124 LearnersLast updated on May 26th, 2025
Square Root of 859
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 859.
What is the Square Root of 859?
The square root is the inverse of the square of the number. 859 is not a perfect square. The square root of 859 is expressed in both radical and exponential form. In the radical form, it is expressed as √859, whereas (859)^(1/2) in the exponential form. √859 ≈ 29.3087, 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 859
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 859 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 859 is broken down into its prime factors.
Step 1: Finding the prime factors of 859
Breaking it down, we get 859 = 7 × 7 × 17.
Step 2: Now we found the prime factors of 859. The second step is to make pairs of those prime factors. Since 859 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating 859 using prime factorization is impossible.
Square Root of 859 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 859, we need to group it as 59 and 8.
Step 2: Now we need to find n whose square is less than or equal to 8. We can say n as ‘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 bring down 59, 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 ≤ 459. Let us consider n as 9, now 49 × 9 = 441.
Step 6: Subtract 441 from 459; the difference is 18, and the quotient is 29.
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 1800.
Step 8: Now we need to find the new divisor, which is 586, because 586 × 3 = 1758.
Step 9: Subtracting 1758 from 1800, we get the result 42.
Step 10: Now the quotient is 29.3
Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose there is no decimal value; continue till the remainder is zero.
So the square root of √859 is approximately 29.31
Square Root of 859 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 859 using the approximation method.
Step 1: Now we have to find the closest perfect square of √859. The smallest perfect square less than 859 is 841, and the largest perfect square greater than 859 is 900. √859 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) Going by the formula (859 - 841) / (900 - 841) = 18/59 ≈ 0.31
Using the formula, we identified the decimal point of our square root. The next step is adding the integer value to the decimal number, which is 29 + 0.31 = 29.31, so the square root of 859 is approximately 29.31.
Common Mistakes and How to Avoid Them in the Square Root of 859
Students do make mistakes while finding the square root, like forgetting about the negative square root or skipping long division methods, etc. 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 will be taking only the positive square root, as it is the required one.
For example, √50 = 7.07, there is also -7.07, which should not be forgotten.
Mistake 2
Not adding square root symbol in proper places
Not using the square root 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 the wrong method of using roots. The right way of using roots 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 wrong; 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 cube root and 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, which results in a wrong answer at the end of the sum. This may either happen in subtracting or elsewhere.
Square Root of 859 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √859?
The area of the square is 859 square units.
Explanation
The area of the square = side^2.
The side length is given as √859.
Area of the square = side^2 = √859 × √859 = 859.
Therefore, the area of the square box is 859 square units.
Problem 2
A square-shaped building measuring 859 square feet is built; if each of the sides is √859, what will be the square feet of half of the building?
429.5 square feet
Explanation
We can just divide the given area by 2, as the building is square-shaped.
Dividing 859 by 2, we get 429.5.
So half of the building measures 429.5 square feet.
Problem 3
Calculate √859 × 5.
146.54
Explanation
The first step is to find the square root of 859, which is approximately 29.31.
The second step is to multiply 29.31 by 5. So 29.31 × 5 = 146.54.
Problem 4
What will be the square root of (859 + 1)?
The square root is 30.
Explanation
To find the square root, we need to find the sum of (859 + 1). 859 + 1 = 860.
The square root of 860 is approximately 29.31, rounded to the nearest whole number, is 30.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √859 units and the width ‘w’ is 38 units.
We find the perimeter of the rectangle as 134.62 units.
Explanation
Perimeter of the rectangle = 2 × (length + width)
Perimeter = 2 × (√859 + 38) = 2 × (29.31 + 38) = 2 × 67.31 = 134.62 units.
FAQ on Square Root of 859
1.What is √859 in its simplest form?
2.Mention the factors of 859.
3.Calculate the square of 859.
4.Is 859 a prime number?
5.859 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 859?
8.How do technology and digital tools in Indonesia support learning Algebra and Square Root of 859?
9.Does learning Algebra support future career opportunities for students in Indonesia?
Important Glossaries for the Square Root of 859
- 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.
- Prime factorization: Prime factorization is expressing a number as the product of its prime factors. Example: The prime factorization of 859 is 7 × 7 × 17.
- Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example: 7.86, 8.65, and 9.42 are decimals.
- Long division method: It is a method used to find the square root of non-perfect square numbers by dividing the number step by step to find the quotient.
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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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