Square Root of 59
The Square Root 59 is a number which when multiplied by itself, results in the number 59. 59 is also a prime number which makes it trickier to calculate its square root. We will learn how to calculate the Square Root of 59 and look at a few problems to help us understand this topic.
Square Root of 59: √59= 7.6811457...
Square of 59: 59^{2} = 3481
1.  What is the Square Root of 59? 
2.  Is Square Root of 59 Rational or Irrational? 
3.  How to Find the Square Root of 59? 
4.  FAQs on Square Root of 59 
What is the Square Root of 59?
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 59. We can express the square root of 59 in different ways
 Decimal form: 7.681
 Radical form: √59
 Exponent form: (59)^{1/2}
Is the Square Root of 59 Rational or Irrational?
 59 is a number that is not a perfect square, meaning it does not have a natural number as its square root.
 Also, its square root cannot be expressed as a fraction of the form p/q which tells us that the square root of 59 is an irrational number.
How to Find the Square Root of 59?
Since we concluded that the square root of 59 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 59 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 59. In this case, that number would be 7.
 Step 3: Dividing by 59 by 7 with quotient set as 7, we get a remainder of 10.
 Step 4: Drag a pair of 0’s down and fill it next to the remainder to make it 1000.
 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 14X × X less than or equal to 1000. In this case, 146 is the required value
 Step 6: The number 6 is placed in the quotient after a decimal place. The new divisor for the next division will be 146 + 6 = 152.
Proceeding in the same manner and repeating from step 4, we can calculate the rest of the decimals.
Therefore, the square root of 59 = 7.681
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 and bigger than 59. In this case, 7 and 8 will work as their squares are 49 and 64.
 Step 2: Writing in terms of inequality 7<√59<8 = 49<59<64
 Step 3: Multiply by 100 and write in terms of square roots √4900<√5900<√6400
 Step 4: Move closer to inequality √5776<√5900<√5929 = 76<10√59<77
= 7.6<√59<7.7  Step 5: Taking average of upper and lower limits we get, (7.6 + 7.7)/2 = 7.65
Therefore, we can estimate the square root of 59 ≅ 7.65
Explore Square roots using illustrations and interactive examples
Challenging Questions
 What is the square root of 59 up to 6 decimal places(Use Long Division)
 What are the roots of 59? Also, find the value of the square of the negative root of 59
Solved Examples

Example 1: Zack wants to calculate the diagonal length of the garden at his home. It has an area of 59 square feet. Help him find the length, rounded to the nearest tenth.
Solution:
To find the side of the garden, we will have to find the square root of 59.
Diagonal Length = √2 × Side Length = √2 × √59 = √2 × 7.681 = 10.862
Therefore, the diagonal length of the garden is 10.9 feet. 
Example 2: What is the square root of √59?
Solution:
The square root of negative numbers is represented by imaginary numbers.
So, the square root of √59 is written as √59 i. Where (i = √1)
FAQs on Square Root of 59
What is the square root of 59?
The square root of 59 is √59= 7.681.
Is the square root of 59 a rational number?
No, the square root of 59 is not a rational number since the square root of 59 is nonterminating and cannot be represented in the form of p/q.
What is the square root of 59 in the simplest radical form?
√59 is the simplest radical form.
What is the square of 59?
The square of 59 is 59^{2} = 3481
Can we find the square root of 59 by the prime factorization method?
No, we can’t find the square root of 59 by the prime factorization method. This is because it is a prime number and its only factors are itself and 1. Therefore we cannot simplify it further.