104 LearnersLast updated on May 26th, 2025
Square Root of 825
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 such as vehicle design, finance, etc. Here, we will discuss the square root of 825.
What is the Square Root of 825?
The square root is the inverse of the square of the number. 825 is not a perfect square. The square root of 825 is expressed in both radical and exponential form.
In the radical form, it is expressed as √825, whereas (825)(1/2) is in exponential form. √825 = 28.72281, 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 825
The prime factorization method is used for perfect square numbers. However, the prime factorization method is not used for non-perfect square numbers, where long-division and approximation methods are used. Let us now learn the following methods:
- Prime factorization method
- Long division method
- Approximation method
Square Root of 825 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 825 is broken down into its prime factors.
Step 1: Finding the prime factors of 825
Breaking it down, we get 3 x 5 x 5 x 11: 31 x 52 x 111
Step 2: Now we found out the prime factors of 825. The second step is to make pairs of those prime factors. Since 825 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating 825 using prime factorization is impossible.
Square Root of 825 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 825, we need to group it as 25 and 8.
Step 2: Now we need to find n whose square is 8. We can say n as ‘2’ because 2 x 2 is lesser than or equal to 8. Now the quotient is 2; after subtracting 4 from 8, the remainder is 4.
Step 3: Now let us bring down 25, which is the new dividend. Add the old divisor with the same number: 2 + 2, we get 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, we need to find the value of n.
Step 5: The next step is finding 4n x n ≤ 425. Let us consider n as 7, now 47 x 7 = 329.
Step 6: Subtract 425 from 329; the difference is 96, and the quotient is 27.
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 9600.
Step 8: Now we need to find the new divisor that is 574 because 574 x 7 = 4018.
Step 9: Subtracting 4018 from 9600, we get the result 5582.
Step 10: Now the quotient is 28.7.
Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue till the remainder is zero.
So the square root of √825 is approximately 28.72.
Square Root of 825 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 825 using the approximation method.
Step 1: Now we have to find the closest perfect square of √825. The smallest perfect square of 825 is 784, and the largest perfect square of 825 is 841. √825 falls somewhere between 28 and 29.
Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (825 - 784) ÷ (841-784) = 0.717.
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 28 + 0.717 = 28.717, so the square root of 825 is approximately 28.717.
Common Mistakes and How to Avoid Them in the Square Root of 825
Students do make mistakes while finding the square root, like forgetting about the negative square root or skipping long division methods. 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.
Square Root of 825 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √825?
The area of the square is 825 square units.
Explanation
The area of the square = side².
The side length is given as √825.
Area of the square = side² = √825 x √825 = 825.
Therefore, the area of the square box is 825 square units.
Problem 2
A square-shaped building measuring 825 square feet is built; if each of the sides is √825, what will be the square feet of half of the building?
412.5 square feet
Explanation
We can just divide the given area by 2 as the building is square-shaped.
Dividing 825 by 2, we get 412.5.
So half of the building measures 412.5 square feet.
Problem 3
Calculate √825 x 5.
143.64
Explanation
The first step is to find the square root of 825, which is approximately 28.72.
The second step is to multiply 28.72 with 5.
So 28.72 x 5 = 143.64.
Problem 4
What will be the square root of (825 + 25)?
The square root is 29.
Explanation
To find the square root, we need to find the sum of (825 + 25). 825 + 25 = 850, and then √850 = 29.
Therefore, the square root of (825 + 25) is approximately ±29.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √825 units and the width ‘w’ is 20 units.
We find the perimeter of the rectangle as approximately 97.44 units.
Explanation
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√825 + 20) = 2 × (28.72 + 20) = 2 × 48.72 = 97.44 units.
FAQ on Square Root of 825
1.What is √825 in its simplest form?
2.Mention the factors of 825.
3.Calculate the square of 825.
4.Is 825 a prime number?
5.825 is divisible by?
Important Glossaries for the Square Root of 825
Square root: A square root is the inverse of a square. Example: 42 = 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, it is always the positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as a principal square root.
Dividend: The number that is being divided in a division operation.
Quotient: The result obtained by dividing one number by another.
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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.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
