In this case, 141 is the required value.
 Step 6: The number 1 is placed in the quotient after a decimal place. The new divisor for the next division will be 14X + X, in this case, 142.
Proceeding in the same manner and repeating from step 4, we can calculate the rest of the decimals.
Therefore, the square root of 51 = 7.141
Estimation and Approximation
The estimation method gives us an approximate answer and is usually not accurate to more than 1 decimal place. However, it is easy to perform as can be seen under.
 Step 1: Find a perfect square that is smaller than 51 and one that is bigger than 51. In this case, 7 and 8 will work as their squares are 49 and 64.
 Step 2: Writing in terms of inequality: 7<√51<8 = 49<51<64
 Step 3: Multiply by 10 and write in terms of square roots: √4900<√5100<√6400
 Step 4: Move closer to inequality: √5041<√5100<√5184 = 71<10√51<72
= 7.1<√51<7.2  Step 5: Taking average of upper and lower limits, we get (7.1 + 7.2)/2 = 7.15
Therefore, we can estimate the square root of 51 ≅ 7.15
Explore square roots using illustrations and interactive examples
Challenging Questions
 Find the value of
 √0.51
 √5100
 What are the roots of 51?
Important Notes
 The real roots of √51 are ± 7.141.
 Square root of a perfect square is always a whole number and that of any other number is always irrational. For example, √16 = 4, whereas √17 = 4.1231…
Solved Examples

Example 1: Ben was wondering whether the value of √51 is the same as √51. What do you think?
Solution:
Square roots of negative numbers cannot be real numbers.
√51 is a real number. However, √51 is an imaginary number.
Hence, they are not the same, and √51 is not the same as √51. 
Example 2: What is the circumference of a circular race track whose area is 51π m²
Solution:
Let circumference be denoted by C, Area by A and radius length by r
C = 2πr
A = πr²
51π = πr²
Simplifying we get, r² = 51 which implies r = √51
C = 2πr = 2π × √51 = 44.87 mTherefore, the perimeter of the land is about 27.13 m
FAQs on Square Root of 51
What is the square root of 51?
The square root of 51 is √51= 7.141.
Can the root of 51 be negative?
The root of the number 51 denotes the positive and negative value of 7.141.
Therefore, the root of 51 can be negative.
What is the square root of 51 in the simplest radical form?
√51 is the simplest radical form.
What is the square of 51?
The square of 51 is 51^{2} = 2601
Can we find the square root of 51 by the prime factorization method?
No, we can’t find the square root of 51 by prime factorization method. This is because its factors are all prime numbers (3 and 17) with power 1 and therefore we cannot simplify it further.