Square Root of 1681
2026-02-28 08:44 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 operation of finding a square is determining its square root. The square root is used in various fields such as vehicle design, finance, etc. Here, we will discuss the square root of 1681.

What is the Square Root of 1681?

The square root is the inverse of the square of a number. 1681 is a perfect square. The square root of 1681 can be expressed in both radical and exponential form. In radical form, it is expressed as √1681, whereas in exponential form, it is expressed as (1681)^(1/2). The square root of 1681 is 41, which is a rational number because it can be expressed as a fraction where both the numerator and the denominator are integers.

Finding the Square Root of 1681

The prime factorization method is commonly used for perfect square numbers. However, for educational purposes, the long division method and approximation method can also be used. Let us now learn these methods:

  • Prime factorization method
  • Long division method
  • Approximation method

Square Root of 1681 by Prime Factorization Method

The prime factorization of a number involves breaking it down into its prime factors. Let's see how 1681 is factored:

Step 1: Finding the prime factors of 1681. Breaking it down, we find 1681 = 41 × 41.

Step 2: Since 1681 is a perfect square, the digits can be grouped into pairs, and each pair repeated.

Therefore, the square root of 1681 using prime factorization is 41.

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Square Root of 1681 by Long Division Method

The long division method is particularly useful for finding the square roots of non-perfect square numbers, but it is shown here for thoroughness.

Step 1: Pair the digits of 1681 from right to left, giving us two pairs: 16 and 81.

Step 2: Find a number whose square is less than or equal to 16. That number is 4, as 4 x 4 = 16. Subtract 16 from 16, leaving 0.

Step 3: Bring down the next pair, 81, to make it 081.

Step 4: Double the current quotient, 4, to make it 8. We now need to find a digit, say n, such that 8n × n ≤ 81. The number 1 works, as 81 x 1 = 81.

Step 5: Subtract 81 from 81, leaving 0.

The quotient is 41, which is the square root of 1681.

Square Root of 1681 by Approximation Method

The approximation method is not necessary for perfect squares but can demonstrate the closeness of the square root for numbers similar to 1681.

Step 1: Identify the closest perfect squares around 1681. Here, 1600 (40^2) and 1764 (42^2) are the nearest perfect squares. √1681 lies between 40 and 42.

Step 2: Check the number halfway between these squares, which is 41.

Since 41 x 41 = 1681, we confirm that the square root of 1681 is 41.

Common Mistakes and How to Avoid Them in the Square Root of 1681

Students often make mistakes while finding square roots, such as forgetting about the negative square root or skipping necessary steps in methods. Let's look at a few common 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 √1681?

Okay, lets begin

The area of the square is 1681 square units.

Explanation

The area of a square is calculated as the side length squared. The side length is given as √1681, which is 41. Area of the square = side^2 = 41 × 41 = 1681 square units. Therefore, the area of the square box is 1681 square units.

Well explained 👍

Problem 2

A square-shaped building measuring 1681 square feet is constructed. If each of the sides is √1681, what will be the square feet of half of the building?

Okay, lets begin

840.5 square feet

Explanation

To find half of the building's area, divide the total area by 2. Dividing 1681 by 2 gives 840.5. So, half of the building measures 840.5 square feet.

Well explained 👍

Problem 3

Calculate √1681 × 3.

Okay, lets begin

123

Explanation

First, find the square root of 1681, which is 41. Then multiply 41 by 3. So, 41 × 3 = 123.

Well explained 👍

Problem 4

What will be the square root of (1600 + 81)?

Okay, lets begin

The square root is 41.

Explanation

First, find the sum of 1600 and 81, which is 1681. Then find the square root of 1681. 1681 = 41^2, so the square root of 1681 is ±41.

Well explained 👍

Problem 5

Find the perimeter of the rectangle if its length ‘l’ is √1681 units and the width ‘w’ is 38 units.

Okay, lets begin

The perimeter of the rectangle is 158 units.

Explanation

Perimeter of a rectangle = 2 × (length + width). Perimeter = 2 × (√1681 + 38) = 2 × (41 + 38) = 2 × 79 = 158 units.

Well explained 👍

FAQ on Square Root of 1681

1.What is √1681 in its simplest form?

Since 1681 is a perfect square, the simplest form of √1681 is 41.

2.Mention the factors of 1681.

The factors of 1681 are 1, 41, and 1681.

3.Calculate the square of 1681.

The square of 1681 is 1681 × 1681 = 2825761.

4.Is 1681 a prime number?

1681 is not a prime number, as it has more than two factors (1, 41, 1681).

5.1681 is divisible by?

1681 is divisible by 1, 41, and 1681.

Important Glossaries for the Square Root of 1681

  • Square root: A square root is the inverse operation of squaring a number. For example, if 5^2 = 25, then √25 = 5.
     
  • Rational number: A rational number is a number that can be expressed as a fraction p/q, where p and q are integers and q ≠ 0.
     
  • Perfect square: A perfect square is a number that is the square of an integer. For example, 36 is a perfect square because it is 6^2.
     
  • Integer: An integer is a whole number that can be positive, negative, or zero. Examples include -3, 0, 4.
     
  • Exponential form: A way of expressing numbers using a base raised to an exponent. For example, the square of 4 can be expressed as 4^2.

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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.