Square Root of 59
2026-02-28 23:55 Diff
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Last updated on August 5, 2025

If a number is multiplied by the same number, the result is a square. The inverse of the square is a square root. The square root is used in the fields of vehicle design, finance, etc. Here, we will discuss the square root of 59.

What is the Square Root of 59?

The square root is the inverse of the square of the number. 59 is not a perfect square. The square root of 59 is expressed in both radical and exponential forms. In the radical form, it is expressed as √59, whereas (59)^(1/2) in the exponential form. √59 ≈ 7.68115, which is an irrational number because it cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0.

Finding the Square Root of 59

The prime factorization method is used for perfect square numbers. However, the prime factorization method is not used for non-perfect square numbers, where the long division method and approximation method are used. Let us now learn the following methods:

  • Prime factorization method
  • Long division method
  • Approximation method

Square Root of 59 by Prime Factorization Method

The product of prime factors is the prime factorization of a number. Now let us look at how 59 is broken down into its prime factors:

Step 1: Finding the prime factors of 59 59 is a prime number, so it cannot be broken down further.

Step 2: Since 59 is not a perfect square, calculating √59 using prime factorization is not possible.

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Square Root of 59 by Long Division Method

The long division method is particularly used for non-perfect square numbers. Let us now learn how to find the square root using the long division method, step by step.

Step 1: To begin with, we need to group the number as a pair of two digits from right to left. In the case of 59, it is already a two-digit number.

Step 2: We find n whose square is less than or equal to 59. We can say n is '7' because 7 × 7 = 49, which is less than 59. The quotient is 7, and the remainder is 59 - 49 = 10.

Step 3: Since the remainder is less than the divisor, add a decimal point and bring down a pair of zeros, making the new dividend 1000.

Step 4: Double the quotient, 7, to get 14. This will be our new divisor's first part.

Step 5: We find a digit 'd' such that 14d × d ≤ 1000. Trying d = 6, we find 146 × 6 = 876.

Step 6: Subtract 876 from 1000 to get a remainder of 124, and the quotient becomes 7.6.

Step 7: Continue this process until the desired accuracy is achieved.

So, √59 ≈ 7.68

Square Root of 59 by Approximation Method

The approximation method is another method for finding square roots. It is an easy method to find the square root of a given number. Now let us learn how to find the square root of 59 using the approximation method.

Step 1: Now we have to find the closest perfect squares of 59. The smallest perfect square less than 59 is 49, and the largest perfect square greater than 59 is 64. Therefore, √59 falls somewhere between 7 and 8.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Using the formula, (59 - 49) / (64 - 49) = 10 / 15 = 0.6667 Adding this decimal to the smaller integer, we get 7 + 0.67 = 7.67.

So, the approximate square root of 59 is 7.67.

Common Mistakes and How to Avoid Them in the Square Root of 59

Students do make mistakes while finding the square root, like forgetting about the negative square root or skipping steps in long division methods. Now let us look at a few of those mistakes that students tend to make in detail.

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Problem 1

Can you help Max find the area of a square box if its side length is given as √58?

Okay, lets begin

The area of the square is approximately 58 square units.

Explanation

The area of the square = side².

The side length is given as √58.

Area of the square = (√58)² = 58.

Therefore, the area of the square box is approximately 58 square units.

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Problem 2

A square-shaped building measuring 59 square feet is built; if each of the sides is √59, what will be the square feet of half of the building?

Okay, lets begin

29.5 square feet

Explanation

We can just divide the given area by 2 as the building is square-shaped.

Dividing 59 by 2 gives us 29.5.

So half of the building measures 29.5 square feet.

Well explained 👍

Problem 3

Calculate √59 × 3.

Okay, lets begin

Approximately 23.04345

Explanation

The first step is to find the square root of 59, which is approximately 7.68115.

The second step is to multiply 7.68115 by 3.

So, 7.68115 × 3 ≈ 23.04345.

Well explained 👍

Problem 4

What will be the square root of (49 + 10)?

Okay, lets begin

The square root is 7.81

Explanation

To find the square root, we need to find the sum of (49 + 10). 49 + 10 = 59, and then √59 ≈ 7.68115.

Therefore, the square root of (49 + 10) is approximately ±7.68115.

Well explained 👍

Problem 5

Find the perimeter of the rectangle if its length ‘l’ is √59 units and the width ‘w’ is 20 units.

Okay, lets begin

We find the perimeter of the rectangle as approximately 55.36 units.

Explanation

Perimeter of the rectangle = 2 × (length + width).

Perimeter = 2 × (√59 + 20) = 2 × (7.68115 + 20) ≈ 2 × 27.68115 ≈ 55.36 units.

Well explained 👍

FAQ on Square Root of 59

1.What is √59 in its simplest form?

Since 59 is a prime number, the simplest form of √59 is itself: √59.

2.What are the factors of 59?

Since 59 is a prime number, its only factors are 1 and 59.

3.Calculate the square of 59.

We get the square of 59 by multiplying the number by itself: 59 × 59 = 3481.

4.Is 59 a prime number?

Yes, 59 is a prime number because it has only two factors: 1 and 59.

5.What numbers is 59 divisible by?

59 is only divisible by 1 and 59, as it is a prime number.

Important Glossaries for the Square Root of 59

  • Square root: A square root is the inverse operation of squaring a number. Example: 4² = 16, and the square root of 16 is √16 = 4.
     
  • Irrational number: An irrational number is a number that cannot be written as a simple fraction (i.e., in the form of p/q where p and q are integers, and q ≠ 0).
     
  • Prime number: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Example: 59.
     
  • Long division method: A step-by-step method used to find the square root of a non-perfect square number by repeatedly dividing and averaging.
     
  • Decimal: A number that consists of a whole number and fractional part separated by a decimal point, such as 7.68, 8.65, or 9.42.

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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.