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 the fields of engineering, finance, etc. Here, we will discuss the square root of 3104.
The square root is the inverse of the square of the number. 3104 is not a perfect square. The square root of 3104 is expressed in both radical and exponential form. In radical form, it is expressed as √3104, whereas in exponential form it is (3104)^(1/2). √3104 ≈ 55.689, 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 long-division and approximation methods 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 3104 is broken down into its prime factors.
Step 1: Finding the prime factors of 3104. Breaking it down, we get 2 x 2 x 2 x 2 x 2 x 97 = 2^5 x 97.
Step 2: Now we found out the prime factors of 3104. The second step is to make pairs of those prime factors. Since 3104 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating √3104 using prime factorization is not straightforward.
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 3104, we need to group it as 04 and 31.
Step 2: Now we need to find n whose square is ≤ 31. We can say n is ‘5’ because 5 x 5 = 25, which is less than 31. Now the quotient is 5, and subtracting gives a remainder of 6.
Step 3: Bring down 04, making it the new dividend of 604. Add the old divisor with the same number: 5 + 5 = 10, which will be our new divisor.
Step 4: The new divisor will be 10n. Now we need to find n such that 10n x n ≤ 604. Let’s consider n as 5, so 105 x 5 = 525.
Step 5: Subtract 525 from 604, the difference is 79, and the quotient is 55.
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 7900.
Step 7: Now we need to find the new divisor. Let’s choose 7 because 1117 x 7 = 7819.
Step 8: Subtracting 7819 from 7900 gives a remainder of 81.
Step 9: Now the quotient is 55.7. Continue doing these steps until we get more decimal places or the remainder is zero. So the square root of √3104 is approximately 55.689.
The approximation method is another method for finding the 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 3104 using the approximation method.
Step 1: Now we have to find the closest perfect squares of √3104. The smallest perfect square less than 3104 is 3025, and the largest perfect square greater than 3104 is 3136. √3104 falls somewhere between 55 and 56.
Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (3104 - 3025) / (3136 - 3025) = 79 / 111 ≈ 0.7126. Adding this decimal to the lower integer: 55 + 0.7126 = 55.7126. Therefore, the square root of 3104 is approximately 55.7126.
Students do make mistakes while finding the square root, such as forgetting about the negative square root. Skipping long division steps, etc. Now let us look at a few of those mistakes that students tend to make in detail.
Can you help Max find the area of a square box if its side length is given as √3104?
The area of the square is approximately 9647.521 square units.
The area of the square = side².
The side length is given as √3104.
Area of the square = side² = √3104 × √3104 = 3104.
Therefore, the area of the square box is approximately 9647.521 square units.
A square-shaped building measuring 3104 square feet is built; if each of the sides is √3104, what will be the square feet of half of the building?
1552 square feet
We can just divide the given area by 2 as the building is square-shaped.
Dividing 3104 by 2 = we get 1552.
So half of the building measures 1552 square feet.
Calculate √3104 x 5.
278.445
The first step is to find the square root of 3104, which is approximately 55.689.
The second step is to multiply 55.689 by 5. So 55.689 × 5 ≈ 278.445.
What will be the square root of (3100 + 4)?
The square root is 56.
To find the square root, we need to find the sum of (3100 + 4).
3100 + 4 = 3104, and then √3104 ≈ 55.689.
Therefore, the square root of (3100 + 4) is approximately ±55.689.
Find the perimeter of the rectangle if its length ‘l’ is √3104 units and the width ‘w’ is 20 units.
We find the perimeter of the rectangle as approximately 151.378 units.
Perimeter of the rectangle = 2 × (length + width). Perimeter = 2 × (√3104 + 20) = 2 × (55.689 + 20) = 2 × 75.689 ≈ 151.378 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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