Square Root of 1729
2026-02-28 13:54 Diff
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Last updated on August 5, 2025

If a number is multiplied by itself, the result is a square. The inverse of the square is a square root. The square root is used in fields such as vehicle design, finance, etc. Here, we will discuss the square root of 1729.

What is the Square Root of 1729?

The square root is the inverse of the square of a number. 1729 is not a perfect square. The square root of 1729 is expressed in both radical and exponential form. In radical form, it is expressed as √1729, whereas in exponential form it is (1729)^(1/2). √1729 ≈ 41.5633, 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 1729

The prime factorization method is used for perfect square numbers. However, for non-perfect square numbers, 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 1729 by Prime Factorization Method

The product of prime factors is the prime factorization of a number. Now let us look at how 1729 is broken down into its prime factors:

Step 1: Finding the prime factors of 1729 Breaking it down, we get 7 x 13 x 19.

Step 2: Now we found out the prime factors of 1729. The second step is to make pairs of those prime factors. Since 1729 is not a perfect square, the digits of the number cannot be grouped in pairs.

Therefore, calculating √1729 using prime factorization is not straightforward.

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Square Root of 1729 by Long Division Method

The long division method is particularly used for non-perfect square numbers. In this method, we check the closest perfect square number to the given number. Let us now learn how to find the square root using the long division method, step by step:

Step 1: Group the numbers from right to left. In the case of 1729, group it as 17 and 29.

Step 2: Find a number whose square is less than or equal to 17. This number is 4, since 4 x 4 = 16. The quotient is 4, and the remainder is 1 (17 - 16).

Step 3: Bring down the next pair, 29, to get the new dividend, 129.

Step 4: Double the previous quotient (4) to get 8, which will be part of the new divisor.

Step 5: Find a number n such that 8n x n is less than or equal to 129. In this case, n is 1, since 81 x 1 = 81.

Step 6: Subtract 81 from 129 to get a remainder of 48, and the quotient is 41.

Step 7: Add decimal points to the quotient and bring down pairs of zeros.

Step 8: Repeat the process to find the next digit(s) after the decimal point.

The square root of 1729 is approximately 41.563.

Square Root of 1729 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 1729 using the approximation method.

Step 1: Find the closest perfect squares around 1729.

The smallest is 1600 (40^2) and the largest is 1764 (42^2).

√1729 falls somewhere between 40 and 42.

Step 2: Apply the formula:

(Given number - smaller perfect square) / (larger perfect square - smaller perfect square).

(1729 - 1600) / (1764 - 1600) = 129 / 164 ≈ 0.786.

Step 3: Add this decimal to the smaller square root: 40 + 0.786 = 40.786.

The square root of 1729 is approximately 41.563, refining the approximation from the previous steps.

Common Mistakes and How to Avoid Them in the Square Root of 1729

Students often make mistakes while finding the square root, such as forgetting about the negative square root or skipping steps in the long division method. Let us look at a few of these 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 √1729?

Okay, lets begin

The area of the square is 1729 square units.

Explanation

The area of the square = side^2. The side length is given as √1729. Area of the square = side^2 = √1729 x √1729 = 1729. Therefore, the area of the square box is 1729 square units.

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Problem 2

A square-shaped building measuring 1729 square feet is built; if each of the sides is √1729, what will be the square feet of half of the building?

Okay, lets begin

864.5 square feet

Explanation

We can divide the given area by 2 as the building is square-shaped. Dividing 1729 by 2, we get 864.5. So half of the building measures 864.5 square feet.

Well explained 👍

Problem 3

Calculate √1729 x 5.

Okay, lets begin

207.8165

Explanation

The first step is to find the square root of 1729, which is approximately 41.563, and then multiply by 5. So, 41.563 x 5 ≈ 207.8165.

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Problem 4

What will be the square root of (1600 + 129)?

Okay, lets begin

The square root is approximately 41.563.

Explanation

To find the square root, find the sum of (1600 + 129) = 1729, then √1729 ≈ 41.563. Therefore, the square root of (1600 + 129) is approximately ±41.563.

Well explained 👍

Problem 5

Find the perimeter of the rectangle if its length ‘l’ is √1729 units and the width ‘w’ is 38 units.

Okay, lets begin

The perimeter of the rectangle is approximately 159.126 units.

Explanation

Perimeter of the rectangle = 2 × (length + width) Perimeter = 2 × (√1729 + 38) ≈ 2 × (41.563 + 38) = 2 × 79.563 ≈ 159.126 units.

Well explained 👍

FAQ on Square Root of 1729

1.What is √1729 in its simplest form?

The prime factorization of 1729 is 7 x 13 x 19, so the simplest form of √1729 is √(7 x 13 x 19).

2.Mention the factors of 1729.

Factors of 1729 are 1, 7, 13, 19, 91, 133, 247, and 1729.

3.Calculate the square of 1729.

We get the square of 1729 by multiplying the number by itself, that is 1729 x 1729 = 2,990,041.

4.Is 1729 a prime number?

1729 is not a prime number, as it has more than two factors.

5.1729 is divisible by?

1729 has several factors; those are 1, 7, 13, 19, 91, 133, 247, and 1729.

Important Glossaries for the Square Root of 1729

  • Square root: A square root is the inverse operation to squaring a number. Example: 4^2 = 16, and the inverse of squaring is the square root, so √16 = 4.
     
  • Irrational number: An irrational number is a number that cannot be written as a simple fraction (p/q), where p and q are integers and q is not zero.
     
  • Approximation: The process of finding a value that is close enough to the right answer, typically with some thought or calculation involved.
     
  • Perfect square: A perfect square is a number that is the square of an integer. For example, 4, 9, and 16 are perfect squares.
     
  • Long division method: A method used to find the square root of a number by performing division and averaging processes iteratively.

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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.