Square Root of 487
2026-02-28 08:24 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 fields such as engineering, architecture, and finance. Here, we will discuss the square root of 487.

What is the Square Root of 487?

The square root is the inverse of squaring a number. 487 is not a perfect square. The square root of 487 is expressed in both radical and exponential form. In radical form, it is expressed as √487, whereas in exponential form, it is expressed as (487)^(1/2). √487 ≈ 22.068, 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 487

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 487 by Prime Factorization Method

The product of prime factors is the prime factorization of a number. Now, let us look at how 487 is broken down into its prime factors:

Step 1: Finding the prime factors of 487. 487 is a prime number, so it cannot be broken down further into other prime factors. Thus, it cannot be simplified using the prime factorization method. Calculating the square root of 487 using prime factorization is not feasible.

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Square Root of 487 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: Start by grouping the digits from right to left in pairs. In the case of 487, we have 4 and 87.

Step 2: Find a number n whose square is less than or equal to 4. This number is 2, because 2 × 2 = 4.

Step 3: Subtract 4 from 4, which leaves a remainder of 0, and bring down the next pair, 87.

Step 4: Double the divisor (2) to get 4, and find a new digit (n) such that 4n × n ≤ 87. This digit is 2, because 42 × 2 = 84.

Step 5: Subtract 84 from 87, resulting in a remainder of 3.

Step 6: Since the dividend is less than the divisor, add a decimal point and bring down a pair of zeros to the remainder to make it 300.

Step 7: The new divisor is 44, and find n such that 44n × n ≤ 300. The number is 6, because 446 × 6 = 2676.

Step 8: Subtract 2676 from 3000 to get a remainder of 324.

Step 9: Continue the process to get the desired precision.

So the square root of √487 ≈ 22.068.

Square Root of 487 by Approximation Method

The approximation method is another method for finding square roots. Now let us learn how to find the square root of 487 using the approximation method.

Step 1: Determine the closest perfect squares surrounding √487. The closest perfect squares are 484 (22^2) and 529 (23^2). Thus, √487 lies between 22 and 23.

Step 2: Apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square). Applying the formula: (487 - 484) ÷ (529 - 484) = 3 ÷ 45 ≈ 0.067. Adding this to 22 gives us 22 + 0.067 ≈ 22.067.

Hence, the square root of 487 is approximately 22.067.

Common Mistakes and How to Avoid Them in the Square Root of 487

Students often make mistakes while finding square roots, such as forgetting about the negative square root or skipping important steps in the long division method. Here are a few common mistakes to avoid:

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Problem 1

Can you help Sarah find the area of a square box if its side length is given as √487?

Okay, lets begin

The area of the square is approximately 487 square units.

Explanation

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

Given the side length is √487, the area is √487 × √487 = 487 square units.

Well explained 👍

Problem 2

A square-shaped garden measures 487 square feet. If each side is √487, what will be the square feet of half of the garden?

Okay, lets begin

243.5 square feet.

Explanation

To find half the area of the garden, divide the total area by 2: 487 ÷ 2 = 243.5 square feet.

Well explained 👍

Problem 3

Calculate √487 × 5.

Okay, lets begin

Approximately 110.34.

Explanation

First, find the square root of 487, which is approximately 22.068.

Then, multiply this by 5: 22.068 × 5 ≈ 110.34.

Well explained 👍

Problem 4

What will be the square root of (484 + 3)?

Okay, lets begin

The square root is approximately 22.07.

Explanation

To find the square root, first calculate the sum: 484 + 3 = 487.

The square root of 487 is approximately 22.07.

Well explained 👍

Problem 5

Find the perimeter of a rectangle if its length ‘l’ is √487 units and the width ‘w’ is 10 units.

Okay, lets begin

The perimeter of the rectangle is approximately 64.136 units.

Explanation

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

Perimeter = 2 × (√487 + 10) ≈ 2 × (22.068 + 10) = 2 × 32.068 ≈ 64.136 units.

Well explained 👍

FAQ on Square Root of 487

1.What is √487 in its simplest form?

Since 487 is a prime number, √487 cannot be simplified and remains as √487.

2.Calculate the square of 487.

The square of 487 is 487 × 487 = 237,169.

3.Is 487 a prime number?

Yes, 487 is a prime number because it has no factors other than 1 and itself.

4.What are the factors of 487?

The factors of 487 are 1 and 487, as it is a prime number.

5.487 is divisible by?

487 is only divisible by 1 and 487.

Important Glossaries for the Square Root of 487

  • Square root: A square root is the inverse of squaring a number. Example: 4^2 = 16, and the inverse is the square root, √16 = 4.
     
  • Irrational number: A number that cannot be expressed as a simple fraction (p/q, where p and q are integers and q ≠ 0), such as √487.
     
  • Prime number: A number greater than 1 with no divisors other than 1 and itself, such as 487.
     
  • Decimal: A number that includes a fractional part, such as 22.068.
     
  • Long division method: A technique for finding the square root of non-perfect squares by dividing the number into smaller, more manageable parts.

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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.

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: He loves to play the quiz with kids through algebra to make kids love it.