120 LearnersLast updated on May 26th, 2025
Square Root of 866
If a number is multiplied by itself, the result is a square. The inverse of the square is a square root. The square root is used in fields like vehicle design, finance, etc. Here, we will discuss the square root of 866.
What is the Square Root of 866?
The square root is the inverse of the square of a number. 866 is not a perfect square. The square root of 866 is expressed in both radical and exponential form. In the radical form, it is expressed as √866, whereas in exponential form it is (866)^(1/2). √866 ≈ 29.4279, 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 866
The prime factorization method is used for perfect square numbers. However, for non-perfect square numbers, methods like 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 866 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 866 is broken down into its prime factors.
Step 1: Finding the prime factors of 866
Breaking it down, we get 2 x 433, where 433 is a prime number.
Step 2: We found the prime factors of 866. Since 866 is not a perfect square, the digits of the number can’t be grouped into pairs. Therefore, calculating 866 using prime factorization is impossible.
Square Root of 866 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, group the digits from right to left. In the case of 866, treat it as 86 and 6.
Step 2: Find the largest integer n such that n² is less than or equal to 86. The integer is 9 because 9² = 81.
Step 3: Subtract 81 from 86, leaving a remainder of 5. Bring down the next pair of digits (66), making the new dividend 566.
Step 4: Double the divisor (9) to get 18. Now, find a digit x such that 18x × x is less than or equal to 566. The number is 3 because 183 × 3 = 549.
Step 5: Subtract 549 from 566, leaving a remainder of 17.
Step 6: Since the remainder is less than the divisor, add a decimal point and bring down two zeros, making the new dividend 1700.
Step 7: Double the previous divisor (183) and append an appropriate digit to find the next part of the quotient. Continue this process to get more decimal places.
Following these steps, the square root of 866 is approximately 29.4279.
Square Root of 866 by Approximation Method
The 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 866 using the approximation method.
Step 1: Identify the closest perfect squares to 866. The smallest perfect square is 841 (29²) and the largest is 900 (30²). Hence, √866 is between 29 and 30.
Step 2: Use the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Applying the formula: (866 - 841) / (900 - 841) = 25 / 59 ≈ 0.4237 Adding this to the smaller perfect square root: 29 + 0.4237 = 29.4237 Thus, the square root of 866 is approximately 29.4237.
Common Mistakes and How to Avoid Them in the Square Root of 866
Students make mistakes while finding the square root, such as overlooking the negative square root or skipping steps in the long division method. Let us look at a few common mistakes in detail.
Mistake 1
Forgetting about the negative square root
It is important to remember that a number has both positive and negative square roots. However, typically only the positive square root is used in practical applications.
For example: √50 = 7.07, but there is also -7.07.
Mistake 2
Not adding the square root symbol in proper places
Incorrect use of the square root symbol is a common error. Ensure that students simplify the expression inside the square root before proceeding.
For example: √(12 + 24) = √36, not √12 + √24.
Mistake 3
Not finding the correct value of a non-perfect square
Students should be taught to simplify numbers inside the square root for accurate approximation.
For example: √50 ≈ 7.071, not 7.
Mistake 4
Confusing the square root symbol with the cube root
Students should distinguish between square roots and cube roots.
For example, √50 and ∛50 are different.
Mistake 5
Making mistakes in long division
Finding a square root using the long division method involves multiple steps. Skipping a step can lead to incorrect results. This may occur during subtraction or elsewhere in the process.
Square root of 866 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √866?
The area of the square is approximately 750.52 square units.
Explanation
The area of the square = side².
The side length is given as √866.
Area of the square = side² = √866 × √866 ≈ 29.4279 × 29.4279 ≈ 866.
Therefore, the area of the square box is approximately 866 square units.
Problem 2
A square-shaped building measuring 866 square feet is built; if each of the sides is √866, what will be the square feet of half of the building?
433 square feet
Explanation
We can just divide the given area by 2 as the building is square-shaped.
Dividing 866 by 2 = 433.
So half of the building measures 433 square feet.
Problem 3
Calculate √866 × 5.
Approximately 147.14
Explanation
The first step is to find the square root of 866, which is approximately 29.4279.
Multiply 29.4279 by 5.
So, 29.4279 × 5 ≈ 147.14.
Problem 4
What will be the square root of (850 + 16)?
The square root is 30.
Explanation
To find the square root, sum (850 + 16) = 866, and then √866 ≈ 29.4279.
Therefore, the square root of (850 + 16) is approximately ±29.4279.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √866 units and the width ‘w’ is 40 units.
The perimeter of the rectangle is approximately 138.86 units.
Explanation
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√866 + 40) ≈ 2 × (29.4279 + 40) ≈ 2 × 69.4279 ≈ 138.86 units.
FAQ on Square Root of 866
1.What is √866 in its simplest form?
2.Mention the factors of 866.
3.Calculate the square of 866.
4.Is 866 a prime number?
5.866 is divisible by?
6.How does learning Algebra help students in United States make better decisions in daily life?
7.How can cultural or local activities in United States support learning Algebra topics such as Square Root of 866?
8.How do technology and digital tools in United States support learning Algebra and Square Root of 866?
9.Does learning Algebra support future career opportunities for students in United States?
Important Glossaries for the Square Root of 866
- Square root: A square root is the inverse operation of squaring a number. Example: 4² = 16 and the inverse is the square root, √16 = 4.
- Irrational number: An irrational number cannot be expressed as a simple fraction. Its decimal representation is non-repeating and non-terminating.
- Principal square root: The non-negative root of a number. For example, the principal square root of 25 is 5.
- Prime number: A number that has no divisors other than 1 and itself. For example, 433 is a prime number.
- Long division method: A technique used to divide numbers and can be adapted to find square roots of non-perfect squares.
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Jaskaran Singh Saluja
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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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