Last updated on May 26th, 2025
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 various fields such as vehicle design, finance, etc. Here, we will discuss the square root of 8725.
The square root is the inverse of the square of the number. 8725 is not a perfect square. The square root of 8725 is expressed in both radical and exponential form. In the radical form, it is expressed as √8725, whereas (8725)^(1/2) in the exponential form. √8725 ≈ 93.3516, 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.
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:
The product of prime factors is the prime factorization of a number. Now let us look at how 8725 is broken down into its prime factors:
Step 1: Finding the prime factors of 8725.
Breaking it down, we get 5 × 5 × 349: 5^2 × 349.
Step 2: Now we found out the prime factors of 8725. The second step is to make pairs of those prime factors. Since 8725 is not a perfect square, it is impossible to pair all digits. Therefore, calculating √8725 using prime factorization is not feasible.
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 8725, we need to group it as 87 and 25.
Step 2: Now we need to find n whose square is less than or equal to 87. We can say n is 9 because 9 × 9 = 81, which is less than 87. Now the quotient is 9, and after subtracting 81 from 87, the remainder is 6.
Step 3: Bring down 25, making the new dividend 625. Add the old divisor (9) to itself to get 18 as part of the new divisor.
Step 4: Determine n such that 18n × n ≤ 625. Let n be 3, then 183 × 3 = 549.
Step 5: Subtract 549 from 625, resulting in a remainder of 76. The quotient so far is 93.
Step 6: Since the remainder is still there, add a decimal point to the quotient and bring down 00, making the new dividend 7600.
Step 7: Determine the next digit for the quotient. The new divisor becomes 1860. Choose n such that 1860n × n ≤ 7600.
Step 8: After determining n and performing the steps, continue until the desired precision is achieved.
So the square root of √8725 is approximately 93.35.
The approximation method is another method for finding square roots, and 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 8725 using the approximation method.
Step 1: Find the closest perfect squares around 8725. The nearest perfect squares are 8464 (92^2) and 8836 (94^2). √8725 falls between 92 and 94.
Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).
Using the formula (8725 - 8464) / (8836 - 8464) = 261 / 372 = 0.701. Add the decimal 0.701 to the lower square root estimate, 92 + 0.701 = 92.701. Rounding gives us approximately 93.35, so √8725 ≈ 93.35.
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 of these common mistakes in detail.
Can you help Max find the area of a square box if its side length is given as √8725?
The area of the square is approximately 76,116.75 square units.
The area of the square = side^2.
The side length is given as √8725.
Area of the square = side^2 = √8725 × √8725 = 8725.
Therefore, the area of the square box is approximately 76,116.75 square units.
A square-shaped building measuring 8725 square feet is built; if each of the sides is √8725, what will be the square feet of half of the building?
4362.5 square feet
We can divide the given area by 2 as the building is square-shaped.
Dividing 8725 by 2 gives us 4362.5.
So half of the building measures 4362.5 square feet.
Calculate √8725 × 5.
Approximately 466.758
The first step is to find the square root of 8725, which is approximately 93.35.
The second step is to multiply 93.35 by 5.
So 93.35 × 5 ≈ 466.758.
What will be the square root of (8725 + 25)?
The square root is approximately 94.
To find the square root, we need to find the sum of (8725 + 25). 8725 + 25 = 8750, and then √8750 ≈ 93.5.
Therefore, the square root of (8725 + 25) is approximately 93.5.
Find the perimeter of the rectangle if its length 'l' is √8725 units and the width 'w' is 50 units.
The perimeter of the rectangle is approximately 386.7 units.
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√8725 + 50) = 2 × (93.35 + 50) ≈ 2 × 143.35 = 286.7 units.
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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