123 LearnersLast updated on May 26th, 2025
Square Root of 857
If a number is multiplied by itself, the result is a square. The inverse operation is finding the square root. The square root is used in various fields like vehicle design, finance, etc. Here, we will discuss the square root of 857.
What is the Square Root of 857?
The square root is the inverse of the square of the number. 857 is not a perfect square. The square root of 857 is expressed in both radical and exponential form. In radical form, it is expressed as √857, whereas (857)^(1/2) in exponential form. √857 ≈ 29.273, 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 857
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 857 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 857 is broken down into its prime factors.
Step 1: Finding the prime factors of 857. 857 is a prime number, so it cannot be broken down into other prime factors.
Step 2: Since 857 is not a perfect square, calculating √857 using prime factorization is not possible.
Square Root of 857 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 group the numbers from right to left. In the case of 857, we need to group it as 57 and 8.
Step 2: Now find n whose square is ≤ 8. We can say n is 2 because 2 × 2 = 4, which is less than 8. Now the quotient is 2, and subtracting 4 from 8 leaves a remainder of 4.
Step 3: Bring down 57, the new dividend becomes 457. Add the old divisor with the same number 2 + 2 = 4, which will be our new divisor prefix.
Step 4: The new divisor is 4n. Find n such that 4n × n is ≤ 457. Let n be 9, thus 49 × 9 = 441.
Step 5: Subtract 441 from 457; the difference is 16, and the quotient is 29.
Step 6: Since the dividend is less than the divisor, we add a decimal point and bring down two zeroes, making the new dividend 1600.
Step 7: Find the new divisor 58n, where n is such that 58n × n ≤ 1600. Try n = 2, giving 582 × 2 = 1164.
Step 8: Subtract 1164 from 1600 to get 436.
Step 9: Continue this process until the desired precision is achieved.
So the square root of √857 ≈ 29.273.
Square Root of 857 by Approximation Method
The approximation method is another way to find square roots. It is an easy method to find the square root of a given number. Let us learn how to find the square root of 857 using the approximation method.
Step 1: Find the closest perfect squares to √857.
The smallest perfect square less than 857 is 841 (29^2) and the largest is 900 (30^2). So √857 is between 29 and 30.
Step 2: Use the formula: (Given number - smallest perfect square) / (Larger perfect square - smallest perfect square). (857 - 841) / (900 - 841) = 16 / 59 ≈ 0.271.
Step 3: Add this decimal to the smaller integer: 29 + 0.271 = 29.271.
So the square root of 857 is approximately 29.273.
Common Mistakes and How to Avoid Them in the Square Root of 857
Students make mistakes while finding square roots, such as forgetting the negative square root, skipping steps in long division, etc. Let's look at a few common mistakes and how to avoid them.
Mistake 1
Forgetting about the negative square root
It is important to note that a number has both positive and negative square roots. However, we generally consider only the positive square root as the principal square root.
For example: √857 ≈ 29.273, but there is also -29.273.
Mistake 2
Not adding the square root symbol in proper places
Using the square root symbol correctly is crucial. Students must simplify numbers inside the square root before proceeding.
For example, √(16 + 9) ≠ √16 + √9. Correct use is √(16 + 9) = √25 = 5.
Mistake 3
Not finding the correct value of a non-perfect square
Students should simplify numbers inside the square root to find correct approximations.
For example, √50 is not 7; it is approximately 7.071.
Mistake 4
Confusing the square root symbol with the cube root
Students should understand the difference between cube root and square root.
For example, √8 = 2.828, but ∛8 = 2.
Mistake 5
Making mistakes in long division
Finding square roots using long division involves many steps. Skipping any step can result in wrong answers, whether in subtraction or elsewhere.
Square Root of 857 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √857?
The area of the square is approximately 857 square units.
Explanation
The area of the square = side^2.
The side length is given as √857.
Area of the square = √857 × √857 = 857.
Therefore, the area of the square box is approximately 857 square units.
Problem 2
A square-shaped building measuring 857 square feet is built; if each of the sides is √857, what will be the square feet of half of the building?
428.5 square feet
Explanation
Dividing the building's area by 2, as it is square-shaped: 857 / 2 = 428.5.
So half of the building measures 428.5 square feet.
Problem 3
Calculate √857 × 5.
Approximately 146.365.
Explanation
First, find the square root of 857, which is approximately 29.273.
Multiply 29.273 by 5 to get: 29.273 × 5 ≈ 146.365.
Problem 4
What will be the square root of (857 + 3)?
The square root is approximately 29.301.
Explanation
To find the square root, first find the sum of (857 + 3): 857 + 3 = 860. √860 ≈ 29.301.
Therefore, the square root of (857 + 3) is approximately ±29.301.
Problem 5
Find the perimeter of a rectangle if its length 'l' is √857 units and the width 'w' is 20 units.
The perimeter of the rectangle is approximately 98.546 units.
Explanation
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√857 + 20) = 2 × (29.273 + 20) = 2 × 49.273 ≈ 98.546 units.
FAQ on Square Root of 857
1.What is √857 in its simplest form?
2.Is 857 a prime number?
3.What is the square of 857?
4.What are the factors of 857?
5.How is 857 divisible?
6.How does learning Algebra help students in Oman make better decisions in daily life?
7.How can cultural or local activities in Oman support learning Algebra topics such as Square Root of 857?
8.How do technology and digital tools in Oman support learning Algebra and Square Root of 857?
9.Does learning Algebra support future career opportunities for students in Oman?
Important Glossaries for the Square Root of 857
- Square root: The square root is the inverse operation of squaring a number. For example, 5^2 = 25, and the inverse is √25 = 5.
- Prime number: A prime number is a natural number greater than 1 that is not divisible by any other numbers except for 1 and itself.
- Irrational number: An irrational number cannot be expressed as a simple fraction or ratio of two integers.
- Long division method: A method used to find the square root of non-perfect squares by dividing the number into groups and performing iterative division steps.
- Radical: The symbol (√) used to denote the square root or nth root of a number.
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About BrightChamps in Oman
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.
