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 field of vehicle design, finance, etc. Here, we will discuss the square root of 3840.
The square root is the inverse of the square of the number. 3840 is not a perfect square. The square root of 3840 is expressed in both radical and exponential form. In the radical form, it is expressed as √3840, whereas (3840)^(1/2) in the exponential form. √3840 ≈ 61.977, 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, for non-perfect squares, 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 3840 is broken down into its prime factors.
Step 1: Finding the prime factors of 3840 Breaking it down, we get 2 × 2 × 2 × 2 × 2 × 2 × 3 × 5: 2^6 × 3^1 × 5^1
Step 2: Now we found out the prime factors of 3840. The second step is to make pairs of those prime factors. Since 3840 is not a perfect square, the digits of the number can’t be fully grouped in pairs.
Therefore, calculating √3840 using prime factorization directly is not possible.
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 3840, we group it as 84 and 38.
Step 2: Now we need to find n whose square is less than or equal to 38. We can say n as ‘6’ because 6 × 6 = 36 is lesser than or equal to 38. Now the quotient is 6.
Step 3: Subtract 36 from 38, the remainder is 2, and bring down the next pair, which is 40, making it 240.
Step 4: The new divisor is 2 times the current quotient, which is 12. We need to find the value of n such that 12n × n ≤ 240.
Step 5: Consider n as 1, 121 × 1 = 121. Step 6: Subtract 121 from 240, the difference is 119, and the quotient is 61.
Step 7: Since the dividend is less than the divisor, add a decimal point and bring down two zeroes to the dividend, making it 11900.
Step 8: Find the new divisor that is 122 times n such that it fits into the new dividend.
Step 9: Continue the process until you get the desired precision.
The square root of √3840 ≈ 61.977.
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 3840 using the approximation method.
Step 1: Find the closest perfect squares around 3840. The smallest perfect square is 3600 (60^2) and the largest perfect square is 4096 (64^2). √3840 falls between 60 and 64.
Step 2: Apply the approximation formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) (3840 - 3600) ÷ (4096 - 3600) = 240 ÷ 496 ≈ 0.484 Step 3: Add this decimal to the smaller integer value: 60 + 0.484 = 60.484 So, the square root of 3840 is approximately 61.977 when calculated more precisely.
Students make mistakes while finding the square root, such as forgetting about the negative square root, and skipping key steps in methods. Let us look at a few of those mistakes in detail.
Can you help Max find the area of a square box if its side length is given as √3840?
The area of the square is approximately 3840 square units.
The area of the square = side^2.
The side length is given as √3840.
Area of the square = side^2 = √3840 × √3840 = 3840.
Therefore, the area of the square box is 3840 square units.
A square-shaped building measuring 3840 square feet is built; if each of the sides is √3840, what will be the square feet of half of the building?
1920 square feet
We can just divide the given area by 2 as the building is square-shaped.
Dividing 3840 by 2, we get 1920.
So half of the building measures 1920 square feet.
Calculate √3840 × 5.
Approximately 309.885
First, find the square root of 3840, which is approximately 61.977.
Multiply this by 5.
So, 61.977 × 5 ≈ 309.885.
What will be the square root of (3840 + 160)?
The square root is approximately 64.
To find the square root, calculate the sum of (3840 + 160).
3840 + 160 = 4000.
The square root of 4000 is approximately 63.2455532.
Therefore, the square root of (3840 + 160) is approximately 64.
Find the perimeter of a rectangle if its length ‘l’ is √3840 units and the width ‘w’ is 60 units.
The perimeter is approximately 244 units.
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√3840 + 60)
= 2 × (61.977 + 60)
= 2 × 121.977 ≈ 243.954.
Therefore, the perimeter is approximately 244 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.
: He loves to play the quiz with kids through algebra to make kids love it.