125 LearnersLast updated on May 26th, 2025
Square Root of 869
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 various fields such as vehicle design, finance, etc. Here, we will discuss the square root of 869.
What is the Square Root of 869?
The square root is the inverse of the square of the number. 869 is not a perfect square. The square root of 869 is expressed in both radical and exponential form. In radical form, it is expressed as √869, whereas in exponential form, it is expressed as (869)^(1/2). √869 ≈ 29.4788, 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 869
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 869 by Prime Factorization Method
The product of prime factors is known as the prime factorization of a number. Now let us look at how 869 is broken down into its prime factors.
Step 1: Finding the prime factors of 869
Breaking it down, we get 869 = 11 x 79
Step 2: Now, we found out the prime factors of 869. The second step is to make pairs of those prime factors. Since 869 is not a perfect square, we cannot group the digits into pairs. Therefore, calculating 869 using prime factorization is not possible.
Square Root of 869 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 869, we need to group it as 69 and 8.
Step 2: Now, we need to find n whose square is ≤ 8. We can say n as ‘2’ because 2 x 2 = 4, which is lesser than or equal to 8. Now the quotient is 2, and after subtracting 4 from 8, the remainder is 4.
Step 3: Now, let us bring down 69, which is the new dividend. Add the old divisor with the same number: 2 + 2 = 4, which will be our new divisor.
Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 4n as the new divisor, and we need to find the value of n.
Step 5: The next step is finding 4n × n ≤ 469. Let us consider n as 1, now 41 x 1 = 41.
Step 6: Subtract 41 from 46, the difference is 5, and the quotient is 21.
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 500.
Step 8: Now, we need to find the new divisor that is 294 because 294 x 1 = 294.
Step 9: Subtracting 294 from 500, we get the result 206.
Step 10: Now, the quotient is 29.4.
Step 11: Continue doing these steps until we get two numbers after the decimal point. If there are no decimal values, continue until the remainder is zero.
So the square root of √869 ≈ 29.48
Square Root of 869 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 869 using the approximation method.
Step 1: Now, we have to find the closest perfect square of √869. The smallest perfect square less than 869 is 841, and the largest perfect square greater than 869 is 900. √869 falls somewhere between 29 and 30.
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 (869 - 841) / (900 - 841) ≈ 0.478
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 + 0.478 = 29.478 Thus, the square root of 869 is approximately 29.478.
Common Mistakes and How to Avoid Them in the Square Root of 869
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. 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 usually consider only the principal (positive) square root, as it is the required one.
For example: √50 ≈ 7.07, but there is also -7.07 which should not be forgotten.
Mistake 2
Not adding the square root symbol in proper places
Not using the square root symbol properly is another mistake that children make. To resolve this, we need to teach children that simplifying the numbers inside the square root is important and then move on to the second step.
For example: √(12 + 24) = √12 + √24 is incorrect. The correct method is √(12 + 24) = √36.
Mistake 3
Not finding the correct value of a non-perfect square
The first step that needs to be taught to children is to simplify the numbers inside the square root. This is important so that students do not make mistakes while identifying the approximate value.
For example: √50 ≈ 7 is incorrect; the correct answer is √50 ≈ 7.071.
Mistake 4
Confusing the square root symbol with the cube root
Students need to be taught the difference between the cube root and the square root constantly as this will help them avoid mistakes.
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. Due to this, many students might skip a step accidentally, resulting in a wrong answer at the end of the calculation. This may happen in subtracting or elsewhere.
Square Root of 869 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √869?
The area of the square is 869 square units.
Explanation
The area of the square = side^2.
The side length is given as √869.
Area of the square = side^2 = √869 x √869 = 869.
Therefore, the area of the square box is 869 square units.
Problem 2
A square-shaped building measuring 869 square feet is built. If each of the sides is √869, what will be the square feet of half of the building?
434.5 square feet
Explanation
We can just divide the given area by 2 as the building is square-shaped.
Dividing 869 by 2, we get 434.5.
So, half of the building measures 434.5 square feet.
Problem 3
Calculate √869 x 3.
88.4364
Explanation
The first step is to find the square root of 869, which is approximately 29.478.
The second step is to multiply 29.478 by 3.
So, 29.478 x 3 ≈ 88.4364.
Problem 4
What will be the square root of (869 + 31)?
The square root is approximately 30.
Explanation
To find the square root, we need to find the sum of (869 + 31). 869 + 31 = 900, and then √900 = 30.
Therefore, the square root of (869 + 31) is ±30.
Problem 5
Find the perimeter of a rectangle if its length ‘l’ is √869 units and the width ‘w’ is 20 units.
We find the perimeter of the rectangle as approximately 98.956 units.
Explanation
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√869 + 20) ≈ 2 × (29.478 + 20) ≈ 2 × 49.478 ≈ 98.956 units.
FAQ on Square Root of 869
1.What is √869 in its simplest form?
2.Mention the factors of 869.
3.Calculate the square of 869.
4.Is 869 a prime number?
5.869 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 869?
8.How do technology and digital tools in Indonesia support learning Algebra and Square Root of 869?
9.Does learning Algebra support future career opportunities for students in Indonesia?
Important Glossaries for the Square Root of 869
- Square root: A square root is the inverse of a square. Example: 4^2 = 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.
- Principal square root: A number has both positive and negative square roots; however, the positive square root has more prominence due to its uses in the real world. This is the reason it is also known as the principal square root.
- Prime factorization: Prime factorization is expressing a number as a product of its prime factors.
- Decimal point: A decimal point separates the whole number from the fractional part in a decimal number. For example, in 29.478, the decimal point separates 29 from 478.
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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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