121 LearnersLast updated on May 26th, 2025
Square Root of 865
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 vehicle design, finance, etc. Here, we will discuss the square root of 865.
What is the Square Root of 865?
The square root is the inverse of the square of the number. 865 is not a perfect square. The square root of 865 is expressed in both radical and exponential form. In radical form, it is expressed as √865, whereas (865)^(1/2) is the exponential form. The square root of 865 is approximately 29.409, 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 865
The prime factorization method is used for perfect square numbers. However, the prime factorization method is not used for non-perfect square numbers, where 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 865 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 865 is broken down into its prime factors.
Step 1: Finding the prime factors of 865
Breaking it down, we get 5 x 173. Both 5 and 173 are prime numbers, so the prime factorization of 865 is 5^1 x 173^1.
Step 2: For non-perfect squares, we cannot group the digits in pairs. Therefore, calculating √865 using prime factorization is not straightforward.
Square Root of 865 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 865, we need to group it as 65 and 8.
Step 2: Now we need to find n whose square is less than or equal to 8. We can say n is 2 because 2 × 2 = 4, which is less than 8. Now the quotient is 2, and after subtracting 4 from 8, the remainder is 4.
Step 3: Now let us bring down 65, which makes the new dividend 465. Add the old divisor with the same number, 2 + 2, to get 4, which will be our new divisor.
Step 4: The new divisor is 4n. We need to find n such that 4n × n ≤ 465. Let n be 9; then, 49 × 9 = 441.
Step 5: Subtract 441 from 465; the difference is 24, and the quotient is 29.
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 2400.
Step 7: Now we need to find the new divisor, which is 8 because 298 × 8 = 2384.
Step 8: Subtracting 2384 from 2400, we get the result 16.
Step 9: Now the quotient is 29.4.
Step 10: Continue doing these steps until we get two numbers after the decimal point or until the remainder is zero.
So the square root of √865 is approximately 29.41.
Square Root of 865 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 865 using the approximation method.
Step 1: Now we have to find the closest perfect squares of √865. The smallest perfect square less than 865 is 841, and the largest perfect square greater than 865 is 900. Thus, √865 falls somewhere between 29 and 30.
Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Plugging in the numbers: (865 - 841) ÷ (900 - 841) = 24 ÷ 59 ≈ 0.41
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 29 + 0.41 = 29.41. Therefore, the square root of 865 is approximately 29.41.
Common Mistakes and How to Avoid Them in the Square Root of 865
Students often 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 mistakes that students tend to make 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, we typically consider only the positive square root, as it is often the required one.
For example, √865 ≈ 29.41, but there is also -29.41, which should not be forgotten.
Mistake 2
Not adding the square root symbol in proper places
Incorrectly using the square root symbol is a common mistake. Teach students to simplify the numbers inside the square root before moving on.
For example, √(12 + 24) ≠ √12 + √24. The correct way is √(12 + 24) = √36.
Mistake 3
Not finding the correct value of a non-perfect square
Students should be taught to simplify numbers inside the square root to avoid mistakes in identifying the approximate value.
For example, √50 ≠ 7; the correct answer is √50 ≈ 7.071.
Mistake 4
Confusing the square root symbol with the cube root
Students need to understand the difference between cube roots and square roots to avoid confusion.
For example, √50 and ∛50 are different.
Mistake 5
Making mistakes in long division
Finding a square root using the long division method involves many steps. Students might skip a step accidentally, leading to a wrong answer. This can happen in subtraction or elsewhere.
Square Root of 865 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √865?
The area of the square is approximately 865 square units.
Explanation
The area of the square = side^2.
The side length is given as √865.
Area of the square = side^2 = √865 × √865 = 865.
Therefore, the area of the square box is approximately 865 square units.
Problem 2
A square-shaped building measuring 865 square feet is built; if each of the sides is √865, what will be the square feet of half of the building?
432.5 square feet
Explanation
We can divide the given area by 2 as the building is square-shaped.
Dividing 865 by 2 = 432.5.
So half of the building measures 432.5 square feet.
Problem 3
Calculate √865 × 5.
147.045
Explanation
First, find the square root of 865, which is approximately 29.409.
Then multiply 29.409 by 5. So, 29.409 × 5 ≈ 147.045.
Problem 4
What will be the square root of (865 + 35)?
The square root is approximately 30.
Explanation
To find the square root, we need to find the sum of (865 + 35). 865 + 35 = 900, and √900 = 30.
Therefore, the square root of (865 + 35) is ±30.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √865 units and the width ‘w’ is 40 units.
We find the perimeter of the rectangle as approximately 138.82 units.
Explanation
Perimeter of the rectangle = 2 × (length + width)
Perimeter = 2 × (√865 + 40) = 2 × (29.409 + 40) = 2 × 69.409 ≈ 138.82 units.
FAQ on Square Root of 865
1.What is √865 in its simplest form?
2.Mention the factors of 865.
3.Calculate the square of 865.
4.Is 865 a prime number?
5.865 is divisible by?
6.How does learning Algebra help students in Indonesia make better decisions in daily life?
7.How can cultural or local activities in Indonesia support learning Algebra topics such as Square Root of 865?
8.How do technology and digital tools in Indonesia support learning Algebra and Square Root of 865?
9.Does learning Algebra support future career opportunities for students in Indonesia?
Important Glossaries for the Square Root of 865
- Square root: A square root is the inverse of a square. Example: 4^2 = 16, and the inverse square is the square root, which is √16 = 4.
- Irrational number: An irrational number cannot be written in the form of p/q, where q is not equal to zero, and p and q are integers.
- Principal square root: A number has both positive and negative square roots; however, the positive square root is often more prominent in real-world applications. Hence, it is known as the principal square root.
- Approximation method: A technique used to find an approximate value of a square root when it cannot be determined exactly.
- Long division method: A step-by-step process used to find the square root of a non-perfect square number accurately.
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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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