146 LearnersLast updated on May 26th, 2025
Square Root of 521
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 architecture, finance, and engineering. Here, we will discuss the square root of 521.
What is the Square Root of 521?
The square root is the inverse of the square of the number. 521 is not a perfect square. The square root of 521 is expressed in both radical and exponential form. In the radical form, it is expressed as √521, whereas (521)(1/2) in the exponential form. √521 ≈ 22.82542, 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 521
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 521 by Prime Factorization Method
The prime factorization of a number is the product of its prime factors. Now let us look at how 521 is broken down into its prime factors.
Step 1: Finding the prime factors of 521 521 is a prime number itself and cannot be broken down further into smaller prime factors.
Step 2: Since 521 is not a perfect square, calculating its square root using prime factorization is not feasible.
Square Root of 521 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 numbers from right to left. In the case of 521, we group it as 21 and 5.
Step 2: Now we need to find n whose square is less than or equal to 5. We can say n as ‘2’ because 2 × 2 = 4, which is less than 5. Now the quotient is 2, and after subtracting 4 from 5, the remainder is 1.
Step 3: Now let us bring down 21, which is the new dividend. Add the old divisor with the same number 2 + 2 to get 4, which will be our new divisor.
Step 4: The new divisor will be 4n. We need to find the value of n.
Step 5: The next step is finding 4n × n ≤ 121. Let us consider n as 2, now 42 × 2 = 84.
Step 6: Subtract 84 from 121, the difference is 37, and the quotient is 22.
Step 7: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 3700.
Step 8: Now we need to find the new divisor, which is 229 because 229 × 9 = 2061.
Step 9: Subtracting 2061 from 3700, we get the result 1639.
Step 10: Now the quotient is 22.8.
Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose there are no decimal values, continue till the remainder is zero.
So the approximate square root of √521 is 22.82.
Square Root of 521 by Approximation Method
The approximation method is another method for finding square roots, and 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 521 using the approximation method.
Step 1: Now we have to find the closest perfect squares surrounding √521. The smallest perfect square less than 521 is 484, and the largest perfect square greater than 521 is 529. √521 falls somewhere between 22 and 23.
Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).
Using the formula (521 - 484) ÷ (529 - 484) = 0.82 Using the formula, we identified the decimal point of our square root.
The next step is adding the value we got initially to the decimal number, which is 22 + 0.82 = 22.82, so the square root of 521 is approximately 22.82.
Common Mistakes and How to Avoid Them in the Square Root of 521
Students do make mistakes while finding the square root, such as forgetting about the negative square root or skipping long division methods, etc. Now let us look at a few of those mistakes that students tend to make in detail.
Mistake 1
Forgetting about the negative square root
It is important to make students aware that a number does have both positive and negative square roots. However, we will be taking only the positive square root, as it is the required one.
For example: √50 = 7.07, there is also -7.07 which should not be forgotten.
Square Root of 521 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √521?
The area of the square is approximately 521 square units.
Explanation
The area of the square = side².
The side length is given as √521.
Area of the square = side² = √521 × √521 = 521.
Therefore, the area of the square box is approximately 521 square units.
Problem 2
A square-shaped building measuring 521 square feet is built; if each of the sides is √521, what will be the square feet of half of the building?
260.5 square feet
Explanation
We can just divide the given area by 2 as the building is square-shaped.
Dividing 521 by 2 = we get 260.5
So half of the building measures 260.5 square feet.
Problem 3
Calculate √521 × 5.
114.127
Explanation
The first step is to find the square root of 521, which is approximately 22.82.
The second step is to multiply 22.82 by 5.
So, 22.82 × 5 ≈ 114.127.
Problem 4
What will be the square root of (500 + 21)?
The square root is approximately 22.82.
Explanation
To find the square root, we need to find the sum of (500 + 21). 500 + 21 = 521, and then √521 ≈ 22.82.
Therefore, the square root of (500 + 21) is approximately ±22.82.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √521 units and the width ‘w’ is 38 units.
The perimeter of the rectangle is approximately 121.64 units.
Explanation
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√521 + 38)
= 2 × (22.82 + 38)
= 2 × 60.82
= 121.64 units.
FAQ on Square Root of 521
1.What is √521 in its simplest form?
2.Mention the factors of 521.
3.Calculate the square of 521.
4.Is 521 a prime number?
5.521 is divisible by?
6.How does learning Algebra help students in Thailand make better decisions in daily life?
7.How can cultural or local activities in Thailand support learning Algebra topics such as Square Root of 521?
8.How do technology and digital tools in Thailand support learning Algebra and Square Root of 521?
9.Does learning Algebra support future career opportunities for students in Thailand?
Important Glossaries for the Square Root of 521
- Square root: A square root is the inverse of squaring a number. For example, 4² = 16 and the inverse of the square is the square root, that is √16 = 4.
- Irrational number: An irrational number is a number that cannot be written in the form of p/q, where q is not equal to zero and p and q are integers.
- Prime number: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
- Decimal: A decimal is a number that has a whole number and a fraction in a single number, for example, 7.86, 8.65, and 9.42.
- Approximation method: The approximation method is a technique used to find an estimated value of the square root of a number, especially when it is not a perfect square.
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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.
