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Last updated on May 26th, 2025

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Square Root of 514

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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 the field of vehicle design, finance, etc. Here, we will discuss the square root of 514.

Square Root of 514 for Indian Students
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What is the Square Root of 514?

The square root is the inverse of the square of the number. 514 is not a perfect square. The square root of 514 is expressed in both radical and exponential form. In the radical form, it is expressed as √514, whereas (514)(1/2) in the exponential form. √514 ≈ 22.67157, 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.

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Finding the Square Root of 514

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: -

 

  1. Prime factorization method 
  2. Long division method 
  3. Approximation method
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Square Root of 514 by Prime Factorization Method

The product of prime factors is the prime factorization of a number. Now let us look at how 514 is broken down into its prime factors.

 

Step 1: Finding the prime factors of 514 Breaking it down, we get 2 x 257.

 

Step 2: Now we found the prime factors of 514.

 

Since 514 is not a perfect square, calculating its square root using prime factorization isn't straightforward.

Professor Greenline from BrightChamps

Square Root of 514 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 514, we need to group it as 14 and 5.

 

Step 2: Now we need to find n whose square is closest to 5 without exceeding it. We can say n as ‘2’ because 2 x 2 = 4, which is less than or equal to 5. Now the quotient is 2. After subtracting 4 from 5, the remainder is 1.

 

Step 3: Bring down 14, making the new dividend 114. Add the old divisor with the same number 2 + 2 = 4, which will be our new divisor.

 

Step 4: Find the value of n such that 4n x n ≤ 114. Let n = 2, so 42 x 2 = 84.

 

Step 5: Subtract 84 from 114, the difference is 30, and the quotient is 22.

 

Step 6: 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 3000.

 

Step 7: The next step is finding the new divisor. Try n = 7, as 447 x 7 = 3129, which exceeds 3000. Try n = 6, as 446 x 6 = 2676.

 

Step 8: Subtracting 2676 from 3000 gives 324.

 

Step 9: The quotient is approximately 22.67. Continue the process to obtain more decimal places if necessary.

 

So the square root of √514 is approximately 22.67157.

Professor Greenline from BrightChamps

Square Root of 514 by Approximation Method

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 514 using the approximation method.

 

Step 1: Now we have to find the closest perfect squares around √514. The smallest perfect square less than 514 is 484 (222), and the largest perfect square greater than 514 is 529 (232). √514 falls somewhere between 22 and 23.

 

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (514 - 484) / (529 - 484) = 30 / 45 = 0.6667.

 

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 + 0.6667 ≈ 22.67.

 

Therefore, the square root of 514 is approximately 22.67157.

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Common Mistakes and How to Avoid Them in the Square Root of 514

Students do 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 those mistakes in detail.

Mistake 1

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Forgetting about the negative square root

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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 ≈ 7.071, there is also -7.071 which should not be forgotten.

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Square Root of 514 Examples

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Max, the Girl Character from BrightChamps

Problem 1

Can you help Max find the area of a square box if its side length is given as √514?

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The area of the square is approximately 514 square units.

Explanation

The area of the square = side2.

The side length is given as √514.

Area of the square = side2 = √514 x √514 = 514.

Therefore, the area of the square box is approximately 514 square units.

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Max, the Girl Character from BrightChamps

Problem 2

A square-shaped field measures 514 square feet; if each of the sides is √514, what will be the square feet of half of the field?

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257 square feet

Explanation

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

Dividing 514 by 2 = we get 257.

So, half of the field measures 257 square feet.

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Max, the Girl Character from BrightChamps

Problem 3

Calculate √514 x 5.

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Approximately 113.35785

Explanation

The first step is to find the square root of 514, which is approximately 22.67157.

The second step is to multiply 22.67157 by 5. So, 22.67157 x 5 ≈ 113.35785.

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Max, the Girl Character from BrightChamps

Problem 4

What will be the square root of (484 + 30)?

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The square root is approximately 22.67.

Explanation

To find the square root,

we need to find the sum of (484 + 30). 484 + 30 = 514, and then √514 ≈ 22.67157.

Therefore, the square root of (484 + 30) is approximately 22.67.

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Max, the Girl Character from BrightChamps

Problem 5

Find the perimeter of the rectangle if its length ‘l’ is √514 units and the width ‘w’ is 38 units.

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The perimeter of the rectangle is approximately 121.34314 units.

Explanation

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

Perimeter = 2 × (√514 + 38)

= 2 × (22.67157 + 38) ≈ 2 × 60.67157 ≈ 121.34314 units.

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FAQ on Square Root of 514

1.What is √514 in its simplest form?

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2.Mention the factors of 514.

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3.Calculate the square of 514.

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4.Is 514 a prime number?

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5.514 is divisible by?

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6.How does learning Algebra help students in India make better decisions in daily life?

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7.How can cultural or local activities in India support learning Algebra topics such as Square Root of 514?

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8.How do technology and digital tools in India support learning Algebra and Square Root of 514?

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9.Does learning Algebra support future career opportunities for students in India?

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Professor Greenline from BrightChamps

Important Glossaries for the Square Root of 514

  • Square root: A square root is the inverse of a square. 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.

 

  • Long Division Method: A step-by-step method used to find square roots of non-perfect squares by dividing the number into smaller parts.

 

  • Approximation Method: A technique used to estimate the square root of a number by finding the nearest perfect squares and using a proportional relationship.

 

  • Decimal: A number that includes a whole number and a fractional part separated by a decimal point, such as 22.67.
Professor Greenline from BrightChamps

About BrightChamps in India

At BrightChamps, we see algebra as more than just symbols—it opens doors to endless opportunities! Our mission is to help children all over India develop vital math skills, focusing today on the Square Root of 514 with special attention to understanding square roots—in a way that’s engaging, lively, and easy to follow. Whether your child is calculating the speed of a passing train, keeping scores during a cricket match, or managing pocket money for the latest gadgets, mastering algebra gives them the confidence needed for everyday situations. Our interactive lessons keep learning simple and fun. As kids in India have varied learning styles, we personalize our approach to match each child. From the busy markets of Mumbai to Delhi’s vibrant streets, BrightChamps brings math to life, making it relatable and exciting throughout India. Let’s make square roots a joyful part of every child’s math journey!
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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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Fun Fact

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

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