Square Root of 709
2026-02-28 08:29 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 various fields such as engineering, finance, etc. Here, we will discuss the square root of 709.

What is the Square Root of 709?

The square root is the inverse of the square of the number. 709 is not a perfect square. The square root of 709 is expressed in both radical and exponential form. In radical form, it is expressed as √709, whereas in exponential form it is expressed as (709)^(1/2). √709 ≈ 26.62705, 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 709

The prime factorization method is used for perfect square numbers. However, the prime factorization method is not used for non-perfect square numbers, where 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 709 by Prime Factorization Method

The product of prime factors is the prime factorization of a number. Now let us look at how 709 is broken down into its prime factors.

Step 1: Finding the prime factors of 709 709 is a prime number, so it cannot be broken down further. The prime factorization of 709 is simply 709^1.

Step 2: Since 709 is not a perfect square, calculating √709 using prime factorization is not feasible. We will use other methods such as the long division method or approximation method.

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Square Root of 709 by Long Division Method

The long division method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square number for the given number. 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 numbers from right to left. In the case of 709, we group it as 09 and 7.

Step 2: Now we need to find n whose square is ≤ 7. We can say n as ‘2’ because 2^2 = 4, which is less than 7. The quotient is 2, and after subtracting 4 from 7, we get a remainder of 3.

Step 3: Bring down 09, making the new dividend 309. Double the quotient (2) and write it as 4_, where _ is the next digit of the divisor. Step 4: Find a digit n such that 4n × n ≤ 309. Using trial, we find n = 6, as 46 × 6 = 276.

Step 5: Subtract 276 from 309, leaving a remainder of 33. The quotient so far is 26.

Step 6: Since the dividend is less than the divisor, we need to add a decimal point and bring down pairs of zeroes. The next dividend is 3300.

Step 7: Find a new divisor by doubling the current quotient (26) and adding the next digit. Repeat the process to find the next digits of the quotient until the desired precision is achieved.

Step 8: Continue until you get two numbers after the decimal point. The approximate square root of √709 is 26.63.

Square Root of 709 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 709 using the approximation method.

Step 1: Find the closest perfect squares surrounding 709. The smallest perfect square less than 709 is 676 (26^2), and the largest perfect square greater than 709 is 729 (27^2). √709 falls between 26 and 27.

Step 2: Use the formula: (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square). (709 - 676) / (729 - 676) = 33 / 53 ≈ 0.6226 Using the formula, we identified the decimal point of our square root. Add this value to the smaller integer square root: 26 + 0.6226 ≈ 26.63. Therefore, the square root of 709 is approximately 26.63.

Common Mistakes and How to Avoid Them in the Square Root of 709

Students do make mistakes while finding the square root, such as forgetting about the negative square root, skipping steps in the long division method, etc. Let's look at a few of those mistakes 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 √709?

Okay, lets begin

The area of the square is approximately 709 square units.

Explanation

The area of a square is calculated as side^2.

The side length is given as √709.

Area of the square = (√709) × (√709) = 709.

Therefore, the area of the square box is 709 square units.

Well explained 👍

Problem 2

A square-shaped garden measuring 709 square feet is built; if each of the sides is √709, what will be the square feet of half of the garden?

Okay, lets begin

354.5 square feet

Explanation

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

Dividing 709 by 2 gives us 354.5.

So, half of the garden measures 354.5 square feet.

Well explained 👍

Problem 3

Calculate √709 × 5.

Okay, lets begin

Approximately 133.13525

Explanation

The first step is to find the square root of 709, which is approximately 26.62705.

The second step is to multiply 26.62705 by 5.

So, 26.62705 × 5 ≈ 133.13525.

Well explained 👍

Problem 4

What will be the square root of (700 + 9)?

Okay, lets begin

The square root is approximately 26.63.

Explanation

To find the square root, we calculate the sum of 700 + 9, which is 709.

The square root of 709 is approximately 26.63.

Well explained 👍

Problem 5

Find the perimeter of a rectangle if its length ‘l’ is √709 units and the width ‘w’ is 38 units.

Okay, lets begin

The perimeter of the rectangle is approximately 129.2541 units.

Explanation

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

Perimeter = 2 × (√709 + 38) = 2 × (26.62705 + 38) = 2 × 64.62705 = 129.2541 units.

Well explained 👍

FAQ on Square Root of 709

1.What is √709 in its simplest form?

709 is a prime number, so its simplest radical form is √709.

2.Is 709 a prime number?

Yes, 709 is a prime number as it has only two factors, 1 and 709.

3.Calculate the square of 709.

We get the square of 709 by multiplying the number by itself, which is 709 × 709 = 502681.

4.709 is divisible by?

709 is a prime number and is only divisible by 1 and 709.

5.What are some uses of square roots?

Square roots are used in various fields like engineering, physics, finance, and statistics for calculations involving quadratic equations, area computations, and data analysis.

Important Glossaries for the Square Root of 709

  • Square root: A square root is the inverse operation of squaring a number. For example, 4^2 = 16, and the square root of 16 is √16 = 4.
  • Irrational number: An irrational number cannot be expressed as a simple fraction. For example, √709 is irrational.
  • Prime number: A prime number has only two distinct positive divisors: 1 and itself. For example, 709 is a prime number.
  • Radical: A symbol (√) used to denote the root of a number. For example, √709 represents the square root of 709.
  • Approximation: The process of finding a value that is close to the actual value. For example, √709 is approximately 26.63.

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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.