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:

• Prime factorization method
• Long division method
• Approximation method

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.

• Square root: A square root is the inverse of a square. For example, 4^2 = 16, and the inverse of the square is the square root, which 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 why it is also known as the principal square root.
   
• Prime factorization: Breaking down a number into the product of its prime factors. For example, the prime factorization of 3840 is 2^6 × 3 × 5.
   
• Decimal: A decimal is a number that contains a whole number and a fractional part, such as 7.86, 8.65, and 9.42.